Abstract - Enumath 2013

ENUMATH 201 3 Hosted by

École Polytechnique Fédérale de Lausanne

Mathematical Institute of Computational Science and Engineering

BOOK OF ABSTRACTS

26 – 30 August 2013

DAILY TIMETABLES

MONDAY 26TH AUGUST

Monday 26th August (Morning)

ENUMATH 2013 07:30 - 08:50

Registration Rolex Learning Center Auditorium (RLC)

08:50 - 09:00

Opening Rolex Learning Center Auditorium (RLC)

09:00 - 09:50 09:50 - 10:40

Rolex Learning

Barbara Wohlmuth (Pg. 416)

Center Auditorium

Ernst Hairer (Pg. 150)

Interfaces, corners and point sources

Organizers Chair

CO017

CO122

PARA: MSMA: SMAP: Bridging software Multiscale methods Surrogate modeling design and for atomistic and continuum models approaches for PDEs performance tuning

FEPD: Finite elements for PDE-constrained optimization

ACDA: Approximation, compression, and data analysis

Abdulle, Ortner

Perotto, Smetana, Veneziani

Engwer, Goeddeke

Rösch, Vexler

Grohs, Fornasier, Ward

Cancès, Després

Abdulle

Smetana

Goeddeke

Vexler

Grohs

Després

Debrabant (Pg. 92)

Shapeev (Pg. 346)

Engwer (Pg. 114)

Ehler (Pg. 105)

Cancès (Pg. 62)

Monotone approximations for Hamilton-Jacobi-Bellman equations

Atomistic-to-Continuum coupling for crystals: analysis and construction

Braack (Pg. 53)

Mini-symposium keynote: Bridging software design and performance tuning for parallel numerical codes

Signal reconstruction from magnitude measurements via semidefinite programming

Monotone corrections for cell-centered Finite Volume approximations of diffusion equations

Golbabaee (Pg. 133)

A monotone nonlinear finite volume method for diffusion equations and multiphase flows

CO1

CO2

LRTT: Low-rank tensor techniques

NMFN: Numerical methods for fully nonlinear PDE’s

Grasedyck, Huckle, Khoromskij, Kressner Khoromskij

Jensen, Lakkis, Pryer Pryer

Lim (Pg. 242) 11:10 - 11:40

Symmetric tensors with positive decompositions

CO3

Kalise (Pg. 191) Entanglement via algebriac geometry

An accelerated semi-Lagrangian/policy iteration scheme for the solution of dynamic programming equations

Uschmajew (Pg. 388)

Lakkis (Pg. 232)

Skowera (Pg. 351) 11:40 - 12:10

12:10 - 12:40

On asymtotic complexity of hierarchical Tucker approximation in L2 Sobolev classes

Khoromskaia (Pg. 200) 12:40 - 13:10

Rolf Rannacher

Coffee Break

10:40 - 11:10

Minisymposia

Chair:

Long-term analysis of numerical and analytical oscillations

Hartree-Fock and MP2 calculations by grid-based tensor numerical methods

Review of Recent Advances in Galerkin Methods for Fully Nonlinear Elliptic Equations

CO016

Model- and mesh adaptivity for transient problems

Deparis (Pg. 98) Ortner (Pg. 289) Optimising Multiscale Defect Simulations

On the continuity of flow rates, stresses and total stresses in geometrical multiscale cardiovascular models

Cecka (Pg. 68) Fast Multipole Method Framework and Repository

Elfverson (Pg. 110) Cances (Pg. 61) Multiscale eigenvalue problems

Smears (Pg. 352) Discontinuous Galerkin finite element approximation of HJB equations with Cordès coefficients

CO015

Discontinuous Galerkin method for convection-diffusion-reaction problems

Rupp (Pg. 325) ViennaCL - Portable High Performance at High Convenience

Stamm (Pg. 356)

Cervone (Pg. 69)

Recent developments of Parallel assembly on Hierarchical Model (HiMod) overlapping meshes using the reduction for boundary value LifeV library problems

Lunch

13:10 - 14:30

3

Nikitin (Pg. 280) Model Selection with Piecewise Regular Gauges

Chrysafinos (Pg. 78)

Schnass (Pg. 342)

Discontinuous time-stepping schemes for the velocity tracking problem under low regularity assumptions

Non-Asymptotic Dictionary Identification Results for the K-SVD Minimisation Principle

Sheng (Pg. 348) The nonlinear finite volume scheme preserving maximum principle for diffusion equations on polygonal meshes

Kirchner (Pg. 205)

Perotto (Pg. 298) Domain decomposition for implicit solvation models

Pieper (Pg. 303) Finite element error analysis for optimal control problems with sparsity functional

CO123 SDIFF: New trends in nonlinear methods for solving diffusion equation

Efficient computation of a Tikhonov regularization parameter for nonlinear inverse problems with adaptive discretization methods

Krahmer (Pg. 216)

Burman (Pg. 55)

The restricted isometry property for random convolutions

Computability of filtered quantities for the Burgers’ equation

Monday 26th August (Afternoon)

ENUMATH 2013 CO1 Minisymposia

Organizers Chair

ANMF: Advanced numerical methods for fluid mechanics Burman, Ern, Fernandez Burman

15:00 - 15:30

16:00 - 16:30

16:30 - 17:00

Abdulle, Ortner

Olshanskii

Ortner

Vassilevski (Pg. 394)

CO016

CO017

CO122

SMAP: Surrogate modeling approaches for PDEs

PARA: Bridging software design and performance tuning

FEPD: Finite elements for PDE-constrained optimization

ACDA: Approximation, compression, and data analysis

Engwer, Goeddeke

Rösch, Vexler

Grohs, Fornasier, Ward

Rösch

Ward

Perotto, Smetana, Veneziani Perotto

A numerical approach to Newtonian and viscoplastic free surface flows using dynamic octree meshes

Bai (Pg. 38)

Falcó (Pg. 117)

Reduced basis finite element heterogeneous multiscale method for quasilinear problems

Proper Generalized Decomposition for Dynamical Systems

Vilmart (Pg. 399)

Kestler (Pg. 199)

Tews (Pg. 374)

Lee (Pg. 236)

Optimal control of incompressible two-phase flows

Numerical simulation of Kaye effects

Numerical homogenization methods for multiscale nonlinear elliptic problems of nonmonotone type

On the adaptive tensor product wavelet Galerkin method in view of recent quantitative improvements

Ehrlacher (Pg. 107)

Schieweck (Pg. 340) 15:30 - 16:00

Olshanskii, Vassilevski

CO015

On stability properties of different variants of local projection type stabilizations

Tobiska (Pg. 378) 14:30 - 15:00

CO2 CO3 FREE: MSMA: Numerical methods for Multiscale methods for fluid flows with free atomistic and boundaries and continuum models interfaces

Aizinger (Pg. 18)

Engwer

Pfefferer (Pg. 301) Jolivet (Pg. 188) How to easily solve PDE with FreeFem++ ?

de la Cruz (Pg. 90)

Wollner (Pg. 418)

Weinmann (Pg. 408)

Adjoint Consistent Gradient Computation with the Damped Crank-Nicolson Method

Jump-sparse reconstruction by the minimization of Potts functionals

Van der Zee (Pg. 389)

Reguly (Pg. 314)

Discontinuous Galerkin method for 3D free surface flows and wetting/drying

Optimization of a structurally graded microstructured material

Adaptive Modeling for Partitioned-Domain Concurrent Continuum Models

OP2: A library for unstructured grid applications on heterogeneous architectures

Sangalli (Pg. 337)

Danilov (Pg. 88)

Makridakis (Pg. 257)

Smetana (Pg. 353)

Wells (Pg. 411)

Isogeometric elements for the Stokes problem

Numerical simulation of large-scale hydrodynamic events

Consistent Atomistic / Continuum approximations to atomistic models.

The Hierarchical Model Reduction-Reduced Basis approach for nonlinear PDEs

Domain-specific languages and code generation for solving PDEs using specialised hardware

4

Davenport (Pg. 89) One-Bit Matrix Completion

Unveiling WARIS code, a parallel and multi-purpose FDM framework

An efficient dG-method for transport dominated problems based on composite finite elements

Coffee Break

On properties of discretized optimal control problems with semilinear elliptic equations and pointwise state constraints

Aßmann (Pg. 34) Regularization in Sobolev spaces with fractional order

Steinig (Pg. 357) Convergence Analysis and A Posteriori Error Estimation for State-Constrained Optimal Control Problems

Peter (Pg. 300) Damping Noise-Folding and Enhanced Support Recovery in Compressed Sensing

Vandergheynst (Pg. 390) Compressive Source Separation: an efficient model for large scale multichannel data processing

Monday 26th August (Late Afternoon)

ENUMATH 2013 CO1

CO2

CO3

Contributed Talks

CT1.1: Treatment of large number of random variables

CT1.2: Interpolation, quadrature and PDEs

CT1.3: Buoyancy driven flows and integration schemes

Chair

Ishizuka

Caboussat

Lukin

Macedo (Pg. 253) 17:00 - 17:30

A low-rank tensor method for large-scale Markov Chains

Berrut (Pg. 46)

Pekmen (Pg. 295)

The linear barycentric rational quadrature method for Volterra integral equations

Steady Mixed Convection in a Heated Lid-Driven Square Cavity Filled with a Fluid-Saturated Porous Medium

Luh (Pg. 249)

Tezer-Sezgin (Pg. 375)

CO015

CO016

CT1.5: CT1.4: A posteriori error Hamiltonian estimates and systems and their adaptive methods integration I Janssen

Greff Grandchamp (Pg. 138) Multi-scale DNA Modelling and Birod Mechanics

Kleiss (Pg. 207) Guaranteed and Sharp a Posteriori Error Estimates in Isogeometric Analysis

CO017 CT1.6: Domain decomposition and parallel methods

CO122 CT1.7: Modeling and simulation of vascular and respiratory systems

Sumitomo

Cattaneo

Christophe (Pg. 76) Mortar FEs on overlapping subdomains for eddy current non destructive testing

Prokop (Pg. 309) Numerical Simulation of Generalized Oldroyd-B Fluid Flows in Bypass

Konshin (Pg. 211) Migliorati (Pg. 269) 17:30 - 18:00

Adaptive polynomial approximation by random projection of multivariate aleatory functions

The Criteria of Choosing the Shape Parameter for Radial Basis Function Interpolations

DRBEM Solution of Full MHD and Temperature Equations in a Lid-driven Cavity

Papez (Pg. 293) D’Ambrosio (Pg. 85) Numerical solution of Hamiltonian systems by multi-value methods

Distribution of the algebraic, discretization and total errors in numerical PDE model problems

Continuous parallel algorithm of the second order incomplete triangular factorization with dynamic decomposition and reordering

Augustin (Pg. 27) Parallel solvers for the numerical simulation of cardiovascular tissues

CO123

CO124

CT1.8: Boundary element and pseudospectral methods

CT1.9: Numerical treatment of boundaries, interfaces and block materials

Weißer

Varygina

af Klinteberg (Pg. 17) Fast simulation of particle suspensions using double layer boundary integrals and spectral Ewald summation

Saffar Shamshirgar (Pg. 333) The Spectrally Fast Ewald method and a comparison with SPME and P3M methods in Electrostatics

Berger (Pg. 45) Ishizuka (Pg. 178) 18:00 - 18:30

Simulating information propagation by near-field P2P wireless communication

Heine (Pg. 155)

Lukin (Pg. 251)

Mean-Curvature Reconstruction with Linear Finite Elements

Mathematical modelling of radiatively accelerated canalized magnetic jets

Caboussat (Pg. 59) 18:30 - 19:00

Numerical solution of a partial differential equation involving the Jacobian determinant

Greff (Pg. 140) Conservation of Lagrangian and Hamiltonian structure for discrete schemes

Pousin (Pg. 306) A posteriori estimate and adaptive partial domain decomposition

Lejon (Pg. 239)

Janssen (Pg. 182)

Higher order projective integration schemes for multiscale kinetic equations in the diffusive limit

The hp-adaptive Galerkin time stepping method for nonlinear differential equations with finite time blow-up

5

Ruprecht (Pg. 327) Convergence of Parareal for the Navier-Stokes equations depending on the Reynolds number

Sumitomo (Pg. 364) GPU accelerated Symplectic Integrator in FEA for solid continuum

Solving the Generalised Large Deformation Poroelastic Equations for Modelling Tissue Deformation and Ventilation in the Lung

Cattaneo (Pg. 66) Computational models for coupling tissue perfusion and microcirculation

Börm (Pg. 52) Fast evaluation of boundary element matrices by quadrature techniques

Weißer (Pg. 409) Challenges in BEM-based Finite Element Methods on general meshes

Kreiss (Pg. 222) Imposing Neumann and Robin boundary conditions with added penalty term

Juntunen (Pg. 189) A posteriori estimate of Nitsche’s method for discontinuous material parameters

Varygina (Pg. 392) Numerical Modeling of Elastic Waves Propagation in Block Media with Thin Interlayers

TUESDAY 27TH AUGUST

Tuesday 27th August (Morning)

ENUMATH 2013 08:20 - 09:10

Rolex Learning

Ruth Baker (Pg. 39)

09:10 - 10:00

Center Auditorium

Eric Cancès (Pg. 63)

Developing multiscale models for exploring biological phenomena

CO1

CO2

FREE: ANMF: methods Advanced numerical Numerical for fluid flows with methods for fluid free boundaries and mechanics interfaces

Organizers

Burman, Ern, Fernandez

Olshanskii, Vassilevski

Chair

Ern

Vassilevski

10:30 - 11:00

Miloslav Feistauer

Coffee Break

10:00 - 10:30

Minisymposia

Chair:

Electronic structure calculation CO3

CO015

CO016

LRTT: Low-rank tensor techniques

CTNL: Current trends in numerical linear algebra

ADFE: Adaptive finite elements

Simoncini

Micheletti, Perotto, Picasso

Grasedyck, Huckle, Khoromskij, Kressner Huckle

Simoncini

CO017 MANT: Modelling, Analysis and Numerical Techniques for Viscoelastic Fluids

Bonito, Nochetto

Chartier, Lemou

Bonito

Chartier

Picasso

Kroll & Turek Sahin (Pg. 334) Parallel Large-Scale Numerical Simulations of Purely-Elastic Instabilities with a Template-Based Mesh Refinement Algorithm

Gross (Pg. 142)

Ehrlacher (Pg. 106)

Powell (Pg. 308)

Fictitious Domain Formulation for Immersed Boundary Method

XFEM for pressure and velocity singularities in 3D two-phase flows

Greedy algorithms for high-dimensional eigenvalue problems

Fast solvers for stochastic FEM discretizations of PDEs with uncertainty

Unified variational multiscale method for compressible and incompressible flows using anisotropic adaptive mesh

Jiranek (Pg. 186)

Henning (Pg. 157)

A general framework for algebraic multigrid methods

Error control for a Multiscale Finite Element Method

GEOP: Geometric Partial Differential Equations

CO123 ASHO: Asymptotic preserving schemes for highly-oscillatory PDEs

Kroll, Turek

Hachem (Pg. 147) Gastaldi (Pg. 125)

CO122

Heine (Pg. 156)

Vilmart (Pg. 400)

Mean-Curvature Reconstruction with Linear Finite Elements

Multi-revolution composition methods for highly oscillatory problems

Caiazzo (Pg. 60) 11:00 - 11:30

An explicit stabilized projection scheme for incompressible NSE: analysis and application to POD based reduced order modeling

Burman (Pg. 56) 11:30 - 12:00

Projection methods for the transient Navier–Stokes equations discretized by finite element methods with symmetric stabilization

Bonelle (Pg. 49) 12:00 - 12:30

Compatible Discrete Operator Schemes on Polyhedral Meshes for Stokes Flows

Basting (Pg. 42) A hybrid level set / front tracking approach for fluid flows with free boundaries and interfaces

Turek (Pg. 386) 3D Level Set FEM techniques for (non-Newtonian) multiphase flow problems with application to pneumatic extension nozzles and micro-encapsulation

Kramer (Pg. 219) Converting Interface Conditions due to Excluded Volume Interactions into Boundary Conditions by FEM-BEM Methods

Schneider (Pg. 343) Convergence of dynamical low rank approximation in hierarchical tensor formats

Badia (Pg. 37) Zulehner (Pg. 427) Tyrtyshnikov (Pg. 387) Operator Adaptive finite element Preconditioning for Tensor decompositions in the simulation of incompressible a Mixed Method of drug design optimization problems

Ballani (Pg. 40) Black box approximation strategies in the hierarchical tensor format

Biharmonic Problems on Polygonal Domains

flows by hybrid continuous-discontinuous Galerkin formulations

Stoll (Pg. 361)

Artina (Pg. 26)

Fast solvers for Allen-Cahn and Cahn-Hilliard problems

Anisotropic mesh adaptation for brittle fractures

Wünsch (Pg. 419)

Caboussat (Pg. 58)

Thalhammer (Pg. 377)

Numerical simulation of viscoelastic fluid flow in confined geometries

Numerical Approximation of Fully Nonlinear Elliptic Equations

Multi-revolution composition methods for time-dependent Schrödinger equations

Keslerova (Pg. 195)

Hintermueller (Pg. 166)

Numerical Simulation of Steady and Unsteady Flows for Viscous and Viscoelastic Fluids

Damanik (Pg. 86)

Lunch

12:30 - 14:00

7

A multigrid LCR-FEM solver for viscoelastic fluids with application to problems with free surface

Optimal shape design subject to elliptic variational inequalities

Bartels (Pg. 41) Projection-free approximation of geometric evolution problems

Possanner (Pg. 305) Numerical integration of the MHD equations on the resistive timescale

Tuesday 27th August (Afternoon)

ENUMATH 2013 CO1

CO2

Contributed Talks

CT2.1: Accurate and reliable matrix computations

CT2.2: Multiscale wave equation

Chair

Pena

Stohrer

Miyajima (Pg. 270) 14:00 - 14:30

Fast verified computation for solutions of generalized least squares problems

Ogita (Pg. 282) 14:30 - 15:00

Backward error bounds on factorizations of symmetric indefinite matrices

CO3

CO015

CT2.3: CT2.4: Advanced A posteriori error methods for fluid estimates and and transport adaptive methods problems II Matthies

Arjmand (Pg. 24)

Linke (Pg. 244)

Analysis of Heterogeneous Multiscale Methods for Long Time Multiscale Wave Propagation Problems

Stabilizing Mixed Methods for Incompressible Flows by a New Kind of Variational Crime

On a posteriori error analyses for generalized Stokes problem using an augmented velocity-pseudostress formulation

Muslu (Pg. 275)

Ojala (Pg. 286)

New Numerical Results on Some Boussinesq-type Wave Equations

Accurate bubble and drop simulations in 2D Stokes flow

Ozaki (Pg. 291)

Nguyen (Pg. 278) Homogenization of the one-dimensional wave equation

Stohrer (Pg. 359) 15:30 - 16:00

16:30 - 17:30 17:30 - 18:30

Accurate computations for some classes of matrices

Rolex Learning Center Auditorium

Hadrava (Pg. 148) Space-time Discontinuous Galerkin Method for the Problem of Linear Elasticity

Micro-Scales and Long-Time Effects: FE Heterogeneous Multiscale Method for the Wave Equation

CO122

CT2.7: CT2.6: Finite volume and Regression and finite difference statistical inverse problems methods Touma

Icardi

Sepúlveda (Pg. 345)

ten Thije Boonkkamp (Pg. 371)

Azijli (Pg. 30)

Gorkem (Pg. 136)

Kucera (Pg. 229)

Jannelli (Pg. 180)

On the use of reconstruction operators in discontinuous Galerkin schemes

Quasi-uniform Grids and ad hoc Finite Difference Schemes for BVPs on Infinite Intervals

Verani (Pg. 395)

Touma (Pg. 380)

Icardi (Pg. 174)

Mimetic finite differences for quasi-linear elliptic equations

Central finite volume schemes on nonuniform grids and applications

Bayesian parameter estimation of a porous media flow model

Harmonic complete flux schemes for conservation laws with discontinuous coefficients

Tryoen (Pg. 384)

Error Estimation for The Convective Cahn – Hilliard Equation

Matthies (Pg. 259)

Frolov (Pg. 123)

A two-level local projection stabilisation on uniformly refined triangular meshes

Reliable a posteriori error estimation for plane problems in Cosserat elasticity

Apéro hosted by MathWorks

8

CT2.9: Flow problems in heterogeneous media

Khoromskij

Yücel

Savostyanov (Pg. 338)

Budac (Pg. 54) An adaptive numerical homogenization method for a Stokes problem in heterogeneous media

Ouazzi (Pg. 290) Newton-Multigrid Least-Squares FEM for V-V-P and S-V-P Formulations of the Navier-Stokes Equations

Dolgov (Pg. 103)

Alternating minimal A semi-intrusive energy methods for linear stochastic inverse method systems in higher for uncertainty dimensions. Part II: characterization and implementation hints propagation in and application to hyperbolic problems nonsymmetric systems

Inverse Problems Regularized by Sparsity

Public Lecture: Martin Vetterli (Pg. 397)

CT2.8: Low-rank tensor techniques

Physics-based Alternating minimal interpolation of energy methods for linear incompressible flow fields systems in higher obtained from dimensions. Part I: the experimental data: a framework and theory Bayesian perspective for SPD systems

Madhavan (Pg. 254)

Finite element methods for transient convection-diffusion equations with small diffusion

CO124

Billaud Friess (Pg. 47)

A new a posteriori error estimator of low computational cost for an augmented mixed FEM in linear elasticity

On a Discontinuous Galerkin Method for Surface PDEs

CO123

Convergent Finite Volume Azzimonti (Pg. 32) A Tensor-Based Mixed Finite Elements for Schemes for Nonlocal Algorithm for the spatial regression with and Cross Diffusion Optimal Model Reduction PDE penalization Reaction Equations. of High Dimensional Applications to biology Problems

Gonzalez (Pg. 134)

Fast Interval Matrix Multiplication by Blockwise Computations

Pena (Pg. 297)

CO017

Verani

Frolov Bustinza (Pg. 57)

Nadir (Pg. 276) 15:00 - 15:30

CO016 CT2.5: Discontinuous Galerkin and mimetic finite difference methods

Khoromskij (Pg. 201) Quantized tensor approximation methods for multi-dimensional PDEs

den Ouden (Pg. 96) Application of the level-set method to a multi-component Stefan problem

Yücel (Pg. 422) Distributed Optimal Control Problems Governed by Coupled Convection Dominated PDEs with Control Constraints

WEDNESDAY 28TH AUGUST

Wednesday 28th August

ENUMATH 2013 08:20 - 09:10

Rolex Learning

Rolf Stenberg (Pg. 358)

09:10 - 10:00

Center Auditorium

Ilaria Perugia (Pg. 299)

Mixed Finite Element Methods for Elasticity

Chair:

Trefftz-discontinuous Galerkin methods for time-harmonic wave problems

Yuri Kuznetsov

CO1

CO2

CO3

Coffee Break CO015

CO016

Minisymposia

UQPD: Uncertainty Quantication for PDE model

ASHO: Asymptotic preserving schemes for highly-oscillatory PDEs

LRTT: Low-rank tensor techniques

CTNL: Current trends in numerical linear algebra

ADFE: Adaptive finite elements

Organizers

Nobile, Schwab

Chartier, Lemou

Grasedyck, Huckle, Khoromskij, Kressner

Simoncini

Micheletti, Perotto, Picasso

Kroll, Turek

Bonito, Nochetto

Chair

Schwab

Lemou

Kressner

Simoncini

Micheletti

Turek & Kroll

Bartels

10:00 - 10:30

Cohen (Pg. 80) 10:30 - 11:00

Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs

Despres (Pg. 99) Uniform convergence of Asymptotic Preserving schemes on general meshes

Crouseilles (Pg. 84) Asymptotic preserving schemes for highly oscillatory Vlasov-Poisson equations

11:00 - 11:30

Scheichl (Pg. 339) 11:30 - 12:00

Hierarchical Multilevel Markov Chain Monte Carlo Methods and Applications to Uncertainty Quantification in Subsurface Flow

Giraud (Pg. 132) Recovery policies for Krylov solver resiliency

Donatelli (Pg. 104) Kazeev (Pg. 194) Tensor-structured approach to the Chemical Master Equation

Lafitte (Pg. 231)

Bachmayr (Pg. 35)

Projective integration schemes for kinetic equations in the hydrodynamic limit

Adaptive methods based on low-rank tensor representations of coefficient sequences

CO122 GEOP: Geometric Partial Differential Equations

Hoffman (Pg. 170)

Hegland (Pg. 154) Solving the chemical master equations for biological signalling cascades using tensor factorisation

CO017 MANT: Modelling, Analysis and Numerical Techniques for Viscoelastic Fluids

Multigrid preconditioning for nonlinear (degenerate) parabolic equations with application to monument degradation

Simoncini (Pg. 350) Solving Ill-posed Linear Systems with GMRES

Adaptive finite element methods for turbulent flow and fluid-structure interaction: theory, implementation and applications

Luce (Pg. 248) Robust local flux reconstruction for various finite element methods

Kroll (Pg. 226) An alternative description of the visko-elastic flow behavior of highly elastic polymer melts

Capatina (Pg. 65) Robust discretization of the Giesekus model

Olshanskii (Pg. 287)

Chen (Pg. 70)

An adaptive finite element method for PDEs based on surfaces

A numerical study of viscoelastic fluid-structure interaction and its application in a micropump

Tobiska (Pg. 379) Influence of surfactants on the dynamics of droplets

Walker (Pg. 405) A New Mixed Formulation For a Sharp Interface Model of Stokes Flow and Moving Contact Lines

Antil (Pg. 23) A Stokes Free Boundary Problem with Surface Tension Effects

Hintermueller (Pg. 165) Harbrecht (Pg. 153) 12:00 - 12:30

12:30 - 14:00 14:00 - 18:30

On multilevel quadrature for elliptic stochastic partial differential equations

Crestetto (Pg. 83) Coupling of an Asymptotic-Preserving scheme with the Limit model for highly anisotropic-elliptic problems

Huckle (Pg. 173) Tensor representations of sparse or structured vectors and matrices

Strakos (Pg. 362) Remarks on algebraic computations within numerical solution of partial differential equations

An adaptive finite element method for variational inequalities of second kind with applications in L2-TV-based image denoising and Bingham fluids

Lunch Excursion: Lavaux, Gruyères / Free Afternoon

10

Picasso (Pg. 302) Numerical simulation of extrusion with viscoelastic flows

Dede (Pg. 93) Numerical approximation of Partial Differential Equations on surfaces by Isogeometric Analysis

THURSDAY 29TH AUGUST

Thursday 29th August (Morning)

ENUMATH 2013 08:20 - 09:10

CO1

Stochastic Newton MCMC Methods for Bayesian Inverse Problems, with Application to Ice Sheet Dynamics

Omar Ghattas (Pg. 131)

High-order accurate reduced basis multiscale finite element methods

Jan Hesthaven (Pg. 161)

09:10 - 10:00 10:00 - 10:30

CO1

CO2

CO3

Minisymposia

UQPD: Uncertainty Quantication for PDE models

NEIG: Numerical methods for linear and nonlinear eigenvalue problems

PSPP: Preconditioners for saddle point problems

Organizers

Nobile, Schwab

Benner, Guglielmi

Chair

Nobile

Guglielmi

Deparis, Klawonn, Pavarino Deparis

10:30 - 11:00

Voss (Pg. 403) Variational Principles for Nonlinear Eigenvalue Problems

Widlund (Pg. 413) Two-level overlapping Schwarz methods for some saddle point problems

Ohlberger (Pg. 284) 11:00 - 11:30

11:30 - 12:00

TIME: Time integration of partial differential equations

CO016 ROMY: Reduced order modelling for the simulation of complex systems

Ostermann

Quarteroni, Rozza

Ostermann

Rozza

Hochbruck (Pg. 167)

Tempone (Pg. 370) Numerical Approximation of the Acoustic and Elastic Wave Equations with Stochastic Coefficients

Coffee Break CO015

Model reduction for nonlinear parametrized evolution problems

Le Maitre (Pg. 235)

Jarlebring (Pg. 185)

Galerkin Method for Stochastic Ordinary Differential Equations with Uncertain Parameters

An iterative block algorithm for eigenvalue problems with eigenvector nonlinearities

Schillings (Pg. 341)

Kressner (Pg. 225)

Sparsity in Bayesian Inverse Problems

Interpolation based methods for nonlinear eigenvalue problems

Wiesner (Pg. 414) Algebraic multigrid (AMG) methods for saddle-point problems arising from mortar-based finite element discretizations

DEIM-based Non-Linear PGD

Grote (Pg. 144)

Amsallem (Pg. 22)

Einkemmer (Pg. 108)

12:00 - 12:30

Isogeometric Schwarz preconditioners for mixed elasticity and Stokes systems

CO017

CO122

MMHD: MAXWELL and MHD

NFSI: Numerics of Fluid-Structure Interaction

Bonito, Guermond

Richter, Rannacher

Bonito

Rannacher

Chinesta (Pg. 73)

Error Estimates for Element-Based Hyper-Reduction of Nonlinear Dynamic Finite Element Models

Analysis of an unconditionally convergent stabilized finite element formulation for incompressible magnetohydrodynamics

Lin (Pg. 243)

L2 projected finite element methods for Maxwell’s equations with low regularity solution

Mehl (Pg. 260) Towards massively parallel fluid-structure simulations – two new parallel coupling schemes

Puscas (Pg. 310) 3d conservative coupling method between a compressible fluid flow and a deformable structure

Mula (Pg. 272) The Generalized Empirical Interpolation Method: Analysis of the convergence and application to the Stokes problem

Kolev (Pg. 209)

Richter (Pg. 320)

Parallel Algebraic Multigrid for Electromagnetic Diffusion

A Fully Eulerian Formulation for Fluid-Structure Interactions

Lassila (Pg. 234) The Crank-Nicolson scheme with splitting and discrete transparent Space-time model reduction for nonlinear time-periodic problems boundary conditions for the using the harmonic balance Schrödinger equation on an reduced basis method infinite strip

A local projection stabilization method for finite element approximation of a magnetohydrodynamic model

A discontinuous Galerkin approximation for Vlasov equations

Zlotnik (Pg. 425)

Pavarino (Pg. 294)

Franco Brezzi

Codina (Pg. 79)

Error analysis of implicit Runge-Kutta methods for discontinuous Galerkin discretizations of linear Maxwell’s equations

Runge-Kutta based explicit local time-stepping methods for wave propagation

Chair:

Lunch

12:30 - 14:00

12

Wacker (Pg. 404)

Wick (Pg. 412) A fluid-structure interaction framework for reactive flows in thin channels

Thursday 29th August (Afternoon)

ENUMATH 2013

CO017

CO122

MMHD: MAXWELL and MHD

NFSI: Numerics of Fluid-Structure Interaction

Quarteroni, Rozza

Bonito, Guermond

Richter, Rannacher

Quarteroni

Codina

Richter

CO1

CO2

CO3

CO015

Minisymposia

STOP: Adaptive stopping criteria

NEIG: Numerical methods for linear and nonlinear eigenvalue problems

PSPP: Preconditioners for saddle point problems

TIME: Time integration of partial differential equations

CO016 ROMY: Reduced order modelling for the simulation of complex systems

Organizers

Ern, Strakos, Vohralik

Benner, Guglielmi

Ostermann

Strakos

Benner

Deparis, Klawonn, Pavarino Pavarino

Ostermann

Chair

Ern (Pg. 116) 14:00 - 14:30

Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs

Smirnova (Pg. 354) 14:30 - 15:00

Michiels (Pg. 266)

Freitag (Pg. 122)

A Novel Stopping Criterion for Iterative Regularization with Undetermined Reverse Connection

Computing Jordan blocks in parameter-dependent eigenproblems

Capatina (Pg. 64)

Multiscale adaptive finite element method for PDE eigenvalue/eigenvector approximations

Miedlar (Pg. 268) 15:00 - 15:30

Stopping criteria based on locally reconstructed fluxes

Vohralik (Pg. 401) 15:30 - 16:00

16:00 - 16:30

Adaptive regularization, linearization, and algebraic solution in unsteady nonlinear problems

Olshanskii (Pg. 288)

Projection based methods for Preconditioners for the linearized nonlinear eigenvalue problems Navier-Stokes equations based on and associated distance problems the augmented Lagrangian

Guglielmi (Pg. 146) Computing the distance to defectivity

Qingguo Hong (Pg. 311) A multigrid method for discontinuous Galerkin discretizations of Stokes equations

Schratz (Pg. 344)

Himpe (Pg. 164)

Efficient numerical time integration of the Klein-Gordon equation in the non-relativistic limit regime

Combined State and Parameter Reduction of Large-Scale Hierarchical Systems

Gauckler (Pg. 127)

Veroy-Grepl (Pg. 396)

Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation

On Synergies between the Reduced Basis Method, Proper Orthogonal Decomposition, and Balanced Truncation

Grandperrin (Pg. 139)

Lang (Pg. 233)

Colciago (Pg. 81)

Multiphysics Preconditioners for Fluid–Structure Interaction Problems

Anisotropic Finite Element Meshes for Linear Parabolic Equations

Reduced Order Models for Fluid-Structure Interaction Problems in Haemodynamics

Koskela (Pg. 215)

A reduced computational and geometrical framework for viscous optimal flow control in parametrized systems

Klawonn (Pg. 206) A deflation based coarse space in dual-primal FETI methods for almost incompressible elasticity

Nore (Pg. 281) Dynamo action in finite cylinders

Yang (Pg. 421) Flueck (Pg. 121) Domain decomposition for computing ferromagnetic effects

Coffee Break

13

Numerical Methods for Fluid-Structure Interaction Problems with a Mixed Elasticity Form in Hemodynamics

Sharma (Pg. 347)

Tricerri (Pg. 382)

Convergence Analysis of an Adaptive Interior Penalty Discontinuous Galerkin Method for the Helmholtz Problem

Fluid-Structure Interaction simulation of cerebral aneurysm using anisotropic model for the arterial wall

Heumann (Pg. 163)

The Geometric Conservation law in Astrophysics: Discontinuous Galerkin Methods on Moving Meshes for the non-ideal Gas Dynamics in Wolf-Rayet Stars

Kramer (Pg. 218)

Rozza (Pg. 323) A moment-matching Arnoldi method for phi-functions

Gerbeau (Pg. 128) Luenberger observers for fluid-structure problems

Stabilized Galerkin for Linear Advection of Differential Forms

Thursday 29th August (Late Afternoon)

ENUMATH 2013

CT3.5: Applications of reduced order models

CO017 CT3.6: Elasticity, plasticity and hysteresis in solid and particles systems

CT3.7: Deterministic methods for uncertainty quantification

CO123 CT3.8: Convergence analysis, minimal energy asymptotics, and data analysis

Hess

Sadovskii

Tamellini

Kirby

CO1

CO2

CO3

CO015

CO016

Contributed Talks

CT3.1: Inverse problems and optimal experimental design

CT3.2: Stochastic simulation: Chemistry and finance

CT3.3: Iterative methods and inexactness

CT3.4: Continuous and discontinuous Galerkin methods for complex flow

Chair

Kray

Engblom

Vannieuwenhoven

Feistauer

CO122

Avalishvili (Pg. 28) Rozgic (Pg. 321) 16:30 - 17:00

Mathematical optimization methods for process and material parameter identification in forming technology

Dementyeva (Pg. 94) 17:00 - 17:30

The Inverse Problem of a Boundary Function Recovery by Observation Data for the Shallow Water Model

Reduced Order Optimal Control of DiffusionConvection-Reaction Equation Using Proper Orthogonal Decomposition

On spectral method of approximation of dynamical dual-phase-lag three-dimensional model for thermoelastic shells by two-dimensional initial-boundary value problems

Dimitriu (Pg. 102)

Sadovskaya (Pg. 329)

Bonizzoni (Pg. 50)

POD-DEIM Approach on Dimension Reduction of a Multi-Species Host-Parasitoid System

Parallel Software for the Analysis of Dynamic Processes in Elastic-Plastic and Granular Materials

Low-rank techniques applied to moment equations for the stochastic Darcy problem with lognormal permeability

Akman (Pg. 19) Gergelits (Pg. 129)

Bause (Pg. 43)

Composite polynomial convergence bounds in finite precision CG computations

Space-time Galerkin discretizations of the wave equation

Karasozen (Pg. 193)

Meinecke (Pg. 262)

Krukier (Pg. 227)

Stochastic simulation of diffusion on unstructured meshes via first exit times

Symmetric - skew-symmetric splitting and iterative methods

Vilanova (Pg. 398)

Idema (Pg. 176)

Chernoff-based Hybrid Tau-leap

On the Convergence of Inexact Newton Methods

Engblom (Pg. 112)

Vannieuwenhoven (Pg. 391)

Adaptive Discontinuous Galerkin Methods for nonlinear DiffusionConvection-Reaction Models

Di Pietro (Pg. 100) Long (Pg. 245) 17:30 - 18:00

A Projection Method for Under Determined Optimal Experimental Designs

A generalization of the Crouzeix–Raviart and Raviart–Thomas spaces with applications in subsoil modeling

Herrero (Pg. 158) The reduced basis approximation applied to a Rayleigh-Bénard problem

Murata (Pg. 274) Analysis on distribution of magnetic particles with hysteresis characteristics and field fluctuations

Chkifa (Pg. 75)

Fishelov (Pg. 119)

High-dimensional adaptive sparse polynomial interpolation and application for parametric and stochastic elliptic PDE’s

Convergence analysis of a high-order compact scheme for time-dependent fourth-order differential equations

18:00 - 18:30

19:00

A new approach to solve the inverse scattering problem for the wave equation

Sensitivity estimation and inverse problems in spatial Parallel tensor-vector stochastic models of chemical multiplication using blocking kinetics

Space-time DGFEM for the solution of nonstationary nonlinear convection-diffusion problems and compressible flow

Hess (Pg. 160) Reduced Basis Methods for Maxwell’s Equations with Stochastic Coefficients

Social Dinner 4th floor of building BC

14

On the asymptotics of discrete Riesz energy with external fields

Mali (Pg. 258) Estimates of Effects Caused by Incompletely Known Data in Elliptic Problems Generated by Quadratic Energy Functionals

Feistauer (Pg. 118) Kray (Pg. 220)

Jaraczewski (Pg. 183)

Sadovskii (Pg. 331)

Tamellini (Pg. 367)

Hyperbolic Variational Inequalities in Elasto-Plasticity and Their Numerical Implementation

Quasi-optimal polynomial approximations for elliptic PDEs with stochastic coefficients

Kirby (Pg. 204) Flag manifolds for characterizing information in video sequences

FRIDAY 30TH AUGUST

Friday 30th August

ENUMATH 2013 CO1

CO2

CO3

CO015

CO016

Contributed Talks

CT4.1: Time integration of stiff/multiscale dynamical systems

CT4.2: Preconditioning

CT4.3: Compressible flows, turbulence and flow stability

CT4.4: New approaches in model order reduction

CT4.5: Numerical computation of external flows

Chair

Samaey

Reimer

Reigstad

Chen

Kosík

Algarni (Pg. 20)

John (Pg. 187)

Maier (Pg. 255)

Numerical Simulation of the Atmospheric Boundary Layer Flow over coal mine in North Bohemia

08:20 - 08:50

Numerical Evolution Methods of Rational Form for Reaction Diffusion Equations

A multilevel preconditioner for the biharmonic equation

Louda (Pg. 246) Numerical simulations of laminar and turbulent 3D flow over backward facing step

08:50 - 09:20

Samaey (Pg. 335) 09:20 - 09:50

A micro-macro parareal algorithm: application to singularly perturbed ordinary differential equations

Krendl (Pg. 224) Efficient preconditioning for time-harmonic control problems

Poˇrízková (Pg. 304)

Zhang (Pg. 423)

Furmanek (Pg. 124)

Compressible and incompressible unsteady flows in convergent channel

Reduced-order modeling and ROM-based optimization of batch chromatography

Numerical Simulation of Flow Induced Vibrations with Two Degrees of Freedom

Reigstad (Pg. 316)

Negri (Pg. 277)

Numerical investigation of network models for Isothermal junction flow

Reduced basis methods for PDE-constrained optimization

Numerical Simulation of Compressible Turbulent Flows Using Modified Earsm Model

Chen (Pg. 72)

Kosík (Pg. 213)

A Weighted Reduced Basis Method for Elliptic Partial Differential Equations with Random Input Data

The Interaction of Compressible Flow and an Elastic Structure Using Discontinuous Galerkin Method

Tani (Pg. 369) CG methods in non-standard inner product for saddle-point algebraic linear systems with indefinite preconditioning

Melis (Pg. 264) 09:50 - 10:20

A relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws

Reimer (Pg. 318)

H2 -matrix arithmetic and preconditioning

10:20 - 10:50 10:50 - 11:40 11:40 - 12:30 12:30 - 12:50 12:50 - 14:20

CO1

Keslerova (Pg. 197)

CO122

CO123

CT4.7: CT4.6: CT4.8: Conforming and A posteriori error Monte Carlo and estimates and nonconforming adaptive methods multi level Monte Carlo methods methods for PDEs III Simian

Mali

Lee (Pg. 238)

Rademacher (Pg. 312)

Hodge Laplacian problems with Robin boundary conditions

Model and mesh adaptivity for frictional contact problems

Blumenthal Tesei (Pg. 372) Multi Level Monte Carlo methods with Control Variate for elliptic SPDEs

CO124 CT4.9: Molecular dynamics and quantum mechanics simulations Kieri Szepessy (Pg. 366) How accurate is molecular dynamics for crossings of potential surfaces? Part I: Error estimates

Repin (Pg. 319)

Kamijo (Pg. 192) Numerical Method for Fractal Analysis on Discrete Dynamical Orbit in n-Dimensional Space Using Local Fractal Dimension

A reduced basis method for domain decomposition problems

CO017

Petr Knobloch (Pg. 208) Dmitri Kuzmin (Pg. 230)

Holman (Pg. 171)

Lilienthal (Pg. 241) Non-Dissipative Space Time Hp-Discontinuous Galerkin Method for the Time-Dependent Maxwell Equations

Simian (Pg. 349) Conforming and Nonconforming Intrinsic Discretization for Elliptic Partial Differential Equations

On Poincaré Type Inequalities for Functions With Zero Mean Boundary Traces and Applications to A Posteriori Analysis of Boundary Value Problems

Kadir (Pg. 190) Haji-Ali (Pg. 151) Optimization of mesh hierarchies for Multilevel Monte Carlo

Walloth (Pg. 406)

Hoel (Pg. 168)

Kieri (Pg. 202)

An efficient and reliable residual-type a posteriori error estimator for the Signorini problem

On non-asymptotic optimal stopping criteria in Monte Carlo simulations

Accelerated convergence for Schrödinger equations with non-smooth potentials

Sandberg (Pg. 336)

Blumenthal (Pg. 48)

An Adaptive Algorithm for Optimal Control Problems

Stabilized Multilevel Monte Carlo Method for Stiff Stochastic Problems

Coffee Break Finite element methods for convection dominated problems Vertex-based limiters for continuous and discontinuous Galerkin methods Closing Remarks CO1 Lunch

16

How accurate is molecular dynamics for crossings of potential surfaces? Part II: numerical tests

Chair: Assyr Abdulle

Ludvig af Klinteberg Numerical Analysis, KTH Royal Institute of Technology, SE Fast simulation of particle suspensions using double layer boundary integrals and spectral Ewald summation Contributed Session CT1.8: Monday, 17:00 - 17:30, CO123 We present a method for simulating periodic suspensions of sedimenting rigid particles, based on a boundary integral solution of the Stokes flow equations. The purpose of our work is to improve the understanding of the large scale properties of suspensions by looking at the microscale interactions between individual particles. Boundary integral methods are attractive for this problem type due to high attainable accuracy, depending on the underlying quadrature method, and a reduction of the problem dimensionality from three to two. However, the resulting discrete systems have full matrices, and require the use of fast algorithms for efficient solution. Our method is based on a periodic version of the completed double layer boundary integral formulation for Stokes flow, which yields a well-conditioned system that converges rapidly when solved iteratively using GMRES. The discrete system is formulated using the Nyström method, and the singular integrals of the formulation are treated using singularity subtraction. The method is accelerated by a spectrally accurate fast Ewald summation method, which allows us to compute the single and double layer potentials of the formulation in O(N log N ) time. By developing accurate estimates for the truncation errors of the Ewald summation, we are able to choose the parameters of the fast method such that the computation time is optimal for a given error tolerance. Joint work with Anna-Karin Tornberg.

17

Vadym Aizinger University of Erlangen-Nuernberg, DE Discontinuous Galerkin method for 3D free surface flows and wetting/drying Minisymposium Session FREE: Monday, 15:30 - 16:00, CO2 The local discontinuous Galerkin method is applied to the numerical solution of the three-dimensional hydrostatic equations of coastal ocean circulation. A wetting/drying algorithm in combination with dynamically varying vertical mesh resolution in the vicinity of the free surface is presented in the talk.

18

Tuğba Akman Middle East Technical University, TR Reduced Order Optimal Control of Diffusion-Convection-Reaction Equation Using Proper Orthogonal Decomposition Contributed Session CT3.5: Thursday, 16:30 - 17:00, CO016 We consider the reduced optimal control of time-dependent convection dominated diffusion-convection-reaction equation by proper orthogonal decomposition. The optimal control problem is discretized by space-time discontinuous Galerkin finite elements. Time discretization is performed by discontinuous Galerkin method with piecewise constant and linear polynomials in time, while symmetric interior penalty Galerkin (SIPG) with upwinding is used for space discretization. It is known that discontinuous Galerkin discretization with weak treatment of the boundary conditions results in an accurate solution in the presence of boundary layers. In the case of nonhomogenous boundary conditions, this property provides a reduced order solution satisfying the boundary conditions without any additional treatment. On the other hand, discontinuous Galerkin time discretization is a strongly A-stable method and demonstrates the advantage of long-time integration. The quality of the reduced order approximation is affected by the number of POD basis functions, the number and the location of the snapshots. To obtain an accurate reduced approximation, we increase the number of POD basis functions by measuring the value of the cost functional or the total energy contained in the system. The POD basis is constructed by using not only the state equation, but also the adjoint equation or a combination of them. We present numerical results with different convection terms to illustrate the efficiency of the method. Joint work with Bülent Karasözen.

19

Said Algarni King Fahd University of Petroleum and Minerals , SA Numerical Evolution Methods of Rational Form for Reaction Diffusion Equations Contributed Session CT4.1: Friday, 08:20 - 08:50, CO1 The purpose of this study is to investigate select numerical methods that demonstrate good performance in solving PDEs that couple diffusion and reaction terms. The simple form of a reaction diffusion equation is the following ut (t, x) = αuxx (t, x) + f (u), where u is an order-parameter field, e.g., population density which depends on space x and time t. The order-parameter may be either scalar or vector, depending on the number of variables that describe the physical system. The order-parameter evolves in time due to a local reaction, described by the nonlinear term f (u), in conjunction with spatial diffusion. The coefficient α can be a scalar or it could be dependent on time and space α(t, x). These types of equations have numerous fields of application such as environmental studies, biology, chemistry, medicine, and ecology. Our aim is to investigate and develop accurate and efficient approaches which compare favourably to other applicable methods. In particular, we investigate and adapt a relatively new class of methods based on rational polynomials. Namely, Padé time stepping (PTS), which is highly stable for the purposes of the present application and is associated with lower computational costs. Furthermore, PTS is optimized for our study to focus on reaction diffusion equations. Due to the rational form of PTS method, as shown in Fig. 1, a local error control threshold (LECT) is proposed. Numerical runs are conducted to obtain the optimal LECT. Fig. 2 illustrates the use of LECT. Based on the results, we find PTS alone and combined via splitting with other approaches provided favourable performance in certain and wide ranging parameter regimes.

20

Figure 1: Singularities of each component in φ, θ-space when iteration of PTS[1/1] approach.

α dx2

= 0.5 of the first

Figure 2: The first iteration of PTS[1/1] approach in φ and error space, h = 0.9, LECT = 1.

21

David Amsallem Stanford University, US Error Estimates for Element-Based Hyper-Reduction of Nonlinear Dynamic Finite Element Models Minisymposium Session ROMY: Thursday, 11:00 - 11:30, CO016 Error estimates for a recently developed model reduction approach for the efficient and fast solution of finite-element based dynamical systems are presented. The model reduction approach relies on two main ingredients: (1) the Galerkin projection of the high-dimensional dynamical system onto a set of Proper Orthogonal Decomposition-based modes and (2) the hyper-reduction of that projected system by the evaluation of the non linear internal forces onto a subset of the finite elements. This hyper-reduction step enables an online evaluation of the reduced-order model that does not scale with the dimension of the underlying high-dimensional model. The contribution of the discarded elements is taken care of by weighting the elementary contribution of the retained elements using appropriate weights. These weights are determined in an offline phase through the solution of a non-negative least-squares problem minimizing the discrepancy between the exact internal forces and their approximation using a subset of the elements, up to a given tolerance. A-posteriori error estimates will show that, when linear dynamical systems are considered, the error between the hyper-reduced an non-hyper-reduced models can be bounded by a quantity proportional to the tolerance used in the minimization problem. This suggests that the tolerance can be a criterion for determining a hyper-reduced model that satisfies a certain accuracy. An appropriate choice of training vectors is also suggested by the error bound derivation. In the case of nonlinear dynamical systems, an error bound proportional to the discrepancy between the exact and approximated internal force term can also be derived. Numerical experiments will illustrate the proposed approach as well as the error estimates derived in this work. Joint work with David Amsallem, and Charbel Farhat.

22

Harbir Antil George Mason University, US A Stokes Free Boundary Problem with Surface Tension Effects Minisymposium Session GEOP: Wednesday, 11:30 - 12:00, CO122 We consider a Stokes free boundary problem with surface tension effects in variational form. This model is an extension of the coupled system proposed by P. Saavedra and L. R. Scott, where they consider a Laplace equation in the bulk with Young-Laplace equation on the free boundary to account for surface tension. The two main difficulties for the Stokes free boundary problem are: the vector curvature on the interface, which causes problem to write a variational form of the free boundary problem and the existence of solution to Stokes equations with 1+1/p0 Navier-slip boundary conditions for Wp domains (minimal regularity). We will demonstrate the existence of solution to Stokes equations with Navier-slip boundary conditions using a perturbation argument for the bended half space fol1+1/p0 lowed by standard localization technique. The Wp regularity of the interface allows us to write the variational form for the entire free boundary problem, we conclude with the well-posedness of this system using a fixed point iteration. Joint work with Ricardo H. Nochetto, and Patrick Sodre.

23

Doghonay Arjmand PhD Student, Applied and Computational Mathematics, KTH, SE Analysis of Heterogeneous Multiscale Methods for Long Time Multiscale Wave Propagation Problems Contributed Session CT2.2: Tuesday, 14:00 - 14:30, CO2 We investigate the properties of a heterogeneous multi-scale method (HMM) type multi-scale algorithm for approximating the solution of the following initial boundary value problem modelling long time wave propagation ∂tt uε (t, x) − ∇ · (A(x/ε)∇uε (t, x)) = 0, in [0, T ε ] × Ω uε (0, x) = q(x), ∂t uε (0, x) = z(x), on {t = 0} × Ω,

(1)

where A(y) is 1-periodic, symmetric and uniformly positive definite, Ω ⊂ Rd , and T ε ≈ O(ε−2 ). We assume that the above equation is equipped with suitable boundary data. As ε → 0, the solution of (1) tends to a solution u ˆ which has no dependence on the small scale parameter ε. For short time scales T ε = T ≈ O(1) , the classical homogenization theory reveals the limiting behavior of the multi-scale solution and the equation. In this setting the solution u ˆ satisfies   ˆ u(t, x) = 0, in [0, T ] × Ω ∂tt u ˆ(t, x) − ∇ · A∇ˆ (2) u ˆ(0, x) = q(x), ∂t u ˆ(0, x) = z(x), on {t = 0} × Ω, where the homogenized coefficient Aˆ is a constant matrix, computation of which involves solving another set of non-oscillatory periodic elliptic problems called cell problems. On the other hand, for time scales T ε = O(ε−2 ), the solution uε (t, x) starts to exhibit O(1) dispersive effects which are not present in the short time homogenized solution. In this setting, Symes and Santosa [4] derive an effective equation for the long time wave propagation. In one dimension the equation has the form   ˆ xu ∂tt u ˆ(t, x) − ∂x A∂ ˆ(t, x) + ε2 β∂xxx u ˆ(t, x) = 0, in [0, T ε ] × Ω (3) u ˆ(0, x) = q(x), ∂t u ˆ(0, x) = z(x), on {t = 0} × Ω, where β is given to be a complicated functional of A. HMM is a general framework for treating multi-scale problems. HMM is often useful when we have the full microscopic model which is not affordable to use throughout the entire domain. The main idea is that one starts with assuming a macro model in which some missing data are upscaled from local microscopic simulations, where the micro model is then restricted by the coarse scale data. The multi-scale problem (1) is within the spectrum of application areas of HMM. A typical HMM algorithm for problem (1) starts with assuming a macro model utt = ∇ ·F where F stands for the missing flux in the model. This quantity is then computed by F = (KA(x/ε)uεx ), where K is an averaging operator in time and space, and uε solves the full microscopic problem (1) in a domain of size η ≈ O(ε), with initial data given by linear interpolation of the current macroscopic state u. For further details of such type of HMM based methods for short time multi-scale wave problems we refer the reader to [5] (for finite element HMM) and [3] (for finite difference HMM). In [2], the short time FD-HMM algorithm from [3] was extended to approximate the solution of long time wave propagation problems. The extension involved only 24

modifying the initial data for the micro problem to a third-order polynomial as well as using a high order averaging kernel K in the upscaling procedure. Numerical evidences were shown to illustrate that the numerical solution captures the dispersive effects, represented by β, seen in (3). In this talk, we give a theoretical foundation of the previous results in [2], by proving that HMM indeed computes the correct flux also for the long time multiscale wave problem (1). With suitable macroscale discretization parameters, it will therefore capture the O(1) dispersive effects in (3). More precisely, let FHM M be the flux computed by HMM when the micro problem is given initial data consistent with a third order polynomial u ˆ(x), i.e. (Kuε ) (0, x) = u ˆ(x), then ˆux + ε2 β u FHM M = Aˆ ˆxxx + O(η p + (ε/η)q ), where η is the size of the micro domain and q and p are parameters representing the smoothness and number of vanishing moments of the kernel, which in principle can be chosen arbitrarily large. Moreover, we give a surprisingly simple expression for the parameter β which was known before to equal a very complicated functional of A. In our proof we use two new ideas; the first idea is to look at the solutions of hyperbolic PDEs with a special form of data known as quasi polynomials, where the polynomial coefficients are replaced by periodic functions. This is useful in unfolding the spatial structure of the solution as well as expressing the locally periodic solutions in terms of combination of much simpler purely periodic functions. Next, we look at the time averages of solution of hyperbolic PDEs and provide general statements which might potentially be applicable to much broader areas. With the help of these two ideas we are able to fully understand the crucial role consistency plays in HMM type algorithms. Finally we present numerical results to support our theoretical statements.

References [1] E. Weinan, B. Engquist, The Heterogeneous Multi-Scale Methods, Comm. Math. Sci., 1(1):87-133, 2003. [2] B. Engquist, H. Holst, and O. Runborg, Multi-Scale Methods for Wave Propagation in Heterogeneous Media Over Long Time, in Lect. Notes Comput. Sci. Eng., Springer Verlag, 82:167-186, 2011. [3] B. Engquist, H. Holst and O. Runborg, Multi-Scale methods for Wave Propagation in Heterogeneous Media, Commun. Math. Sci., 9(1):33-56, 2011. [4] F. Santosa, W. W. Symes, A Dispersive Effective Medium For Wave Propagation in Periodic Composites , Siam J. Appl. Math., 51 (4):984-1005, 1991. [5] A. Abdulle and M. J. Grote, Finite Element Heterogeneous Multi-Scale Method for the Wave Equation, SIAM J. Multiscale Model. and Simul., 9(2):766-792, 2011. Joint work with Prof. Olof Runborg.

25

Marco Artina Technische Universität München, DE Anisotropic mesh adaptation for brittle fractures Minisymposium Session ADFE: Tuesday, 12:00 - 12:30, CO016 The study of the evolution of brittle fractures is a very challenging continuum mechanics problem, which requires an interdisciplinary study, from physics to mathematical analysis, and computations. The propagation of a fracture can be mathematically described as a rate independent evolution, where nonconvex and nonsmooth energies are instantaneously minimized under forcing constraints. One of the most advocated model for describing fractures is the Francfort-Marigo model. It is particularly interesting because it is well defined in a rather general setting and does not require any predefined path for the crack. To numerically approximate the problem one needs first to Gamma-approximate the nonsmooth energy, which depends on the displacement and its discontinuity set, by using a smoother version as proposed by Ambrosio and Tortorelli where a smooth indicator function is used to identify the discontinuity set. Then, we resort to an adaptive Finite Element approach based on P1 elements. However, similarly to early work by Chambolle et al., but differently from recent approaches by Süli et al. where isotropic meshes are used, in this work we wish to investigate how designing anisotropic meshes can lead to dramatic improvements in terms of the balance between accuracy and complexity. Indeed the main advantage which can be achieved is the significant reduction of the number of elements required to capture with good confidence the expected fracture path. The employment of anisotropic grids allows us to follow very closely the propagation of the fracture, refining it only in a very thin neighborhood of the crack. In this talk, we first present the derivation of a novel a posteriori error estimator driving the mesh adaptation. Then, we provide several numerical results which assess both the accuracy and the computational saving associated with an anisotropic adapted grid. Joint work with Massimo Fornasier, Stefano Micheletti, and Simona Perotto.

26

Christoph Augustin Institut für Biophysik, Medizinische Universität Graz, AT Parallel solvers for the numerical simulation of cardiovascular tissues Contributed Session CT1.7: Monday, 17:30 - 18:00, CO122 For the numerical simulation of the elastic behavior of biological tissues, such as the artery or the myocardium, we consider the stationary equilibrium equations div σ(u, x) + f (x) = 0

for x ∈ Ω.

(1)

In addition we have to incorporate boundary conditions to describe the displacements or the boundary stresses on Γ = ∂Ω. For the derivation of the constitutive equation of the stress tensor σ, we introduce the strain energy function Ψ(C). From this we obtain the constitutive equation σ = J −1 F

∂Ψ(C) > F , ∂C

(2)

where J = det F is the Jacobian of the deformation gradient F = ∇x ϕ(x), and C = F > F is the right Cauchy-Green tensor. The specific form of the strain energy function now varies from material to material, e.g. for the artery we use the well known Holzapfel model o κ c k1 X n Ψ(C) = (J − 1)2 + (J −2/3 I1 − 3) + exp[k1 (J −2/3 Ii − 1)2 ] − 1 , 2 2 2k2 i=4,6 where κ, c, k1 and k2 are positive parameters, I1 = tr(C) and I4 and I6 are invariants representing the stretch in fiber direction. Due to preferential orientations of fibers, such as collagen, the modeling of biological tissues leads to an anisotropic and highly nonlinear material model. In order to obtain a numerical solution of eq. (1) we use variational and finite element techniques. For the linearization of the resulting system Newton’s method is applied. However, such detailed multiphysics simulations are computationally vastly demanding. While current trends in high performance computing (HPC) hardware promise to alleviate this problem, exploiting the potential of such architectures remains challenging for various reasons. On one hand, strongly scalable algorithms are necessitated to achieve a sufficient reduction in execution time by engaging a large number of cores, and, on the other hand, alternative acceleration technologies such as graphics processing units (GPUs) are playing an increasingly important role which imposes further constraints on design and implementation of solver codes. We discuss two different parallel approaches to solve the non-linear elasticity problems arising from the simulation of the mechanical behavior of cardiovascular tissues. The finite element tearing and interconnecting (FETI) methods, in particular classical and allfloating FETI, and a proper domain decomposition algebraic multigrid. Scalability results for these mechanical simulations will be presented and we discuss advantages and limitations of the particular numerical methods. We will also show first results of weakly and strongly coupled electro-mechanical problems and discuss challenges that need to address with regard to highly scalable parallel implementations. Joint work with Gernot Plank. 27

Gia Avalishvili Iv. Javakhishvili Tbilisi State University, GE On spectral method of approximation of dynamical dual-phase-lag three-dimensional model for thermoelastic shells by two-dimensional initial-boundary value problems Contributed Session CT3.6: Thursday, 16:30 - 17:00, CO017 The present paper is devoted to construction and investigation of spectral algorithm of approximation of three-dimensional initial-boundary value problem corresponding to dynamical model for thermoelastic shells by two-dimensional ones in the context of Chandrasekharaiah-Tzou nonclassical theory of thermoelasticity. We study dynamical problems for thermoelastic shells within the framework of the nonclassical theory of thermoelasticity with two phase-lags, which was proposed to eliminate shortcomings of the classical thermoelasticity, such as infinite velocity of thermoelastic disturbances, unsatisfactory thermoelastic response of a solid to short laser pulses, and poor description of thermoelastic behavior at low temperatures. In the paper [1] Tzou proposed a dual-phase-lag heat conduction model, where the phase-lag corresponding to temperature gradient is caused by microstructural interactions such as phonon scattering or phonon-electron interactions, while the second phase-lag is interpreted as the relaxation time due to fast-transient effects of thermal inertia. Further, Chandrasekharaiah [2] constructed nonclassical model for thermoelastic bodies, where the classical Fourier’s law of heat conduction was replaced with its generalization proposed by Tzou. In this model the equation describing the temperature field involves the third order derivative with respect to the time variable of the temperature and divergence of the third order derivative with respect to the time variable of the displacement. Note that the Chandrasekharaiah-Tzou model is an extension of the Lord-Shulman [2] nonclassical model for thermoelastic bodies, which depends on one phase-lag. Spatial behavior of solutions of the dual-phase-lag heat conduction equation and problems of stability of dual-phase-lag heat conduction models have been investigated and particular one-dimensional initial-boundary value problems have been analysed in the Chandrasekharaiah-Tzou theory [3-5]. In this paper we investigate general three-dimensional initial-boundary value problem with mixed boundary conditions corresponding to Chandrasekharaiah-Tzou nonclassical model. Applying variational approach and suitable a priori estimates, we obtain the existence and uniqueness of solution in corresponding Sobolev spaces. In order to simplify algorithms of numerical solution of three-dimensional problem we construct a sequence of two-dimensional initial-boundary value problems applying spectral approximation method, which is a generalization and extension of the dimensional reduction method suggested by I. Vekua [6] in the classical theory of elasticity for plates with variable thickness. Note that the classical Kirchhoff-Love and Reissner-Mindlin models can be incorporated into the hierarchy obtained by I. Vekua. Static two-dimensional models constructed by I. Vekua for general shells first were investigated in [7] and in the case of plate the rate of approximation of the exact solution of the three-dimensional problem by the vector-functions of three space variables restored from the solutions of the reduced two-dimensional problems in the spaces of classical smooth functions was estimated in [8]. Later on, various two-dimensional and one-dimensional models were constructed and investigated for problems of the theory of elasticity and mathematical physics applying I. Vekua’s reduction method and similar spectral methods (see [9] and references

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given therein). We construct algorithm of approximation of the three-dimensional initial-boundary value problem corresponding to Chandrasekharaiah-Tzou dynamical model by a sequence of two-dimensional problems for thermoelastic shells with variable thickness, which may vanish on a part of the boundary. Applying semidiscretization of the three-dimensional problem in the transverse direction of the shell, we construct a hierarchy of two-dimensional initial-boundary value problems and investigate them in suitable weighted Sobolev spaces. Moreover, we study the relationship between the constructed problems and the original three-dimensional one. We prove convergence in suitable spaces of the sequence of approximate solutions of three space variables, constructed by means of the solutions of the reduced problems, to the exact solution of the original three-dimensional problem and under additional regularity conditions we estimate the rate of convergence. Note that the constructed algorithm of approximation can be used not only for simplification of algorithms for numerical solution of three-dimensional problems, but also the first approximations of the constructed hierarchy of two-dimensional initialboundary value problems can be considered as independent nonclassical models for thermoelastic shells and can be used for mathematical modeling of engineering structures. Acknowledgment. The work of Gia Avalishvili has been supported by the Presidential Grant for Young Scientists (Contract No. 12/62). References [1] D.Y. Tzou, A unified approach for heat conduction from macro to micro-scales, J. Heat Transfer, 117 (1995), 8-16. [2] D.S. Chandrasekharaiah, Hyperbolic thermoelasticity: a review of recent literature, Appl. Mech. Review, 51 (1998), 705-729. [3] D.S. Chandrasekharaiah, One-dimensional wave propagation in the linear theory of thermoelasticity without energy dissipation, J. Thermal Stresses, 19 (1996), 695-710. [4] R. Quintanilla, A condition on the delay parameters in the one-dimensional dual-phase-lag thermoelastic theory, J. Thermal Stresses, 28 (2005), 43-57. [5] R. Quintanilla, R. Racke, Qualitative aspects in dual-phase-lag heat conduction, Proc. Royal Society A, 463 (2007), 659-674. [6] I.N. Vekua, Shell theory: General methods of construction, Pitman Adv. Publ. Program, Boston, 1985. [7] D.G. Gordeziani, On the solvability of some boundary value problems for a variant of the theory of thin shells, Dokl. Akad. Nauk SSSR, 215 (1974), 12891292. [8] D.G. Gordeziani, To the exactness of one variant of the theory of thin shells, Dokl. Akad. Nauk SSSR, 216 (1974), 751-754. [9] G. Avalishvili, M. Avalishvili, D. Gordeziani, B. Miara, Hierarchical modeling of thermoelastic plates with variable thickness, Anal. Appl., 8 (2010), 125-159. Joint work with Mariam Avalishvili.

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Iliass Azijli Delft University of Technology, NL Physics-based interpolation of incompressible flow fields obtained from experimental data: a Bayesian perspective Contributed Session CT2.7: Tuesday, 14:30 - 15:00, CO122 We introduce a statistical interpolation method that uses a physical model in the reconstruction of velocity fields obtained from experiments. It is applicable to incompressible flows. Due to the inclusion of physical knowledge, the spatial resolution of the data can be increased. We formulate the method within the Bayesian framework because it allows for a natural inclusion of measurement uncertainty [1]. Our method is therefore a generalization of the works of Narcowich and Ward [2] and Handscomb [3], who followed a deterministic derivation path. The method is applied to a two-dimensional synthetic test case and to real experimental data of a circular jet in water [4]. Tomographic particle image velocimetry was used to extract the three components of velocity in a three-dimensional space [5]. Bayesian inference consists of three steps. First, one states a prior belief for a state f . Assuming Gaussian processes, f ∼ N (µ, P ). µ is the prior mean and P is the prior covariance matrix. Then, data (y) is collected: y | f ∼ N (Hf, R), where H is the observation operator that converts the unobserved state into the observed set of data points. R is the observation error covariance matrix. Finally, Bayes’ Theorem is used to obtain the posterior distribution, which is also normally distributed due to the assumption of Gaussian processes. From the continuity equation it follows that the velocity field of an incompressible flow has zero divergence. From vector calculus it is known that such a field can be obtained by taking the curl of a vector potential. The state vector f is therefore defined to consist of the components of the potential and their first partial derivatives. The correlation between a vector potential component and its derivatives, and the correlation between these derivatives, is incorporated in the prior covariance matrix [6]. However, it is assumed that the different potential components are uncorrelated, because there is no prior physical knowledge to assume that they are. The observation operator is constructed such that it returns the curl of the potential, being the observed velocity field. Table 1 compares physics-based interpolation (pb) with standard interpolation (st) for the synthetic test case. By standard interpolation, we mean the assumption of uncorrelated velocity components [7]. The results show that physics-based interpolation indeed increases the spatial resolution of the data, even in the presence of measurement uncertainty.

References [1] Wikle, C.K., and Berliner, L.M., A Bayesian tutorial for data assimilation, Physica D: Nonlinear Phenomena, Vol.230, pp. 1-16, 2007. [2] Narcowich, F.J., and Ward, J.D., Generalized Hermite interpolation via matrix-valued conditionally positive definite functions, Mathematics of Computation, Vol. 63, No. 208, pp. 661-687, 1994.

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N 1/2 3 6 9 12 15

NE (st) 2.77e-1 6.62e-3 1.77e-4 4.74e-5 1.76e-5

NE (pb) 2.08e-1 6.64e-4 3.08e-5 1.04e-5 4.54e-6

GE (st) 2.77e-1 7.71e-3 3.57e-3 1.85e-3 1.03e-3

GE (pb) 2.08e-1 5.69e-3 2.29e-3 1.27e-3 7.94e-4

LE (st) 2.77e-1 1.07e-2 4.85e-3 1.79e-3 1.02e-3

LE (pb) 2.08e-1 5.24e-3 1.23e-3 4.60e-4 3.01e-4

Table 1: RMSE as a function of the number of sample points (N ) for the synthetic test case. NE: perfect measurement, no measurement uncertainty. GE: spatially varying measurement uncertainty, but constant uncertainty assumed. LE: spatially varying measurement uncertainty, and correct measurement uncertainty used. [3] Handscomb, D., Local recovery of a solenoidal vector field by an extension of the thin-plate spline technique, Numerical Algorithms, Vol. 5, pp. 121-129, 1993. [4] Violato, D., and Scarano, F., Three-dimensional evolution of flow structures in transitional circular and chevron jets, Physics of Fluid, Vol. 23, 2011. [5] Elsinga, G.E., Scarano, F., Wieneke, B., and van Oudheusden, B.W., Tomographic particle image velocimetry, Exp Fluids, Vol. 41, pp. 933-947, 2006. [6] Rasmussen, C., and Williams, C., Gaussian processes for machine learning, MIT Press, 2006. [7] de Baar, J.H.S., Percin, M., Dwight, R.P., van Oudheusden, B.W., and Bijl, H., Kriging regression of PIV data using a local error estimate, submitted 2012. Joint work with Richard Dwight, and Hester Bijl.

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Laura Azzimonti MOX - Department of Mathematics, Politecnico di Milano, IT Mixed Finite Elements for spatial regression with PDE penalization Contributed Session CT2.7: Tuesday, 14:00 - 14:30, CO122 In this work we study the properties of a non-parametric regression technique for the estimation of bidimensional or three dimensional fields on bounded domains from some pointwise noisy evaluations. We focus on applications in physics, engineering, biomedicine, etc. where a prior knowledge on the field might be available from physical principles and should be taken into account in the smoothing process. We consider in particular phenomena where the field is described by a partial differential equation (PDE) and has to satisfy some known boundary conditions. Spatial regression with PDE penalization (SR-PDE) has been developed in [1] for the estimation of the blood velocity field on the section of an artery from EchoDoppler data. This technique has very broad applicability. Many applications of particular interest can be named: the estimation of the concentration of pollutant released in water or in the air transported by the stream or by the wind, the estimation of temperature or pressure fields from electronic control units or sensors in environmental sciences and many other phenomena in physics, biology and engineering. In this work we focus on phenomena that are well described by linear second order elliptic PDEs, typically transport-reaction-diffusion problems. The field is estimated minimizing a penalized least squares functional that generalizes classical smoothing techniques such as thin-plate splines. Thin-plate splines and, more recently, the spatial spline regression models described in [3] estimate bidimensional surfaces penalizing a measure of the local curvature. We propose instead to minimize the functional Z n 1X 2 (f (pi ) − zi ) + λ (Lf − u)2 (1) J(f ) = n i=1 Ω where pi are the observation sites, zi the observations and f the field to be estimated. The penalty term involves the misfit of a second order PDE, Lf = u, modeling the phenomenon under study. This, in turns, corresponds to assuming that the forcing term in the PDE is not exactly known. On the other hand, we assume here that all the parameters appearing in the PDE (except for the forcing term) and the boundary conditions are completely determined. This approach is similar to the one used in control theory when a distributed control is considered. The main difference from classical results in control theory is that the observations are pointwise and affected by noise. For this reason it is necessary to require higher regularity to the field to ensure that the functional J(f ) is well defined. In [2] we prove the existence and the uniqueness of the estimator in the Sobolev space H 2 , in the described pointwise framework. In particular, minimizing the functional J(f ) is equivalent to solving a fourth order problem; we resort to a mixed approach for fourth order problems in order to prove the existence and the uniqueness of the estimator. Accordingly, a mixed equal order Finite Element method is used for discretizing the estimation problem; the proposed method is similar to some classical methods used for the discretization of fourth order problems. The well-posedness of the discrete problem is also proved. Both the continuous and the discrete estimators have a bias induced by the presence of the penalty term in the minimized functional. We obtain a bound for the bias of the continuous estimator and we study the convergence of the bias of the 32

discrete estimator when the mesh size goes to zero. The proposed mixed equal order Finite Element discretization is known to have sub-optimal convergence rate when applied to fourth order problems with arbitrary boundary conditions and, in particular, the first order approximation might not converge to the exact solution. However we are able to prove optimal convergence of the proposed discretization method for the specific set of boundary conditions that are naturally associated to the smoothing problem (1), whenever the true underlying field satisfies exactly those conditions. Finally the smoothing technique is extended to the case of data distributed on some subdomains, particularly interesting in many applications. For instance in the case of the driving problem considered in [1] concerning the velocity field estimation, the Echo-Doppler data represent the mean velocity of blood on some subdomains on the section of an artery. The properties of the estimator in the areal setting are obtained following the approach used in the pointwise framework.

References [1] Azzimonti L. Blood flow velocity field estimation via spatial regression with PDE penalization. PhD Thesis, Politecnico di Milano, PhD School “Mathematical models and methods in engineering”, 2013. [2] Azzimonti L., Nobile F., Sangalli L.M., Secchi P. Mixed Finite Elements for spatial regression with PDE penalization. In preparation. [3] Sangalli L.M., Ramsay J.O., Ramsay T. Spatial spline regression models. J. R. Stat. Soc. Ser. B Stat. Methodol., (75,4):1—23, 2013. Joint work with Fabio Nobile, Laura Maria Sangalli, and Piercesare Secchi.

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Ute Aßmann Universität Duisburg-Essen, DE Regularization in Sobolev spaces with fractional order Minisymposium Session FEPD: Monday, 15:30 - 16:00, CO017 We study the minimization of a quadratic functional subject to a nonlinear elliptic PDE where the Tichonov regularization term is given in H s with a fractional parameter s > 0. Moreover, pointwise control constraints are given. In order to allow a numerical treatment of this problem we introduce a multilevel approach as an equivalent norm concept. Furthermore, the existence of regular Lagrange multipliers can be shown. At the end of the talk we will present some numerical calculations. Joint work with Arnd Rösch.

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Markus Bachmayr RWTH Aachen, DE Adaptive methods based on low-rank tensor representations of coefficient sequences Minisymposium Session LRTT: Wednesday, 11:30 - 12:00, CO3 We consider a framework for the construction of iterative schemes for high-dimensional operator equations that combine adaptive approximation in a basis and low-rank approximation in tensor formats. Our starting point is an operator equation Au = f , where A is a bounded and elliptic linear operator mapping a separable Hilbert space H – for instance, a function space on a high-dimensional product domain – to its dual H 0 . Assuming that a Riesz basis of H is available, the original problem can be rewritten equivalently as a biinfinite linear system on `2 , where the system matrix is bounded and continously invertible. Under the given assumptions, a simple Richardson iteration on the infinite-dimensional problem converges, but of course cannot be realized in practice. This is the starting point for adaptive wavelet methods as introduced by Cohen, Dahmen and DeVore, which dynamically approximate such an ideal iteration by finite quantities. Such methods exploit the approximate sparsity of coefficient sequences. The new aspect here is that, in order to significantly reduce computational complexity in a high dimensional context, we make use of an additional tensor product structure of the problem. For this discussion, we assume H = H1 ⊗ · · · ⊗ Hd , i.e., that H is a tensor product Hilbert space, and that we have a tensor product Riesz basis of H. We now use a structured tensor format for the corresponding sequence of basis coefficients. Examples of suitable tensor structures are the Tucker format or the Hierarchical Tucker format, where the latter can also be used for problems in very high dimensions. A crucial common feature of both formats is that quasibest approximations by lower-rank tensors, with controlled error in `2 -norm, can be computed by procedures implementable by standard linear algebra routines. We are thus considering a highly nonlinear type of approximation: besides the multiplicative nonlinearity in the tensor representation, we aim to adaptively determine simultaneously suitable finite approximation ranks, the active indices for the basis expansions in the lower-dimensional spaces Hi , and corresponding coefficients. We accomplish this by a perturbed Richardson iteration, where approximation ranks and active basis indices are adjusted implicitly in a sufficiently accurate approximation of the residual. The resulting growth in the complexity of iterates is kept in check by combining a tensor recompression operation, which yields an approximation with lower ranks up to a specified error, with a coarsening operation that eliminates negligible coefficients in the lower-dimensional basis expansions. In the efficient realization of the latter, the special orthogonality properties of the considered tensor formats play a central role. Under the present quite general assumptions, we can then identify a choice of parameters for the resulting iterative scheme that ensures its convergence and produces approximations with near-minimal ranks. To our knowledge this is the first convergence result of this type. Under suitable further approximability conditions on the problem, we also obtain estimates for the total number of operations required for reaching an approximate solution with a certain target accuracy. Furthermore, we discuss the additional difficulties related to the preconditioning of problems posed on Sobolev spaces in this setting. We consider some possible applications and illustrate our theory by numerical experiments.

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Joint work with Wolfgang Dahmen.

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Santiago Badia CIMNE and UPC, ES Adaptive finite element simulation of incompressible flows by hybrid continuousdiscontinuous Galerkin formulations Minisymposium Session ADFE: Tuesday, 11:30 - 12:00, CO016 Conforming cG formulations are preferred over dG formulations when we focus on CPU cost (at the same convergence order). For simplicial meshes, dG formulations involve around 14 times more degrees of freedom than cG ones in dimension three (6 in two dimensions); those are the values obtained for a structured mesh with periodic boundary conditions. For hexahedral meshes this ratio is around 8 and 4 for quadrilateral meshes. Certainly, these numbers cannot be ignored when simulating complex and realistic phenomena. However, the solution of many problems of interest often exhibit sharp layers or strong singularities. The use of locally refined meshes in these regions is required in order to get good results, since uniformly refined meshes can be prohibitive. dG formulations are better suited to adaptive refinement, because they can easily deal with non-conforming meshes with hanging nodes, e.g. using local mesh refinement, compared to cG formulations. The redgreen mesh refinement strategy for cG formulations, which keeps the conformity of the mesh but not the aspect ratio. Alternatively, non-conforming refined meshes can be used together with cG formulations, by constraining the hanging nodes in order to keep continuity. This approach is certainly involved in terms of implementation and it is usually restricted to 1-irregular meshes, i.e. two neighboring elements can only differ in at most one level of refinement. The motivation of this work is a hybrid method that combines the low CPU cost of cG formulations with the capabilities of dG formulations when dealing with adaptive refinement, naturally denoted as continuous-discontinuous Galerkin (cdG) formulation. In particular, we design an equal-order cdG numerical method for the approximation of incompressible flows, due to the superior efficiency and simplicity both in the cG and dG case. The cdG formulation is designed in such a way that the method is stable and optimally convergent for this particular type of FE spaces. The resulting methods is a suitable combination of the cG variational multiscale (VMS) formulation and an equal-order symmetric interior penalty dG formulation with upwind for the convective term. Optimal stability and convergence results are obtained. For the adaptive setting, we use an standard error estimator and marking strategy. Numerical experiments show the optimal accuracy of the hybrid algorithm both for uniformly and adaptively refined non-conforming meshes. The outcome of this work is a finite element formulation that can naturally be used on non-conforming meshes, as discontinuous Galerkin formulations, while keeping the much lower CPU cost of continuous Galerkin formulations. Joint work with Joan Baiges.

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Yun Bai ANMC MATHICSE EPFL, CH Reduced basis finite element heterogeneous multiscale method for quasilinear problems Minisymposium Session MSMA: Monday, 14:30 - 15:00, CO3 In this talk, we introduce the reduced basis finite element heterogeneous multiscale method (RB-FE-HMM) based on an offline-online strategy for quasilinear problems [2]. In this approach, a small number of representative micro problems selected by a greedy algorithm are computed in an offline stage. Missing data in the homogenized equation are efficiently recovered by the linear combinations of those precomputed micro solutions in an online stage. Thanks to a new a posteriori error estimator, the result of [2] can be extended to quasilinear problem. A priori error estimates and convergence of the Newton method can be established. Numerical experiments show that the RB-FE-HMM considerably reduces the cost of the classical FE-HMM for quasilinear multiscale problems [3] originating from a large number of micro FE problems in each iteration of the Newton method.

References [1] A. Abdulle, Y. Bai and G. Vilmart, Reduced basis finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems. Preprint, submitted for publication, 2013. [2] A. Abdulle and Y. Bai, Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems. J. Comput. Phys., 231(21) (2012), 7014-7036. [3] A. Abdulle and G. Vilmart, Analysis of the finite element heterogeneous multiscale method for nonmonotone elliptic homogenization problems. To appear in Math. Comp., 2013. Joint work with Assyr Abdulle, and Gilles Vilmart.

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Ruth Baker University of Oxford, Mathematical Institute, England Developing multiscale models for exploring biological phenomena Plenary Session: Tuesday, 08:20 - 09:10, Rolex Learning Center Auditorium Epithelial tissues consist of one or more layers of closely packed cells and their dynamical behaviour plays a central role during development of the embryo. Epithelial cell sheets line the surfaces and cavities of organs throughout the body where they act as a protective layer, regulating the passage of chemicals to and from underlying tissues and restricting the invasion of pathogens and harmful substances. The highly organised nature of epithelial sheets means they can achieve complex morphogenetic processes involving folding or bending, through the coordinated movement and rearrangement of individual cells. Mechanics plays a key role in driving epithelial morphogenesis and various forms of mechanical feedback, including mechanotransduction, play a role in regulating and ‘fine tuning’ growth during development. A second key player is the dynamics of signalling networks which, for example, regulate cell death and cell size, and control of cell proliferation. Recent advances in our understanding have been facilitated by new imaging tools and fluorescent probes to measure tissue deformation and the dynamics of key signalling proteins within cells and tissues. Mathematical and computational modelling offer a complimentary tool with which to study these processes. Models can be used to develop abstract representations of biological systems, test competing hypotheses and generate new predictions that can then be validated experimentally. In this talk a comprehensive computational framework will be presented within which the effects of chemical signalling factors on growing epithelial tissues can be studied. The method incorporates a vertex-based cell model, in which cells are represented as polygons whose edges are shared with other cells in the tissue. Node movements are determined by simple force laws, and cell proliferation and junctional rearrangements can be incorporated by changes in the shared-edge configuration. The evolution of chemical signalling dynamics is modelled using a system of nonlinear partial differential equations (PDEs). The vertex model provides a natural mesh for the finite element method which is used to solve the PDEs governing the chemical signalling. As the tissue evolves and the cells rearrange an arbitrary Lagrangian-Eulerian framework is used. The method we describe may be adapted to a range of potential application areas, and even to other cell-based models with designated node movements, to accurately probe the role of chemical signalling in epithelial tissues. We demonstrate the potential uses of our model framework by showing its application to a number of areas in development.

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Jonas Ballani RWTH Aachen, DE Black box approximation strategies in the hierarchical tensor format Minisymposium Session LRTT: Tuesday, 12:00 - 12:30, CO3 The hierarchical tensor format allows for the low-parametric representation of tensors even in high dimensions d. The efficiency of this representation strongly relies on an appropriate hierarchical splitting of the different directions 1, . . . , d such that the associated ranks remain sufficiently small. This splitting can be represented by a binary tree which is usually assumed to be given. In this talk, we address the question of finding an appropriate tree from a subset of tensor entries without any a priori knowledge on the tree structure. The proposed strategy can be combined with rank-adaptive cross approximation techniques such that tensors can be approximated in the hierarchical format in an entirely black box way. Numerical examples illustrate the potential and the limitations of our approach. Joint work with Lars Grasedyck.

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Soeren Bartels University of Freiburg, DE Projection-free approximation of geometric evolution problems Minisymposium Session GEOP: Tuesday, 12:00 - 12:30, CO122 Geometric evolution problems are nonlinear parabolic or hyperbolic partial differential equations that involve a pointwise constraint described by a smooth submanifold. Typical examples include the harmonic map heat flow and wave maps. Closely related are problems in nonlinear elasticity that result from a dimension reduction and then involve a pointwise constraint on the gradient that may model the inextensibility of an elastic rod. Numerical methods based on a linearization of the constraint and a subsequent projection of the update to satisfy the constraint require restrictive conditions on the discretizations to guarantee numerical stability and convergence. We will demonstrate that in many evolution problems the projection step can be omitted leading to a violation of the constraint that is controlled by the step size and is independent of the number of iterations.

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Steffen Basting University Erlangen-Nuremberg, DE A hybrid level set / front tracking approach for fluid flows with free boundaries and interfaces Minisymposium Session FREE: Tuesday, 11:00 - 11:30, CO2 We present a hybrid level set / front tracking approach for the representation of sharp interfaces in finite element discretizations of two-phase flow models. The hybrid approach makes use of an implicit representation of the interface by means of a level set function. The computational mesh is obtained by deforming a simplicial reference mesh such that the mesh is aligned to the implicitly described geometry. The resulting meshes provide an additional explicit representation of the interface while guaranteeing optimality of the mesh quality. The proposed method is based on a variational approach to optimal meshes and leads to a fully automated mesh optimization procedure. Because mesh connectivity is retained, the proposed approach can be easily integrated into existing finite element codes. Due to the hybrid interface representation, the geometrical flexibility of conventional front tracking / moving mesh approaches is enhanced. We present and evaluate the proposed framework in the context of particulate flows and two-phase flow applications with free interfaces. Joint work with M. Weismann, and R. Prignitz.

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Markus Bause Helmut Schmidt University, University of the Federal Armed Forces Hamburg, DE Space-time Galerkin discretizations of the wave equation Contributed Session CT3.4: Thursday, 16:30 - 17:00, CO015 1. Motivation The accurate and reliable numerical approximation of the hyperbolic wave equation is of fundamental importance to the simulation of acoustic, electromagnetic and elastic wave propagation phenomena. Our interest in the numerical simualation of wave propagation phenomena comes from material inspection of lightweight materials (e.g. carbon fibre reinforced plastics) by piezoelectric induced ultrasonic waves. This is a relatively new and an intelligent technique to monitor the health of the structure, for damage detection and non-destructive evaluation. For this it is strictly necessary to understand wave propagation in such materials and the influence of the geometrical and mechanical properties of the system; cf. Fig. 1. 2. Variational time discretization Galerkin-type discretization schemes for the temporal variable were recently proposed and studied for the parabolic heat equation and the Stokes system. In this contribution we will present continuous and discontinuous variational time discretization schemes for the hyperbolic wave equation. For the discretization in space a symmetric interior penalty discontinuous Galerkin (SIPG) method is used. In the field of numerical wave propagation the spatial discretization by the discontinuous Galerkin finite element method (DGM) has attracted the interest of researchers. Advantages of the DGM are the flexibility with which it can accommodate discontinuities in the model, material parameter and boundary conditions and the ability to approximate the wavefield with high degree polynomials. The DGM has the further advantage that it can be energy conservative, and it is suitable for parallel implementation. The mass matrix of the DGM is block-diagonal, with block size equal to the number of degrees of freedom per element, such that its inverse is available at low computational cost. We show that the resulting block matrix system can be condensed algebraically by eliminating internal degrees of freedom. Using further the block diagonal structure of the mass matrix of the discontinuous Galerkin discretization in space, then allows us to solve the algebraic system of equations efficiently. The performance properties of the scheme are illustrated by numerical convergence studies. Moreover, the schemes are applied to wave propagation phenomena in heterogeneous media admitting multiple sharp wave fronts; cf. Fig. 1. Here, we briefly present our family of continuous variational time discretization schemes for the acoustic wave equation, as a prototype model, ∂t v(x, t) − ∇ · (c(x)∇u(x, t)) = f (x, t) ,

∂t u(x, t) = v(x, t) ,

written as a first order system of equations and equipped with the initial conditions u(0) = u0 , v(0) = v0 and homogeneous Dirichlet boundary conditions. Its counterpart of discontinuous variational times discretizations is not given here, but it will also be addressed in the presentation. We decompose I = [0, T ] into N subintervals In = (tn−1 , tn ]. For some Hilbert space H, let Xτr (H) = {u ∈ C(I, H) | u|In ∈ Pr (In , H)} and Yτr (H) = {w ∈ L2 (I, H) | w|In ∈ Pr−1 (In , H)}, where 43

 Pr Pr (In , H) = u : In 7→ H | u(t) = j=0 ξnj tj , ξnj ∈ H . Our continuous variational approximation of () then reads as: Find uτ ∈ Xτr (H01 (Ω)), vτ ∈ Xτr (L2 (Ω)) such that uτ (0) = u0 , vτ (0) = v0 and Z

T

0

Z T {h∂t vτ , ϕτ i + hc∇uτ , ∇ϕτ i} dt = hf, ϕτ idt 0 Z T {h∂t uτ , ψτ i − hvτ , ψτ i} dt = 0 0

for all ϕτ ∈ Yτr (H01 (Ω)) and ψτ ∈ Yτr (L(Ω)).

Here, h·, ·i denotes the inner product in L2 (Ω). Precisely, we have a PetrovGalerkin method, since the discrete time space Xτr (H) and discrete time test space Yτr (H) for the unknown uτ differ. We call this approach a cGP(r) method. Since Yτr (H) imposes no continuity constraint on its elements, the variational problem can be rewritten as a time marching scheme. We choose Lagrange basis functions to represent uτ , vτ and apply the Gauß-Lobatto quadrature rule to compute the integrals. Finally, the resulting semidiscrete approximation scheme is combined with an interior penalty discontinuous Galerkin method for the spatial discretization. For the cGP(2) method we observe numerically superconvergence of fourth order at the end points of the time intervals. 3. Future prospects By using the Galerkin method for the time discretization of the wave equation we have a uniform variational approach in space and time which may be advantageous for the future analysis of the fully discrete problem and the construction of simultaneous space-time adaptive methods. Further, it is very natural to construct methods of higher order and the well-known finite element stability concepts of the Galerkin-Petrov or discontinuous Galerkin methods can be applied. For future developments, the well-known adaptive finite element techniques can be applied for changing the polynomial degree and the length of the time steps.

Figure 1: Structural health monitoring by ultrasonic waves (left) and complex wave propagtion phenomena in heterogeneous media (right). Joint work with Uwe Köcher.

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Lorenz Berger University of Oxford, GB Solving the Generalised Large Deformation Poroelastic Equations for Modelling Tissue Deformation and Ventilation in the Lung Contributed Session CT1.7: Monday, 18:00 - 18:30, CO122 Gas exchange in the lungs is optimised by ensuring efficient matching between ventilation and blood flow, the distributions of which are largely governed by tissue deformation, gravity and branching structure of the airway and vascular trees. In this work, we aim to develop a 3D organ scale lung model that tightly couples the deformation of the tissue with the ventilation. Such a fully coupled model is needed to accurately model tissue deformation and ventilation in the lung, especially in the diseased case, where there is a strong interplay between both these components. To achieve this tight coupling we propose a novel multiscale model that approximates lung parenchyma by a biphasic (air and tissue) poroelastic finite deformation model. Briefly, the poroelastic equations are given by, ρs (1 − φ)

∂2u = ∇ · (σ e (u) − pI) + ρs (1 − φ)b + ρf φb ∂t2

w 1 = kf · (−∇p + ρf b + ∇ · (φσ vis )) f ρ φ ∂u ) = 0 in Ωt , ∂t 1 − φ0 J(u) = in Ωt . 1−φ

in Ωt ,

in Ωt ,

∇ · w + ∇ · (ρf

The primary variables of this system of equations are the deformation u, the fluid pressure p, the fluid flux w and the porosity φ. The other terms: σ e is the nonlinear effective stress of the solid skeleton valid for large deformations, b is an external body force and ρf and ρs are the densities of the fluid and solid, respectively. The permeability tensor is given by kf , the viscous stress of the fluid is given by σ vis and the determinant of the deformation gradient is denoted by J. The above model extends the classic linear poroelastic equations commonly used within the geomechanics community to uses in biology, specifically to model ventilation in the lung. Nonlinear elasticity theory that includes the effect of inertia is used to model the large deformations during breathing. Due to the high porosity and relative high fluid velocities in the lung we use a generalised Darcy flow model also known as the Brinkman model to allow for viscous effects in the fluid. Due to the size and nonlinearity of the model we propose an operator splitting scheme to solve the equations using the finite element method. To the best of our knowledge this is the first method that solves the fully incompressible large deformation poroelastic equations using an operator splitting approach. By solving the solid (deformation) and fluid equations separately, well developed preconditioners can be applied to each system. We present simulations that highlight the importance of including inertia forces and viscous stresses in the model for particular choices of parameters and show numerical experiments that demonstrate the convergence and stability of the algorithm. Finally, we present results on coupling a 1D fluid airway network to the 3D poroelastic medium. Joint work with Dr David Kay, and Dr Rafel Bordas. 45

Jean-Paul Berrut Université de Fribourg, CH The linear barycentric rational quadrature method for Volterra integral equations Contributed Session CT1.2: Monday, 17:00 - 17:30, CO2 We shall first introduce linear barycentric rational interpolation to the unaware audience : it can be viewed as a small modification of the classical interpolating polynomial. Then we present two direct quadrature methods based on linear rational interpolation for solving general Volterra integral equations of the second kind. The first, deduced by a direct application of linear barycentric rational quadrature given in former work, is shown to converge at the same rate, but is costly on long integration intervals. The second, based on a composite version of the rational quadrature rule, looses one order of convergence, but is much cheaper. Both require only a sample of the involved functions at equispaced nodes and yield a stable, infinitely smooth solution of most classical examples with machine precision. Joint work with Georges Klein, and Seyyed Ahmad Hosseini.

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Marie Billaud Friess Ecole Centrale de Nantes - GeM, FR A Tensor-Based Algorithm for the Optimal Model Reduction of High Dimensional Problems Contributed Session CT2.8: Tuesday, 14:00 - 14:30, CO123 Due to the need of more realistic numerical simulations, models presenting uncertainties or either numerous parameters are receiving a growing interest. To solve such high dimensional problems, one has to circumvent the so called curse of dimensionality when using classical numerical approaches. To overcome such an issue, model reduction approaches have became popular these last years. This presentation is concerned with the resolution of high dimensional linear equations by means of approximations in low-rank tensor subset. To compute the optimal approximation of the solution in this tensor subset, an ideal best approximation problem which consists in minimizing the distance to the exact solution for a given norm || · || is introduced. Since the exact solution is not available, such a problem cannot be directly solved. However, it can be replaced by computing a low rank tensor approximation of the unknown that minimizes the equation residual, which is computable, measured with another norm || · ||∗ . Nevertheless, if || · ||∗ is chosen in usual way, the resulting approximation may be far from the one expected by solving the initial best approximation problem with respect to ||·||. Here, we present an ideal minimal residual method that relies on an ideal choice for || · ||∗ and that can apply to high dimensional weakly coercive problems. Especially, || · ||∗ is chosen to ensure the equivalence between the best approximation problem for || · || and the residual minimization problem with || · ||∗ . Yet, the computation of the residual norm with || · ||∗ is not affordable in practice. Here, the residual norm is not exactly computed but estimated with a controlled precision δ. We thus propose a perturbed minimization algorithm of gradient type that provides an approximation of the optimal approximation of the solution with an error depending on δ. A progressive construction of the low-rank approximate solution of the initial problem is also introduced by means of greedy corrections computed with the proposed iterative algorithm. The resulting weak greedy algorithm is proved to be convergent under some assumptions on δ and is successfully validated to numerically solve stochastic partial differential equations. Joint work with Anthony Nouy, and Olivier Zahm.

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Adrian Blumenthal ANMC MATHICSE EPF Lausanne, CH Stabilized Multilevel Monte Carlo Method for Stiff Stochastic Problems Contributed Session CT4.8: Friday, 09:50 - 10:20, CO123 A new stabilized multilevel Monte Carlo (MLMC) method is presented which can be used for mean square stable stochastic differential equations with multiple scales. For problems where such stiffness occurs the performance of the standard MLMC approach based on classical explicit numerical integrators degrades. In fact due to the time step restriction on the fastest scales not all levels of the MLMC method can be exploited. In this talk we introduce a new stabilized MLMC approach based on explicit stabilized numerical schemes [1]. It is shown that balancing the stabilization procedure simultaneously with the hierarchical sampling strategy of MLMC methods reduces the computational cost for stiff systems significantly. Due to the explicit time stepping in our algorithm the simplicity of the MLMC implementation is preserved [2].

References [1] A.Abdulle and T.Li, S-ROCK methods for stiff Itô SDEs, Commun. Math. Sci., 6(4):845–868, 2008. [2] A.Abdulle and A.Blumenthal, Stabilized Multilevel Monte Carlo Method for Stiff Stochastic Differential Equations, accepted in J. Comput. Phys., 2013. Joint work with A. Abdulle.

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Jerome Bonelle EDF R&D - Univ. Paris-Est, CERMICS, FR Compatible Discrete Operator Schemes on Polyhedral Meshes for Stokes Flows Minisymposium Session ANMF: Tuesday, 12:00 - 12:30, CO1 Compatible Discrete Operator (CDO) schemes belong to the class of compatible (or mimetic, or structure-preserving) schemes. Their aim is to preserve key structural properties of the underlying PDE. This is achieved by distinguishing topological laws and constitutive relations. CDO schemes are formulated using discrete differential operators for the topological laws and discrete Hodge operators for the constitutive relations. CDO schemes have been recently analyzed in [1] for elliptic problems. We first review the main results in this case. Then, we derive CDO schemes for Stokes flows that are closely related to the recent work of Kreeft and Gerritsma [2]. We discuss analytical results and present numerical tests.

References [1] J. Bonelle and A. Ern, Analysis of Compatible Discrete Operator scheme for Elliptic Problems on Polyhedral Meshes, Available from: http://hal.archives-ouvertes.fr/hal-00751284, 2012. [2] J. Kreeft and M. Gerritsma, A priori error estimates for compatible spectral discretization of the Stokes problem for all admissible boundary conditions, arxiv:1206.2812 [cs.NA]. Joint work with Alexandre ERN.

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Francesca Bonizzoni CSQI - MATHICSE, EPF Lausanne, MOX - Dip. di Matematica, Politecnico di Milano, CH Low-rank techniques applied to moment equations for the stochastic Darcy problem with lognormal permeability Contributed Session CT3.7: Thursday, 17:00 - 17:30, CO122 Many natural phenomena and engineering applications are modeled by deterministic boundary value problems for partial differential equations where all the input data are assumed to be perfectly known. Thanks to the recent developments in scientific computing, it is now possible to efficiently and accurately compute the numerical solution of these problems. However, in reality, the problem data are either incompletely known or contain a certain level of uncertainty due to the material properties, boundary conditions, loading terms, domain geometry, etc. One way to overcome this is to describe the problem data as random variables or random fields, so that the deterministic problem turns into a stochastic differential equation. Stochastic models are employed in many areas such as financial mathematics, seismology and bioengineering. The solution of a stochastic differential equation is itself a random field u(ω) with values in a suitable function space V . The description of this random field requires the knowledge of its k-points correlation E[u⊗k ]. The simplest approach is Monte Carlo Method. Generally, its convergence rate is slow and this method is very costly. An alternative approach consists in deriving the so called moment equations, that is the deterministic equations solved by the probabilistic moments of the stochastic solution. We are interested in studying the fluid flow in a heterogeneous porous domain, with randomly varying permeability. We model this phenomenon using the Darcy law with lognormal permeability coefficient: − divx (a(ω, x)∇x u(ω, x)) = f (x)

a.s.

(1)

where the forcing term is deterministic and a(ω, x) = eY (ω,x) , Y (ω, x) Gaussian random field with standard deviation σ. The aim of the work is the computation of the statistical moments of u. Under the assumption of small standard devotion σ, we expand the random solution u(Y, x) in Taylor series in a neighborhood of E[Y ], and approximate u using its Taylor polynomial T K u. This approach is known as perturbation technique. We predict the divergence of the Taylor series, and the existence of an optimal order σ such that adding further terms to the Taylor polynomial will deteriorate the Kopt accuracy instead of improving it. The Taylor polynomial is directly computable only in the finite-dimensional setting, that is when Y (ω, x) is parametrized by a finite number of random variables. In the infinite-dimensional setting the Taylor polynomial involves uk , the k-th Gateaux derivative of u with respect to Y , for k = 0, . . . , K, which is not computable. However, it is possible to derive the deterministic equations solved by E[uk ]. Starting from the stochastic problem (1) we derive the problem solved by E[uk ] for k = 0, . . . , K, and state its well-posedness. The solution of this k-th order correction problem requires the solution of a recursion on the (l + 1)-points correlations E[uk−l ⊗ Y ⊗l ], for l = 1, . . . , k. Each correlation E[uk−l ⊗ Y ⊗l ] is defined on the tensorized domain D×(l+1) , and solves a high dimensional problem.

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In the discrete setting, each correlation E[uk−l ⊗ Y ⊗l ] is represented by a tensor of order l+1. The curse of dimensionality affects the recursion we are studying, since the number of entries of a tensor grows exponentially in its order. To overcome this problem, we propose to store and make computations between tensors in a data-sparse or low-rank format. Of particular interest is the Tensor Train format. A tensor in TT-format is represented as a sequence of order three tensors whose dimensions are called TT-ranks. We represent all correlations E[uk−l ⊗ Y ⊗l ] in TT-format and show that the number of entries grows almost linearly on the order l + 1, and the curse of dimensionality is greatly reduced. We develop an algorithm in TT-format able to compute E[T K u], the K-th order approximation of E[u]. In the simple one-dimensional case D = [0, 1] we perform some numerical tests both to study the complexity of the algorithm and the accuracy of the TT-solution. We compare the TT-solution with a collocation or Monte Carlo solution. Joint work with Kumar R., Nobile F., and Tobler C..

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Steffen Börm Christian-Albrechts-Universität zu Kiel, DE Fast evaluation of boundary element matrices by quadrature techniques Contributed Session CT1.8: Monday, 18:00 - 18:30, CO123 The boundary integral method is frequently used to treat partial differential equations, e.g., to handle unbounded domains. Discretizing the corresponding integral operators by a boundary element method leads to matrices G of the form Z Z gij = ϕi (x)γ(x, y)ψj (y) dy dx, ∂Ω

∂Ω

where γ is related to the differential operator’s fundamental solution and ϕi and ψj are suitable basis functions. If a standard discretization scheme is used, most of the entries of G can be expected to be non-zero, therefore standard sparse matrix formats cannot be used to represent the matrix efficiently. A viable approach is to use a data-sparse matrix e that approximation, i.e., to replace the n × n matrix G by an approximation G α requires only ∼ n log n units of storage. We propose a new approximation scheme that relies on the same representation formula as the boundary integral method, e.g., Green’s identity in the case of Laplace’s equation: restricted to subdomains τ + × σ with dist(τ + , σ) > 0, the fundamental solution is itself a solution of the homogeneous partial differential equation and can therefore be represented by a boundary integral Z Z ∂γ ∂γ (z, y) dz − (x, z)γ(z, y) dz x ∈ τ + , y ∈ σ. γ(x, y) = γ(x, z) ∂n ∂n + + z z ∂τ ∂τ If we choose τ ⊆ τ + such that dist(τ, ∂τ + ) > 0, the integrands are smooth enough to allow us to approximate the integrals by quadrature and obtain γ(x, y) ≈

q X ν=1

wν γ(x, zν )

q X ∂γ ∂γ (zν , y) − wν (x, zν )γ(zν , y) x ∈ τ, y ∈ σ ∂nz ∂nz ν=1

with quadrature points zν ∈ ∂τ + and quadrature weights wν , i.e., we can approximate the fundamental solution by a sum of tensor products. This approximation translates directly into an approximation of the matrix G by low-rank blocks that can be handled efficiently. For a d-dimensional problem, the boundary is a (d − 1)-dimensional manifold, therefore q ∼ md−1 quadrature points are sufficient for an m-th order quadrature rule. We can prove that the approximation converges exponentially if m is increased. In order to improve the compression ratio, we combine our approach with a cross approximation scheme. The resulting hybrid method starts by constructing local interpolation-type operators for all subdomains τ and then only has to compute a small number of matrix coefficients for each block to obtain the final approximation. Since only a small number of operations are required for each block, the hybrid algorithm is very efficient, and since we can afford to use cross approximation with full pivoting, it is also very robust. Joint work with Jessica Gördes, and Sven Christophersen.

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Malte Braack University of Kiel, DE Model- and mesh adaptivity for transient problems Minisymposium Session SMAP: Monday, 11:10 - 11:40, CO015 We propose a dual weighted error estimator with respect to modeling and discretization error based on time-averages for evolutionary partial differ- ential equations. This goal-oriented estimator measures the error of linear functionals averaged in time. We take advantage of time averages and circumvents the solution of a transient adjoint problem. We use the proposed estimator to solve convectiondiffusion-reaction equations containing e.g. atmospheric chemistry models as commonly used in meteorology. Joint work with Nico Taschenberger.

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Ondrej Budac EPFL, CH An adaptive numerical homogenization method for a Stokes problem in heterogeneous media Contributed Session CT2.9: Tuesday, 14:00 - 14:30, CO124 A finite element heterogeneous multiscale method is proposed for solving the Stokes problem in porous media. The method is based on the coupling of an effective Darcy equation on a macroscopic mesh, whose a priori unknown permeability is recovered from microscopic finite element approximations of Stokes problems on sampling domains. The computational work is independent of the smallness of the pore structure. A priori estimates are obtained and fully resolved for a locally periodic pore structure. Realistic micro structures lead to non-convex micro domains which significantly decrease convergence rates when uniform microscopic refinement is used. For complicated macroscopic domains, uniform macroscopic refinement also yields poor convergence rates. We therefore propose an adaptive multiscale strategy on both micro and macro scale based on a posteriori error indicators and derive an a posteriori error analysis of the coupled problem. Two and three-dimensional numerical experiments confirm the derived convergence rates. Joint work with Assyr Abdulle.

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Erik Burman University College London, GB Computability of filtered quantities for the Burgers’ equation Minisymposium Session SDIFF: Monday, 12:40 - 13:10, CO123 In this talk we will discuss finite element discretizations of the viscous Burgers’ equation. Stability will be ensured by a nonlinear stabilization term that switches on automatically where the solution exhibits oscillations at under resolved layers. For this method we consider estimates in weak norms, that can be interpreted as measuring the error in filtered quantities or local averages. Both a posteriori and a priori error estimates will be discussed, where the latter are derived from the former using the stability properties of the nonlinear scheme. An important property of these estimates is that the error constant is independent both of the Reynolds number and the Sobolev regularity of the exact solution, but depends on the initial data only. We will give a detailed exposition on the results on the Burgers’ equation and then discuss possible extensions to higher dimension. In particular we will discuss the Navier-Stokes’ equation in two space dimensions and how the present theory can be applied to the numerical analysis of large eddy simulation in a model situation.

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Erik Burman University College London, GB Projection methods for the transient Navier–Stokes equations discretized by finite element methods with symmetric stabilization Minisymposium Session ANMF: Tuesday, 11:30 - 12:00, CO1 We consider the transient Navier–Stokes equations discretized in space by finite elements with symmetric stabilization and in time by a projection method. We focus on the implicit Euler scheme for the time derivative of the velocity and a semi-implicit treatment of the convective term. The stabilization of velocities and pressures can be treated explicitly or implicitly. The analysis is performed for the Oseen equations. Stability estimates are derived under a CFL condition, leading to quasi-optimal error estimates for smooth solutions. The estimates do not depend explicitly on the viscosity, but, as usual, on the regularity of the exact solution. The analysis is illustrated by some numerical experiments. Joint work with Erik Burman, Alexandre Ern, and Miguel A. Fernandez.

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Rommel Bustinza Universidad de Concepcion, CL On a posteriori error analyses for generalized Stokes problem using an augmented velocity-pseudostress formulation Contributed Session CT2.4: Tuesday, 14:00 - 14:30, CO015 We develop two a posteriori error analyses for an augmented mixed method for the generalized Stokes problem. The stabilized scheme is obtained by adding suitable least squares terms to the velocity-pseudostress formulation of the generalized Stokes problem. Then, in order to approximate its solution applying an adaptive mesh refinement technique, we derive two reliable a posteriori error estimators of residual type, and study their efficiency. To this aim, we include two different analyses: the standard residual based approach, and an unusual one, based on the Ritz projection of the error. The main difference of both approaches relies on the way we treat the nonhomogeneous boundary condition. Finally, we present some numerical examples that confirm the theoretical properties of our approach and estimators.

References [1] T.P. B ARRIOS , R. B USTINZA , G. C. G ARCÍA , AND E. H ERNÁNDEZ, On stabilized mixed methods for generalized Stokes problem based on the velocity-pseudostress formulation: A priori error estimates. Computer Methods in Applied Mechanics and Engineering, vol. 237-240, pp. 78-87, (2012). [2] G.N. G ATICA , L. F. G ATICA AND A. M ARQUEZ, Analysis of a pseudostress based mixed finite element method for Brinkman model of porous media flow. Preprint 2012-02, Centro de investigación en Ingeniería Matemática, Universidad de Concepción, (2012). [3] G.N. G ATICA , A. M ÁRQUEZ AND M.A. S ÁNCHEZ, Analysis of a velocity-pressurepseudostress formulation for the stationary Stokes equations. Computer Methods in Applied Mechanics and Engineering, vol. 199, 17-20, pp. 1064-1079, (2010). [4] S. R EPIN AND R. S TENBERG, A posteriori error estimates for the generalized Stokes problem. Journal of Mathematical Sciences, vol. 142, 1, pp. 1828-1843, (2007). Joint work with Tomás P. Barrios, Galina C. García, and María González.

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Alexandre Caboussat Geneva School of Business Administration, CH Numerical Approximation of Fully Nonlinear Elliptic Equations Minisymposium Session GEOP: Tuesday, 11:00 - 11:30, CO122 Fully nonlinear elliptic equations have many applications in geometry, finance, mechanics or physics. Among them, the Monge-Ampère equation is the most well-known and the one that has gathered most of the attention for several years already. In this talk, we present some numerical methods for the solution of the Dirichlet problem for fully nonlinear elliptic equations. We focus in particular on the cases when no classical solutions exist or when solutions exhibit some non-smooth properties. We focus first on the Monge-Ampère equation in two dimensions of space, and then on the (sigma-2) equation in three dimensions of space. Both problems correspond to finding a function defined by some kind of given curvature. We detail a relaxation method, using a least squares approach, well-suited to the particular structure of these problems. This iterative method allows to decouple the differential operators from point-wise nonlinear problems, and provide a flexible computational framework. Classical variational PDE techniques and mixed finite element approximations are used to solve the differential operators. Mathematical programming techniques are used to solve the nonlinear optimization problems. Numerical experiments are presented for various examples in two and three dimensions of space, in particular when non-smooth solutions are expected. This is a joint work with Roland Glowinski (Univ. of Houston) and Danny C. Sorensen (Rice University).

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Alexandre Caboussat Geneva School of Business Administration, CH Numerical solution of a partial differential equation involving the Jacobian determinant Contributed Session CT1.2: Monday, 18:30 - 19:00, CO2 We address the numerical approximation of a fully nonlinear partial differential equation that involves the Jacobian determinant and that reads as follows: Find u : Ω → R2 satisfying det∇u = f u=g

in Ω on ∂Ω

where Ω ⊂ R2 is a two-dimensional domain, and f, g are given, sufficiently regular, data. This example of fully nonlinear equation has been studied from the theoretical viewpoint, starting in, e.g., [Dacorogna, Moser (1990)]. We present here a numerical framework relying on variational arguments together with an adequate high-order regularization. Based on previous works on fully nonlinear equations, we advocate an augmented Lagrangian method to provide an approximation of the solution of this problem. An iterative, Uzawa-type, algorithm is used to solve the corresponding saddle-point problem, and decouples the local nonlinearities from the differential operators arising in the variational framework. Piecewise linear finite elements are used for the space discretization. The discrete iterative algorithm consists in solving alternatively a boundary-value elliptic problem involving a biharmonic operator, and a sequence of local constrained optimization problems that arise on each grid element. Numerical experiments show the efficiency, and robustness and the algorithm, as well as its convergence properties when the problem admits a classical soluti on. Finally, we numerically investigate the cases when the problem is not necessarily well-posed. This is a joint work with Prof. Roland Glowinski (University of Houston) and Prof. Bernard Dacorogna (Ecole Polytechnique Fédérale de Lausanne). Keywords: Fully nonlinear equation, Jacobian determinant, Volume preserving mapping, Augmented Lagrangian, Finite element approximation.

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Alfonso Caiazzo WIAS Berlin, DE An explicit stabilized projection scheme for incompressible NSE: analysis and application to POD based reduced order modeling Minisymposium Session ANMF: Tuesday, 11:00 - 11:30, CO1 In this talk we propose a splitting scheme with a full explicit treatment of the convection for the numerical resolution of incompressible Navier-Stokes equations. The scheme is based on a Chorin-Temam projection method, combined with a recently proposed explicit stabilized treatment of advection equations [Burman, Ern, Fernandez, 2010]. The analysis of the method shows that the explicit stabilized advection is stable under a superlinear CFD condition. The method is tested on several problems, comparing the accuracy against the standard ChorinTemam scheme with semi-implicit advection. Furthermore, we show applications in the context of model order reduction based on Proper Orthogonal Decomposition (POD), where the explicit nature of the scheme allows to pre-compute the reduced matrix. Joint work with Miguel A. Fernandez, and Jean-Frederic Gerbeau.

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Eric Cances Ecole des Ponts and INRIA, FR Multiscale eigenvalue problems Minisymposium Session MSMA: Monday, 12:10 - 12:40, CO3 The numerical computation of the eigenvalues of a self-adjoint operator on an infinite dimensional separable Hilbert space, is a standard problem of numerical analysis and scientific computing, with a wide range of applications in science and engineering. Such problems are encountered in particular in mechanics (vibrations of elastic structures), electromagnetism and acoustics (resonant modes of cavities), and quantum mechanics (bound states of quantum systems). Galerkin methods provide an efficient way to compute the discrete eigenvalues of a bounded-frombelow self-adjoint operator A laying below the bottom of the essential spectrum of A. On the other hand, Galerkin methods may fail to approximate discrete eigenvalues located in spectal gaps, that is between two points of the essential spectrum. Such situations are encountered in multiscale eigenvalue problems when localized bound states are trapped by local defects in infinite periodic media (quantum dots in semi-conductors, defects in photonic crystals, atoms in the QED vacuum, ...). In some cases, the Galerkin method cannot find some of the eigenvalues of A located in spectral gaps (lack of approximation); in other cases, the limit set of the spectrum of the Galerkin approximations of A contains points which do not belong to the spectrum of A (spectral pollution). I will present recent results on the numerical analysis of these problems. Joint work with Virginie Ehrlacher, and Yvon Maday.

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Clément Cancès LJLL - UPMC Paris 6, FR Monotone corrections for cell-centered Finite Volume approximations of diffusion equations Minisymposium Session SDIFF: Monday, 11:10 - 11:40, CO123 We present a nonlinear technique to correct a general Finite Volume scheme for anisotropic diffusion problems, which provides a discrete maximum principle. We point out general properties satisfied by many Finite Volume schemes and prove the proposed corrections also preserve these properties. We then study two specific corrections proving, under numerical assumptions, that the corresponding approximate solutions converge to the continuous one as the size of the mesh tends to 0. Finally we present numerical results showing that these corrections suppress local minima produced by the original Finite Volume scheme. This work results from a collaboration with M. Cathala and Ch. Le Potier.

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Eric Cancès Ecole des Ponts and INRIA, FR Electronic structure calculation Plenary Session: Tuesday, 09:10 - 10:00, Rolex Learning Center Auditorium Electronic structure calculation is one of the main field of applications of quantum mechanics. It has become an essential tool in physics, chemistry, molecular biology, materials science, and nanosciences. In this talk, I will review the main numerical methods to solve the electronic Schrödinger equation and the Kohn-Sham formulation of the Density Functional Theory (DFT). The electronic Schrödinger equation is a high-dimensional linear elliptic eigenvalue problem, whose solutions can be numerically approximated either by stochastic methods. Sparse tensor product techniques can also be considered. Kohn-Sham models are constrained optimization problems, whose Euler-Lagrange equations have the form of nonlinear elliptic eigenvalue problems. Recent progress has been made in the analysis of these mathematical models and of the associated numerical methods, which paves the road to certified numerical simulations (with a posteriori error bounds) of the electronic structure of large molecular systems.

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Daniela Capatina University of Pau, FR Stopping criteria based on locally reconstructed fluxes Minisymposium Session STOP: Thursday, 15:00 - 15:30, CO1 A posteriori error estimators based on locally reconstructed H(div)-fluxes are nowadays well-established. Since they provide sharp upper bounds, it seems appropriate to use them to define stopping criteria for iterative solution algorithms. We consider a unified framework for local flux reconstruction, covering conforming, nonconforming and discontinuous Galerkin finite element methods. For this reconstruction, it is supposed that the discrete equations are satisfied. However, in the context of stopping criteria, this assumption is no longer verified. In this talk, we propose a generalization, where the local conservation property of the H(div)-fluxes is not fulfilled. It leads to a simple stopping criterion, balancing the discretization and the iteration errors. Joint work with Roland Becker, and Robert Luce.

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Daniela Capatina University of Pau, FR Robust discretization of the Giesekus model Minisymposium Session MANT: Wednesday, 11:00 - 11:30, CO017 We consider a discontinuous Galerkin discretization of a matrix-valued nonlinear transport equation, which arises in the modeling of viscoelastic fluids. More precisely, it describes the constitutive law of the conformation tensor for certain polymeric liquids. A challenging question from a numerical point of view is the positivity of the solution. We prove existence and uniqueness of the discrete maximal solution, as well as convergence of a modified Newton method and positive definiteness of the discrete solution. Applications to Giesekus and Oldroyd-B models for polymer flows are discussed. The positivity of the conformation tensor is crucial for the derivation of energy estimates and for the robustness of numerical schemes at large Weissenberg numbers. Numerical simulations will be presented. Joint work with Roland Becker, and Didier Graebling.

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Laura Cattaneo Politecnico di Milano, IT Computational models for coupling tissue perfusion and microcirculation Contributed Session CT1.7: Monday, 18:30 - 19:00, CO122 Reduced models of fluid flows and mass transport in heterogeneous media are often adopted in the computational approach when the geometrical configuration of the system at hand is too complex. A paradigmatic example in this respect is blood flow through a network of capillaries surrounded by a porous interstitium. We numerically address this biological system by a computational model based on the Immersed Boundary (IB) method, a technique originally proposed for the solution of complex fluid-structure interaction problems [Liu et al., Comput. Methods Appl. Mech. Engrg. 195 (2006)]. Exploiting the large aspect ratio of the system, we avoid resolving the complex 3D geometry of the submerged vessels by representing them with a 1D geometrical description of their centerline and the resulting network. The IB method then gives rise to an asymptotic problem, obtained applying a suitable rescaling and replacing the immersed interface and the related conditions by means of an equivalent concentrated source term. The advantage of such an approach relies in its efficiency, because it does not need a full description of the real geometry allowing for a large economy of memory and CPU time and it facilitates handling fully realistic networks. The analysis of perfusion and drug release in vascularized tumors is a relevant application of such techniques. Delivery of diagnostic and therapeutic agents differs dramatically between tumor and normal tissues. Blood vessels in tumors are substantially leakier than in healthy tissue and they are tortuous. These vascular abnormalities lead to an impaired blood supply and abnormal tumor microenvironment characterized by hypoxia and elevated interstitial fluid pressure that reduces the distribution of macromolecules through advection [Chapman, S. et al., Bulletin of Mathematical Biology, 2008]. The aforementioned multiscale approach enables us to develop a simple computational model that retains the fluid dynamics characteristics of the microvasculature at the macroscale and describe the transport of macromolecules in the vascular structure and in the tumor interstitium. Fluid and mass transport within a tumor mass is governed by a subtle interplay of sinks and sources, such as the leakage of the capillary bed, the lymphatic drainage, the exchange of fluid with the exterior volume and the interstitial fluid pressure. To better characterize the microenvironment, we develop a resistance model for lymphatic drainage [Baxter, L.T. and Jain, R. K., Microvascular Research, 1990]. Regarding the boundary conditions on the outer surface of the tissue region, they are frequently not determined by available experimental information and additional assumptions must be made so that the problem is completely specified [Secomb, T.W. et al., Annals of Biomedical Engineering, 2004]. For interstitial perfusion, the flow conditions enforced at the boundary of the domain significantly determine how the model interacts with the exterior. We aim to model the in-vivo configuration, where the tumor, or a sample of it, is embedded into a similar environment. To represent this case, we believe that the most flexible option is to use Robin-type boundary conditions for the interstitial pressure. Tissue perfusion is particularly relevant because it directly affects how efficiently the microcirculation can bring nutrients and drugs to the cells permeating the interstitial tissue and simultaneously remove metabolic wastes. To study these

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effects we introduce two key indicators: the fluid flux from the capillary network to the interstitial volume, f t , and the equivalent conductivity of the tissue construct, kKkF . These indicators are affected by both the capillary conductivity and the interstitial fluid pressure. Understanding which of these two last factors dominates is the key point to determine what is the effect of enhanced permeability and retention over tissue perfusion. Finally we discuss the application of the model to delivery nanoparticles. In particular, transport of nanoparticles in the vessels network, their adhesion to the vessel wall and the drug release in the surrounding tissue will be adressed. Joint work with D. Ambrosi, L.Cattaneo, R. Penta, A. Quarteroni, and P. Zunino.

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Cris Cecka IACS, Harvard University, US Fast Multipole Method Framework and Repository Minisymposium Session PARA: Monday, 11:40 - 12:10, CO016 Fast multipole methods (FMM) are a general strategy for accelerating dense matrix-vector products of the form X φi = K(xi , yj ) σj j

where K is the interaction kernel, xi and yj are source and target values (usually points or functions in Rd ), and σj and φi are the source charges and target fields. The kernel may be a Green’s function from an N-body interaction, a integral operator from a boundary element method (BEM), or a radial basis function for weighting or interpolation. The FMM accelerates the O(N 2 ) matrix-vector product to O(N logα N ) and finds a wide range of applications in mechanics, fluid dynamics, acoustics, electromagnetics, N-body problems, machine learning, computer vision, and interpolation. FMMs require multiple, carefully optimized steps and numerical analysis in order to achieve the improved asymptotic performance and required accuracy. These research areas span tree generation, tree traversal, numerical and functional analysis, and the complex heterogeneous parallel computing strategies for each stage. Unfortunately, many FMM codes are written with a particular application (an interaction kernel and/or compute environment) in mind and optimized around it. It is often difficult to extract out advances from one research area and apply them to another code or application. I will present recent work on a new framework and repository for the generalized matrix-vector product above which attempts to abstract each stage of the FMM for independent development. This allows us to develop the code at a high level and collect a repository of interaction kernels for rapid application development in any of the above domains. We will present of number of use cases and test applications including a simple Poisson problem, more advanced BEM solvers for molecular dynamics and/or electromagnetic PDEs, and the use of FMM as a preconditioner for related PDEs. In addition, I will present recent results for general optimization strategies for the FMM and the abstracted kernels. This includes using a runtime system for parallel scheduling and resolution of the complex dependencies within the tree structure, use of GPU computing for accelerating the more structured operations, and aggregating like-transformations to improve data locality. This is joint work with Lorena Barba, Simon Layton, Aparna Chandramowlishwaran, Rio Yokota, and Louis Ryan.

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Antonio Cervone ENEA - UTFISSM/SICSIS, IT Parallel assembly on overlapping meshes using the LifeV library Minisymposium Session PARA: Monday, 12:40 - 13:10, CO016 Matrices are required for the solution of PDEs when some implicit term is involved. Tipically, the system of (non-linear) differential equations is discretized into a linear system, that couples all the degrees of freedom associated to the problem. In the Finite Element Method (FEM) this matrix is filled using a procedure usually called "assembly", where the mesh that discretizes the domain of interest is traveled element by element and its contribution is added to the matrix. This step in the solution of the PDE can be a large part in terms of CPU time, especially when solving time dependent simulations or non-linear systems where the matrix terms must be assembled at every iteration. Unstructured meshes are very common when dealing with the simulation of large scale and realistic domains. This kind of meshes increase the computational time since the topology of the elements cannot be computed a priori, but every element must know which elements are its neighbors. In parallel codes that rely on domain decomposition, the assembly is performed on each subdomain separately. However, there are degrees of freedom that lie on the separation line between subdomains, and the corresponding set of support elements is split between processes. A widely used strategy to fill this contribution from different processes requires a communication between them. It is well known that the communication is one of the principal bottlenecks in codes. In order to avoid this communication we introduce a novel approach that avoids it by using overlapping subdivisions of the mesh. This means that, when the mesh is cut in subdomains that are assigned to each process, each degree of freedom that is associated with any process will have all the elements of the mesh that are in its support in the local mesh. This clearly generates overlapping meshes, that must be constructed accurately and efficiently when dealing with unstructured grids. The assembly procedure on this kind of meshes requires a larger computational cost, as the duplicated element contributions are computed more then once, but removes the need for any communication between processes. This work will describe the implementation techniques implemented in the LifeV library to perform this type of assembly procedure. An analysis and comparison of this approach is also shown, in order to assess the performance of this approach in comparison with the traditional ones. Joint work with Nur A. Fadel, and Luca Formaggia.

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Xingyuan Chen Center of Smart Interfaces, Technische Universität Darmstadt, DE A numerical study of viscoelastic fluid-structure interaction and its application in a micropump Minisymposium Session MANT: Wednesday, 11:30 - 12:00, CO017 Micropumps play an important role in recent years in biomedical applications, such as sampling and drug delivery. In these applications fluids normally have nonNewtonian, in particular viscoelastic, features. They flow in deformable domains along with the interaction with elastic solids. Micropumps delivering Newtonian fluids have been thoroughly studied in literature, but not much work is associated with viscoelastic fluids. In the present work we use the recent techniques for simulation of viscoelastic fluid flow and fluid-structure interaction (FSI) to investigate this complex problem. We study the interaction between an Oldroyd-B fluid and an elastic solid using the partitioned implicit coupling approach. Our in-house code FASTEST serves as the flow solver, which is based on the block-structured collocated finitevolume method. To cope with the high Weissenberg number problem (HWNP) in simulation of viscoelastic fluid flow, we apply two stabilization approaches based on the so-called Log-Conformation Representation (LCR) [1] and Square RootConformation Representation (SRCR) [2]. We do a comprehensive comparison study of these two approaches and the standard approach, i.e. devoid of any stabilization approaches. Therefore we examine the test cases lid-driven cavity, 4:1 contraction flow and flow past a cylinder. We find that LCR and SRCR are not only more stable, that is they can predict flows with much higher Weissenberg number (Wi) than the standard approach, but they also allow the use of a larger grid spacing without loss of accuracy. The latter can be seen e.g. in the mesh study of the normal stress along the lid in lid-driven cavity (Figure 1). The solid part is solved by the finite-element method program FEAP developed by U.C. Berkeley. The MpCCI Coupling Environment is used as an interface for code coupling. A two-dimensional collapsible channel, i.e. the middle part of the upper wall is replaced by an elastic solid (Figure 2 (a)), is chosen as the first test case to investigate the difference between Newtonian and viscoelastic FSI. We find that the growing elastic effect of fluid increases the pressure on the membrane (Figure 2 (b)), which results in pushing the membrane upwards (Figure 2 (d)). Interestingly, we also find that the pressure drop along the channel decreases with growing Wi (Figure 2 (c)). As reported in [3], the HWNP occurs in this case. This problem is not observed when LCR and SRCR are applied. In the second case, we study a valveless micropump. The fluid flow in this type of micropump is driven by a vibrating membrane. The effect of fluid flow on membrane is not negligible in actual applications. We are investigating the different effects between Newtonian and viscoelastic fluids on the performance and efficiency of the micropump. This work is in progress. [1] R. Fattal et al., J. Non-Newton. Fluid Mech. 126 (2005) 23-37 [2] N. Balci et al., J. Non-Newton. Fluid Mech. 166 (2011) 546-553. [3] D. Chakraboty et al., J. Non-Newton. Fluid Mech. 165 (2010) 1204-1218

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Figure 1: A mesh study of the normal stress along the lid. ErM 1 is the relative error of values of the coarsest mesh M1 compared with the extrapolated values.

Figure 2: (a) Geometry of the 2D collapsible channel; (b) Pressure on the membrane at different Wi; (c) Pressure drop along the channel at different Wi; (d) Displacement of the middle point of the membrane at different Wi. Joint work with Holger Marschall, Michael Schäfer, and Dieter Bothe.

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Peng Chen EPFL, CH A Weighted Reduced Basis Method for Elliptic Partial Differential Equations with Random Input Data Contributed Session CT4.4: Friday, 09:50 - 10:20, CO015 In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equation (PDE) with random input data. The PDE is first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance at different values of the parameters are taken into account by assigning different weight to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and stochastic collocation method in both univariate and multivariate stochastic problems. Joint work with Alfio Quarteroni, and Gianluigi Rozza.

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Francisco Chinesta ECN - IUF, FR DEIM-based Non-Linear PGD Minisymposium Session ROMY: Thursday, 10:30 - 11:00, CO016 The efficient resolution of complex models (in the dimensionality sense) is probably the essential objective of any model reduction method. This objective has been clearly reached for many linear models encountered in physics and engineering [2,4]. However, model order reduction of nonlinear models, and specially, of parametric nonlinear models, remains as an open issue. Using classic linearization techniques such Newton method, both the nonlinear term and its Jacobian must be evaluated at a cost that still depends on the dimension of the non-reduced model [1]. The Discrete Empirical Interpolation Method (DEIM), which the discrete version of the Empirical Interpolation Method (EIM) [3], proposes to overcome this difficulty by using the reduced basis to interpolate the nonlinear term. The DEIM has been used with Proper Orthogonal Decomposition (POD) [1,4] where the reduced basis is a priori known as it comes from several pre-computed snapshots. In this work, we propose to use the DEIM in the Proper Generalized Decomposition (PGD) framework [2], which is an a priori model reduction technique, and thus the nonlinear term is interpolated using the reduced basis that is being constructed during the resolution. Consider a certain model in the general form: L(u) + FN L (u) = 0

(1)

where L is a linear differential operator and FN L is a nonlinear function, both applying over the unknown u(x), x ∈ Ω = Ω1 × . . . × Ωd ⊂ Rd , which belongs to the appropriate functional space and respects some boundary and/or initial conditions. Using the PGD method implies constructing a basis B = {φ1 , . . . , φN } such that the solution can be written as: u(x) ≈

N X i=1

αi · φi (x)

(2)

where αi are coefficients, and φi (x) = Pi1 (x1 ) · . . . · Pid (xd ) , i = 1, . . . , N

(3)

being Pij (xj ), j = 1, . . . , d, functions of a certain coordinate xj ∈ Ωj . In the linear case, the basis B can be constructed sequentially by solving a nonlinear problem at each step in order to find functions Pij . In the nonlinear case a linearization scheme for Eq. 1 is compulsory, but evaluating the nonlinear term is still as costly as in the non-reduced model. The DEIM method proposes to circumvent this inconvenient by performing an interpolation of the nonlinear term using the basis functions. In a POD framework, these functions come from the precomputed snapshots, but in a PGD framework these functions are constructed by using the PGD algorithm. Here we propose to proceed as follows: I - Solve the linear problem: find u0 such that L(u0 ) = 0 → B 0 = {φ01 , . . . , φ0N0 } II - Select a set of points X 0 = {x01 , . . . , x0N0 }. Later on we explain how to make an appropriate choice.

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III - Interpolate the nonlinear term FN L using functions B 0 in the points X 0 . Or in other words, find the coefficients ϕ0i such as: N0

X FN L u0m ≡ FN L u0 (x0m ) = ϕ0i · φ0i (x0m ) , m = 1, . . . , N0

(4)

i=1

IV - Once we have computed {ϕ01 , . . . , ϕ0N0 }, the interpolation of the nonlinear term reads: N0 X FN L (u) ≈ b0 = − ϕ0i · φ0i (5) i=1

and therefore, the linearized problem writes: L(u) = b0

(6)

V - At this point, three options can be thought: (i) Restart the separated representation, i.e., find u1 ; (ii) Reuse the solution u0 , i.e. u1 = u0 + u e and (iii) Reuse by projecting. VI - From this point we repeat the precedent steps: let us assume that we have already computed uk . Then select a set of points X k = {xk1 , . . . , xkNk }, interpolate the nonlinear term using B k , and find uk+1 , until a certain convergence criteria is reached. [1] Chaturantabut, S. and Sorensen, D.C. SIAM J. Sci. Comput. (2010) 32 27372764. [2 ]Chinesta, F., Leygue, A., Bordeu, F., Aguado, J.V., Cueto, E., Gonzalez, D., Alfaro, I., Ammar, A. and Huertal. Arch. Comput. Methods Eng. (2013) 20 31-59 [3] Barrault, M., Maday, Y., Nguyen, N.C. and Patera, A.T. Comptes Rendus Mathematique (2004) 339/9 667-672. [4] Chinesta, F., Lavedeze, P. and Cueto, E. Archives of Computational Methods in Engineering (2011) 18 395-404. Joint work with J.V. Aguado, A. Leygue, E. Cueto, and A. Huerta.

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Moulay Abdellah Chkifa Laboratoire Jacques Louis Lions, FR High-dimensional adaptive sparse polynomial interpolation and application for parametric and stochastic elliptic PDE’s Contributed Session CT3.7: Thursday, 16:30 - 17:00, CO122 The numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number of involved parameter is large. We considers a model class of second order, linear, parametric, elliptic PDEs on a bounded domain D with diffusion coefficients depending on the parameters in an affine manner. For such models, it was shown in [1] that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated in the Hilbert space V = H01 (D) by multivariate sparse polynomials in the parameter vector y with a controlled number N of terms. The convergence rate in terms of N does not depend on the number of parameters in V , which may be arbitrarily large or countably infinite, thereby breaking the curse of dimensionality. However, these approximation results do not describe the concrete construction of these polynomial expansions, and should therefore rather be viewed as benchmark for the convergence analysis of numerical methods. We present the polynomial interpolation process in high dimension proposed in [4]. We explain how this process allows an easy Newton-like interpolation formula for constructing polynomial interpolants and that is has a proven moderate Lebesgue constant for well located interpolation points based on the results in [2]. As for the application to parametric PDEs, we show that sequences of sparse polynomials constructed by the interpolation process are proved to converge toward the solution with the optimal benchmark rate. Numerical experiments are presented in large parameter dimension, which confirm the effectiveness of the process.

References [1] A. C OHEN , R. D E V ORE , AND C. S CHWAB, Analytic regularity and polynomial approximation of parametric and stochastic PDE’s, Analysis and Applications (Singapore) 9, 1-37 (2011). [2] A. C HKIFA , A. C OHEN , R. D E V ORE , AND C. S CHWAB, Sparse Adaptive Taylor Approximation Algorithms for Parametric and Stochastic Elliptic PDEs, M2AN, volume 47-1, pages 253-280, 2013. [3] A. C HKIFA, On the Lebesgue constant of Leja sequences for the complex unit disk and of their real projection, volume 166, pages 176-200, 2013. [4] A. C HKIFA , A. C OHEN , AND C. S CHWAB, High-dimensional adaptive sparse polynomial interpolation and applications to parametric PDEs, To appear in JfoCM 2013 [5] A. C HKIFA , A. C OHEN, Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs., Submitted Joint work with Albert Cohen, and Christoph Schwab.

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Alexandra Christophe Laboratoire de Génie Electrique de Paris (LGEP) and Univ. Nice, FR Mortar FEs on overlapping subdomains for eddy current non destructive testing Contributed Session CT1.6: Monday, 17:00 - 17:30, CO017 The modelisation in eddy current (EC) non destructive testing (NDT) aims at reproducing the interaction between a sensor and a conductor in order to localize possible defects in the latter without damaging it. The finite element (FE) method is frequently used in this context as well suited to treat problems with complex geometries while keeping a simplicity in the implementation. However, in NDT, the modelisation has to be realized for different positions of the sensor, thus requiring a global remeshing of the problem domain. Different techniques to take into account the movement of a sensor avoiding remeshing have been studied (see e.g., [1]-[4]). The mortar element method (MEM), a variational non-conforming domain decomposition approach [5, 6], offers attractive advantages in terms of flexibility and accuracy. In its original version for non-overlapping subdomains, the information is transferred through the skeleton of the decomposition by means of a suitable L2 -projection of the field trace from the master to the slave subdomains. At the occasion of Enumath 2001, a MEM with overlapping subdomains has been proposed to couple a global scalar potential defined everywhere in the considered domain and a local vector potential defined only in (possibly moving) conductors [7], and later applied to study electromagnetic brakes [8]. In this paper, a new FE-MEM able to deal with moving non-matching overlapping grids is introduced, which realizes the bidirectional transfer of information between the fixed subdomain (including the conductor and the air) and the moving one (represented by the sensor). The field source is in the moving part. This is indeed what occurs in EC-NDT, as the alternative current alimented inductive coils move over the conductors to detect possible defects on them (visible as a perturbation of the EC distribution). Two numerical examples are presented to support the theory. The first, an electrostatic problem with known solution, to state the optimality of the method. The second, an EC-NDT application, to underline the flexibility and efficiency of the proposed approach. This work has the financial support of CEA-LIST.

References [1] C.R.I. Emson, C.P. Riley, D.A. Walsh, K. Ueda, T. Kumano, “Modelling eddy currents induced by rotating systems,” IEEE Trans. Mag., vol.34, No.5, pp. 2593-2596, 1998. [2] S. Kurz, J. Fetzer, G. Lehner, W.M. Ricker, “A novel formulation for 3D eddy current problems with moving bodies using a Lagrangian description and BEM-FEM coupling,” IEEE Trans. Mag., vol.34, No.5, pp. 3068-3073, 1998. [3] D. Rodger, H.C. Lai, P.J. Leonard, “Coupled elements for problems involving movement,” IEEE Trans. Mag., vol.26, No.2, pp. 548-550, 1990.

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[4] H. Zaidi, L. Santandrea, G. Krebs, Y. Le Bihan, E. Demaldent, “Use of Overlapping Finite Elements for Connecting Arbitrary Surfaces With Dual Formulations”, IEEE Trans. Mag., vol. 48, No. 2, pp. 583-586, 2012. [5] C. Bernardi, Y. Maday, A. Patera, “A new non-Conforming approach to domain decomposition: the mortar element method”, Seminaire XI du College de France, Brezis & Lions eds., in Nonlinear partial differential equations and their applications, Pitman, pp. 13-51, 1994. [6] B.I. Wohlmuth, “Discretization methods and iterative solvers based on domain decomposition, Lecture Notes in Computational Science and Engineering, vol. 17, Springer, 2001. [7] Y. Maday, F. Rapetti, B. I. Wohlmuth, “Mortar element coupling between global scalar and local vector potentials to solve eddy current problems”, dans “Numerical mathematics and advanced applications”, Enumath 2001 proc., Brezzi F. et al. eds., Springer-Verlag Italy (Milan) pp. 847–865, 2003. [8] B. Flemisch, Y. Maday, F. Rapetti, B. I. Wohlmuth, “Scalar and vector potentials’ coupling on nonmatching grids for the simulation of an electromagnetic brake”, COMPEL (Int. J. for Comp. and Math. in Electric and Electronic Eng.), vol. 24, No. 3, pp. 1061-1070, 2005. Joint work with F. Rapetti, L. Santandrea, G. Krebs, and Y. Le Bihan.

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Konstantinos Chrysafinos National Technical University of Athens, Greece, GR Discontinuous time-stepping schemes for the velocity tracking problem under low regularity assumptions Minisymposium Session FEPD: Monday, 12:10 - 12:40, CO017 The velocity tracking problem for the evolutionary Stokes and Navier-Stokes flows is examined. The scope of the optimal control problem under consideration is to match the velocity vector field to a given target, using distributed controls. In this talk, we present some results related to the analysis of suitable fully-discrete schemes under low regularity assumptions on the given data of the prescribed flows. The schemes are based a discontinuous (in time) Galerkin approach combined with standard conformning (in space) finite elements. Error estimates for the state, adjoint and control variables are presented in case of the evolutionary Stokes flows. In addition, stability estimates and related convergence results are discussed for the tracking problem related to Navier-Stokes flows under low regularity assumptions.

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Ramon Codina Universitat Politècnica de Catalunya, ES Analysis of an unconditionally convergent stabilized finite element formulation for incompressible magnetohydrodynamics Minisymposium Session MMHD: Thursday, 10:30 - 11:00, CO017 In this work, we analyze a numerical formulation for the approximation of the incompressible visco-resistive magnetohydrodynamics (MHD) system, which models incompressible viscous and electrically conducting fluids under electromagnetic fields. Many conforming numerical approximations to this problem have been proposed so far. There are different equivalent formulations of the continuous magnetic sub-problem, namely saddle-point and (weighted) exact penalty formulations. The first one leads to a double-saddle-point formulation for the MHD system. It is well-known that saddle-point formulations require to choose particular mixed FE spaces satisfying discrete versions of the so-called inf-sup conditions. Instead, a weighted exact penalty formulation allows to simplify implementation issues but introduces a new complication, the definition of the weight function. Alternative formulations have been proposed for a regularized version of the system, based on an exact penalty formulation. These methods must be used with caution, since they converge to spurious solutions when the exact magnetic field is not smooth. Non-conforming approximations of discontinuous Galerkin type have also been designed. They have good numerical properties, but the increase in CPU cost (degrees of freedom) of these formulations (with respect to conforming formulations) is severe for realistic large-scale applications. Since the resistive MHD system loses coercivity as the Reynolds and magnetic Reynolds numbers increase, i.e. convection-type terms become dominant, the previous formulations are unstable unless the mesh size is sufficiently refined, which is impractical. In order to treat the problems described, some stabilized FE formulations have been proposed for resistive MHD. These formulations are appealing in terms of implementation issues, since arbitrary order Lagrangian FE spaces can be used for all the unknowns and include convection-type stabilization. However, these formulations are based on the regularized functional setting of the problem, and so, restricted to smooth or convex domains. They are accurate for regular magnetic solutions but tend to spurious (unphysical) solutions otherwise. A further improvement is the formulation we propose, which always converges to the exact (physical) solution, even when it is singular. In this work, we carry out a numerical analysis of this formulation in order to prove stability and unconditional convergence in the correct norms while keeping optimal a priori error estimates for smooth solutions. We first describe the MHD problem of interest and then recall the stabilized FE formulation. We will then present a detailed stability and convergence analysis for the stationary and linearized problem. The possible extension of these results to nonlinear problem will also be discussed. Joint work with Santiago Badia, and Ramon Planas.

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Albert Cohen Université Pierre et Marie Curie, FR Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs Minisymposium Session UQPD: Wednesday, 10:30 - 11:30, CO1 The numerical approximation of parametric partial differential equations D(u, y) = 0 is a computational challenge when the dimension d of of the parameter vector y is large, due to the so-called curse of dimensionality. It was recently shown that, for a certain class of elliptic PDEs with diffusion coefficients depending on the parameters in an affine manner, there exists polynomial approximations to the solution map y 7→ u(y) with an algebraic convergence rate that is immune to the growth in the parametric dimension d, in the sense that it holds in the case d = ∞. This analysis is however heavily tied to the linear nature of the considered diffusion PDE and to the affine parameter dependence of the operator. The present talk proposes a general strategy in order to establish similar results for parametric PDEs that do not necessarily fall in this category. Our approach is based on building an analytic extension z 7→ u(z) of the solution map on certain tensor product of ellipses in the complex domain, and using this extension to estimate the Legendre coefficients of u. The varying radii of the ellipses in each coordinate zj reflect the anisotropy of the solution map with respect to the corresponding parametric variables yj . This allows us to derive algebraic convergence rates for tensorized Legendre expansions in the case d = ∞. We also show that such rates are preserved when using certain interpolation procedures, which is an instance of a non-intrusive method. As examples of parametric PDE’s that are covered by this approach, we consider (i) diffusion equations with uniformly elliptic coefficients that depend on y in a non-affine manner, (ii) nonlinear monotone elliptic PDE’s with coefficients parametrized by y, and (iii) elliptic equations set on a domain that is parametrized by the vector y. While for the first example (i) the validity of the analytic extension follows by straightforward arguments, we give general strategies that allows us to derive it in a simple abstract way for examples (ii) and (iii), in particular based on the holomorphic version of the implicit function theorem in Banach spaces. We expect that this approach can be applied to a large variety of parametric PDEs, showing that the curse of dimensionality can be overcome under mild assumptions. Joint work with Abdellah Chkifa, and Christoph Schwab.

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Claudia Colciago EPFL, CH Reduced Order Models for Fluid-Structure Interaction Problems in Haemodynamics Minisymposium Session ROMY: Thursday, 15:00 - 15:30, CO016 The modelling of the haemodynamics in an arterial vessel requires the coupling of the equation of the blood flow and the one for the vessel wall movement through suitable conditions. This type of problems has high computational costs in terms of time and memory storage. Our aim is to provide a reduced order Fluid-Structure Interaction model (FSI-ROM) which allows to speed up the computations and, at the same time, to lower the memory storage costs. The FSI-ROM is based on two levels of reduction: we firstly perform a model reduction and then a discretization one. In many cases, we are interested in the blood flow dynamics in compliant vessels, whereas the displacement of the domain is small and the structure dynamics is less relevant. In these situations, techniques to reduce the complexity of the model can be used. In particular we focus our attention on two sources of complexity that arise in a FSI problem. The first one is represented by the time dependent fluid domain. A possible solution to overcome this difficulty is using transpiration condition for the fluid model as surrogate for the wall displacement, thus allowing to keep the domain fixed [2]. The second source of complexity is the coupling between two different physical systems. We choose to model the arterial wall as a thin membrane under specific assumptions and, using suitable coupling conditions, we express the structural equation in terms of the blood velocity. This strategy allows to integrate the dynamics of the vessel motion in the fluid equations [1] . The resulting model is a Navier-Stokes system in a fixed domain where the embedding of structural equation yields specific stiffness integrals on the boundaries of the fluid domain. The second level of reduction is achieved through the implementation of a Proper Orthogonal Decomposition (POD) technique. Using a Galerkin projection, the finite element discretization space is reduced to a low dimensional space that can be solved in real time [5, 4]. We apply the FSI-ROM on a realistic case of a femoropopliteal bypass where patient-specific boundary conditions are imposed at the inlet and outlet sections. We first compare the finite element solution of the reduced FSI model with the one of a 3D-3D FSI model [3]. The reference finite element solution has 106 degrees of freedom, while the thanks to the POD we end up with a reduced system of about 30 degrees of freedom.

References [1] C. A. Figueroa et al., Comput. Methods in Applied Mechanics and Engineering 195 (2006). [2] S. Deparis et al., ESAIM:Mathematical Modelling and Numerical Analysis 37 (2003). [3] C. M. Colciago et al., submitted, 33 (2011). [4] A.-L. Gerner et al., arXiv:1208.5010, (2012).

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[5] L. Grinberg et al., Annals of Biomedical Engineering, 37 (2009). Joint work with Simone Deparis, and Alfio Quarteroni.

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Anais Crestetto University Paul Sabatier Toulouse 3, FR Coupling of an Asymptotic-Preserving scheme with the Limit model for highly anisotropic-elliptic problems Minisymposium Session ASHO: Wednesday, 12:00 - 12:30, CO2 We are interested in the numerical simulation of 2D highly anisotropic-elliptic problems, like the ones encountered in strongly magnetized ionospheric plasmas. The anisotropy is parameterized by ε and leads to a multiscale problem, called Singular-Perturbation (SP) problem. In previous works of Degond et al.1 and Besse et al.2 , an Asymptotic-Preserving (AP) reformulation was used in order to obtain an accurate scheme, whatever the value of ε. This formulation is based on the decomposition of the unknown u into its mean part along the anistropy direction (corresponding to the z-axis) and a perturbation. For the applications we consider, ε  1 in a large range of the computational domain. In this part of the domain, we can assume that the solution does not depend on the z-coordinate. That is why we propose a strategy for the spatial coupling of the AP reformulation and its limit (L) model. The obtained AP-L scheme, based on finite elements discretization, is practically available and accurate in the whole domain. Moreover, its cost is reduced in the region where ε  1, which increases the performance of the scheme. We will present some numerical results, for which ε depends on z and presents a high gradient. Our coupling will be compared (accuracy, cost) to the AP reformulation and the SP problem. Joint work with Fabrice Deluzet, Jacek Narski, and Claudia Negulescu.

1 P. Degond, F. Deluzet, C. Negulescu, An Asymptotic Preserving scheme for strongly anisotropic elliptic problem, SIAM-MMS (Multiscale Modeling and Simulation) (2010). 2 C. Besse, F. Deluzet, C. Negulescu, C. Yang, Efficient numerical methods for strongly anisotropic elliptic equations, Journal of Scientific Computing (2012).

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Nicolas Crouseilles inria, FR Asymptotic preserving schemes for highly oscillatory Vlasov-Poisson equations Minisymposium Session ASHO: Wednesday, 11:00 - 11:30, CO2 This work is devoted to the numerical simulation of a Vlasov-Poisson model describing a charged particle beam under the action of a rapidly oscillating external field. We construct an Asymptotic Preserving numerical scheme for this kinetic equation in the highly oscillatory limit. This scheme enables to simulate the problem without using any time step refinement technique. Moreover, since our numerical method is not based on the derivation of the simulation of asymptotic models, it works in the regime where the solution does not oscillate rapidly, and in the highly oscillatory regime as well. Our method is based on a "two scale" reformulation of the initial equation, with the introduction of an additional periodic variable. Joint work with mohammed lemou, and florian méhats.

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Raffaele D’Ambrosio Department of Mathematics, University of Salerno, IT Numerical solution of Hamiltonian systems by multi-value methods Contributed Session CT1.4: Monday, 17:30 - 18:00, CO015 The recent literature regarding geometric numerical integration of ordinary differential equations has given special emphasis to the employ of multi-value methods: in particular, some efforts have been addressed to the construction of general linear methods with the aim of achieving an excellent long-time behavior for the integration of Hamiltonian systems. In this talk we present the analysis and derivation of G-symplectic and symmetric multi-value methods with zero growth parameter for the parasitic components and test their effectiveness on a selection of Hamiltonian problems. A backward error analysis is also presented, which permits to get sharp estimates for the parasitic solution components and for the error in the Hamiltonian. For carefully constructed methods (symmetric and zero growth parameters) the error in the parasitic components typically grows like hp+4 exp(h2 Lt), where p is the order of the method, and L depends on the problem and on the coefficients of the method. This is confirmed by numerical experiments.

References [1] J. C. Butcher, R. D’Ambrosio, Partitioned general linear methods for separable Hamiltonian problems, submitted. [2] R. D’Ambrosio, G. De Martino, B. Paternoster, Numerical integration of Hamiltonian problems by G-symplectic integrators, submitted. [3] R. D’Ambrosio, B. Paternoster, Long-term stability of multi-value methods for ordinary differential equations, submitted.

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Hogenrich Damanik TU Dortmund, DE A multigrid LCR-FEM solver for viscoelastic fluids with application to problems with free surface Minisymposium Session MANT: Tuesday, 12:00 - 12:30, CO017 In this talk, we shall discuss discretization and solution approaches for viscoelastic fluid flow problems. We consider viscoelastic models which are based on upperconvective differential forms. The proposed numerical method is a combination between good remodeling of the upper-convected viscoelastic models, strong discretization techniques and efficient solvers. More specific, we use the Log-Conformation Reformulation (LCR) to remodel the upper-convected viscoelastic materials, which is able to capture high stress gradients with exponential growth at critical Weissenberg numbers. The LCR technique separates the velocity gradient into translational and rotational matrices. This allows to take the logarithmic of the conformation stress in the original equation. Thus, the LCR technique preserves the positivity of the conformation tensor inside, and the conformation tensor can be easily obtained by taking the exponential of the LCR components. We apply high order biquadratic conforming FEM (Q2) to discretize the viscoelastic models given in their LCR form. This Q2 element with discontinuous pressure element (P1) is, in our experience, one of the best choices for velocity-pressure spaces and satisfies the LBB condition. Unfortunately, we are faced with the same stability condition also for the velocity-stress approximation. As a remedy, we penalize the discrete system with a jump over the edges which is known as edge-oriented FEM. This provides stable discrete systems. Next, we solve the total discrete systems in a monolithic coupled way. Here, all unknowns are solved simultaneously with Newton iteration. Inside one Newton step, a geometric multigrid solver takes care of the linearized discrete systems. In our case, the multigrid solver uses a full Vanka-smoother together with the Q2P1 canonical grid transfer routines. The local system inside the smoother utilizes a direct solver for solving small matrices. This guarantees the fully coupled characteristic of viscoelastic problems. Finally, we present mesh convergence studies for a well-known flow around cylinder benchmark configuration to validate our methodology, and present interesting viscoelastic flow applications with multiphysics character, including multiphase flow problems and film casting, see figures. Keywords: Viscoelastic, LCR, FEM, Multigrid, Film casting, rising bubble.

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Figure 1: Film casting configuration

Figure 2: Rising bubbles surrounded by different fluids Joint work with Dr. Otto Mierka, Dr. Abderrahim Ouazzi, and Prof. Dr. Stefan Turek.

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Alexander Danilov Institute of Numerical Mathematics of Russian Academy of Sciences, RU Numerical simulation of large-scale hydrodynamic events Minisymposium Session FREE: Monday, 16:00 - 16:30, CO2 We present basic components of the computational technology for the simulation of complex hydrodynamic events, such as a break of a dam, a wave pileup and run-up, a landlside, or a mud flow. The mathematical model is based on the Navier-Stokes equations and the transport equation for level set function. The relation between the stress tensor and the rate of strain tensor may be nonlinear which results in non-Newtonian flows. The numerical method uses adaptively refined octree meshes and the finite volume discretization of the differential equations. The efficiency of the technology is illustrated by simulations of hydrodynamic events in areas with actual 3D topology. Yu.Vassilevski, K.Nikitin, M.Olshanskii, K.Terekhov. CFD technology for 3D simulation of large-scale hydrodynamic events and disasters. Rus. J. Numer. Anal. Math. Model. 27(4), (2012), 399-412. Joint work with K.Nikitin, and K.Terekhov.

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Mark Davenport Georgia Institute of Technology, US One-Bit Matrix Completion Minisymposium Session ACDA: Monday, 14:30 - 15:00, CO122 In this talk I will describe a theory of matrix completion for the extreme case of noisy 1-bit observations. Instead of observing a subset of the real-valued entries of a matrix M , we obtain a small number of binary (1-bit) measurements generated according to a probability distribution determined by the real-valued entries of M . The central question I will discuss is whether or not it is possible to obtain an accurate estimate of M from this data. In general this would seem impossible, but we show that the maximum likelihood estimate under a suitable constraint returns an accurate estimate of M under certain natural conditions. If the log-likelihood is a concave function (e.g., the logistic or probit observation models), then we can obtain this estimate by optimizing a convex program. Joint work with Yaniv Plan, Ewout van den Berg, and Mary Wootters.

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Raúl de la Cruz Barcelona Supercomputing Center, ES Unveiling WARIS code, a parallel and multi-purpose FDM framework Minisymposium Session PARA: Monday, 15:00 - 15:30, CO016 WARIS is an in-house multi-purpose framework focused on solving scientific problems using Finite Difference Methods (FDM) as numerical scheme. Its framework was designed from scratch to solve in a parallel and efficient way Earth Science and Computational Fluid Dynamic problems among a wide variety of architectures. Structured meshes are employed to represent the problem domains, which are better suited to be optimized in accelerator-based architectures. To succeed in such challenge, WARIS framework was initially designed to be modular in order to ease development cycles, portability, reusability and future extensions of the framework. Our framework is composed of two primary systems, the Physical Simulator Kernel (PSK) and the Workflow Manager (WM). The PSK system is in charge of providing the spatial and temporal discretization scheme code for the simulated physics. Its aim is also to provide a base for the specialization of physical problems (i.e. Advection-Diffusion-Reaction, Navier-Stokes governing equations) on any forthcoming architecture (i.e. general purpose processors, GPGPUs, Intel Xeon Phi). So, this module is basically a template that provides the appropriate framework for implementing a specific simulator. As a consequence, flexibility in design must be attained to let the specialization accommodate any kind of physics by reusing as much code as possible. This approach will minimize the development cycle by reducing the code size and the debugging efforts. In order to provide such a system, the PSK is divided in two components: the host and the device. The former is the part of the framework responsible for the general issues about any simulator kernel, such as domain decomposition, neighbor communications and I/O operations (PSK framework subsystem). The latter is composed of a set of specializations that are used to configure the framework in order to have a functional simulator (PSK Specialization subsystem). The specialization framework may depend on many aspects, such as the physical problem to simulate, the hardware platform target and the numerical method (explicit, implicit, low-order or high-order accuracy schemes). Host and device components are interrelated through the computational architecture model of the PSK. Figure 1 shows the Computational Node (CN) concept considered in such model and their relation. A CN is composed of host and device components (computational elements), which are attached to a common address space (CAS) memory. The host component may be assigned to a general purpose processor that runs the PSK framework subsystem. On the other hand, the device component can be assigned either to a general purpose processor or an accelerator device (i.e. GPGPU, Intel Xeon Phi, FPGA or any other specific processor) running the specialized functions of the PSK Specialization subsystem. The communication among all the computational elements in the same CN are conducted using the CAS memory, whereas the MPI standard is used to communicate several CNs along the network. Besides, the WM system is in charge of providing a framework that allows to process several physical problems in parallel. This framework includes all the necessary components to provide a distributed application in the sense that is capable of process several independent problems using different computational nodes. This

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approach enables to tackle a statistical study or an optimal parameter search in a massively parallel way for a given physical problem. Such designed system architecture implements a Master-Worker pattern. The Master node manages, schedules and commands the Workers’ sets, which are allocated in several CN, by assigning them new tasks and including also fault-tolerance features. Then, each set of Workers run a specific PSK framework with a given input parameter configuration for a physical problem. The Workers’ executions can include a kernel computation or a data post-process step. Finally, at the end of the execution, the Master collect the whole information left by Workers as the result of the computation. To summarize, WARIS framework has shown appealing capabilities by providing successful support for scientific problems using FDM. In the foreseeable future, as the amount of computational resources will increase, more sophisticated physics may be simulated. Furthermore, it provides support for a wide-range of hardware platforms. Therefore, as the computational race keep the hardware changing everyday, support for specific platforms that will give the best performance results will be supplied for the different simulated physics.

Figure 1: Architecture Model that interrelates the components of Computational Nodes. Joint work with Mauricio Hanzich, Arnau Folch, Guillaume Houzeaux, and José Maria Cela.

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Kristian Debrabant University of Southern Denmark, Department of Mathematics and Computer Science, DK Monotone approximations for Hamilton-Jacobi-Bellman equations Minisymposium Session NMFN: Monday, 11:10 - 11:40, CO2 In this talk we consider the numerical solution of diffusion equations of HamiltonJacobi-Bellman type n o ut − inf Lα [u](t, x) + cα (t, x)u + f α (t, x) = 0 in (0, T ] × RN , α∈A

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where Lα [u](t, x) = tr[aα (t, x)D2 u(t, x)] + bα (t, x)Du(t, x). The solution of such problems can be interpreted as value function of a stochastic control problem. We introduce a class of monotone approximation schemes relying on monotone interpolation. Besides providing a unifying framework for several known first order accurate schemes, the presented class of schemes includes new first and higher order approximation methods. Some stability and convergence results are given, as well as numerical examples.

References [1] Kristian Debrabant and Espen Robstad Jakobsen. Semi-Lagrangian schemes for linear and fully non-linear diffusion equations. Math. Comp., 82(283):1433– 1462, 2013. Joint work with Espen Robstad Jakobsen.

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Luca Dede CMCS-MATHICSE-Ecole Polytechnique Federale de Lausanne, CH Numerical approximation of Partial Differential Equations on surfaces by Isogeometric Analysis Minisymposium Session GEOP: Wednesday, 12:00 - 12:30, CO122 We consider the numerical approximation of Partial Differential Equations (PDEs) on surfaces by means of Isogeometric Analysis, an approximation method based on the isoparametric concept for which the basis functions used to represent the computational domain are then used for the approximation of the unknown solutions of the PDEs (Hughes, Cottrell, Bazilevs, Comput. Methods Appl. Mech. Eng. 2005). The method facilitates the encapsulation of the exact geometrical description of lower dimensional manifolds in the analysis when these are represented by B–splines or NURBS. In particular, since NURBS allow to represent a wide range of geometries, including conic sections, we consider the approximation of the PDEs on surfaces by means of NURBS–based Isogeometric Analysis. In this work, we solve linear, nonlinear, time dependent, and geometric PDEs involving the second order Laplace-Beltrami and high-order operators on surfaces. Moreover, we propose a priori error estimates under h–refinement which confirm the accuracy properties of the NURBS–based Isogeometric Analysis. Joint work with Alfio Quarteroni.

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Ekaterina Dementyeva Institute of Computational Modeling SB RAS, RU The Inverse Problem of a Boundary Function Recovery by Observation Data for the Shallow Water Model Contributed Session CT3.1: Thursday, 17:00 - 17:30, CO1 Shallow water models adequately describe a large class of natural phenomena such as large-scale free surface waves arising in seas and oceans, tsunamis, flood currents, surface and channel run-offs, gravitation oscillation of the ocean surface. In this paper the problem of long-wave propagation in a large water area is considered. The mathematical model of the shallow water equations on a spherical surface is used. A boundary of a numerical domain consists of a coastline (“hard”) part and an open-water (“liquid”) part. In general case the influence of the ocean through an open-water part of a boundary is uncertain. Therefore at “liquid” part of a boundary the boundary conditions contain a special unknown function d which should be determined together with velocity and free surface level. Thus, the ill-posed inverse problem of reconstruction of the boundary function is considered. To solve this problem we use additional information, e.g. observation of free surface level on a part of a “liquid” boundary. We investigate three different approaches to regularization of our ill-posed problem using adjoint operators and optimal control theory. The advantages and disadvantages of each regularizer, which uses norm in a search space of d, are researched. As a result, the numerical solving of the inverse problem is a iterative process on alternate solutions of direct and adjoint equations and d refinement equation. The differential problems are reduced to algebraic ones by the finite element method. Numerical experiments of data recovery are carried out on the Sea of Okhotsk region. We use the model observation data of different smoothness — smooth, with white noise (Fig. 1 a), with gaps (Fig. 2 a). Some results of numerical recovery of the unknown boundary function d are represented on Fig. 1, 2. Parallel software using MPI and OpenMP technologies is developed. Considerable attention is paid to the description of effective parallel numerical algorithms based on the MPI. The work was supported by Russian Foundation of Fundamental Researches (grant 11-01-00224-a) and by SB RAS (Project 130).

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Dennis den Ouden Materials innovation institute, NL Application of the level-set method to a multi-component Stefan problem Contributed Session CT2.9: Tuesday, 15:00 - 15:30, CO124 This study focuses on the dissolution and growth of small particles within a matrix phase. The interface between the particle and the matrix phase can have a non-smooth shape. The dissolution or growth of the particle is assumed to be affected by concentration gradients of several chemical elements within the matrix phase at the particle/matrix boundary and by an interface reaction, resulting into a mixed-mode formulation. The mathematical formulation of the dissolution is described by a Stefan problem, in which the location of the interface changes in time. At the interface two conditions are present for each chemical element, one governs the mass balance at the interface and results into an equation of motion, and another condition describes the reaction at the interface which results into a Robin boundary condition. Within the matrix phase we assume that the standard diffusion equation applies to the concentration of the considered chemical elements. The formulated Stefan problem is solved using a level-set method by introducing a time-dependent signed-distance function for which the zero-level contour describes the particle/matrix interface. The evolution of this signed distance function is described by a standard convection equation in which the convection speed is derived from the interface velocity. To ensure the signed-distance property of the level-set function we employ a novel pde-free technique for reinitialization of the level-set function. The equilibrium concentrations of all chemical elements are influenced by each other and the solutions to the moving boundary problem using local equilibria. This leads to a highly non-linear root-finding problem which is solved using an adapted form of Broyden’s Method, which for our problem minimises the number of function calls. Furthermore are the number of function calls in the first iteration independent of the mesh size, which are used in the estimation of the Jacobian matrix. Both the convection equation for the signed-distance function and the diffusion equations are discretised by the use of finite-element techniques. The convection equations for evolution of the signed-distance function are solved on a pre-defined grid using a Streamline Upwind Petrov Galerkin finite-element method. The diffusion equations are solved on a part of the pre-defined grid, which is determined by the negative value of the signed-distance function. The convection equation for the computation of the convection speed is solved on the pre-defined grid which is enriched with extra nodes, which are located on the zero-level of the signed-distance function. Simulations with the implemented methods for the dissolution and growth of various particle shapes show that the methods employed in this study correctly capture the evolution of the particle/matrix interface, especially for non-smooth interfaces and breaking and merging of particles. In the early stages of dissolution and growth the results show that our numerical methods are in good agreement with analytical similarity solutions, where at later stages of growth and dissolution physical equilibrium is attained. We have also seen that our solutions show mass conservation when we let the time-step and mesh-coarseness tend to zero. The most important innovation is the extension of the existing method [1] to the simulation of growth

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and dissolution of particles in multi-component alloys.

References [1] D. Ouden den, A. Segal, F.J. Vermolen, L. Zhao, C. Vuik & J. Sietsma, Application of the level-set method to a mixed-mode driven Stefan problem in 2D and 3D, Computing, (2012), DOI:10.1007/s00607-012-0247-3 Joint work with D. den Ouden, A. Segal, F.J. Vermolen, L. Zhao, C. Vuik, and J. Sietsma.

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Simone Deparis CMCS - MATHICSE - EPFL, CH On the continuity of flow rates, stresses and total stresses in geometrical multiscale cardiovascular models Minisymposium Session SMAP: Monday, 11:40 - 12:10, CO015 After a short revision of the geometric multiscale modeling for the cardiovscular system, we present an algorithm for the implicit coupling of average or mean quantities over the interface between models. The more common ones are the flow rate and the average stress. Recently it has been pointed out the the latter shall be replaced by the average total stress. Indeed, conservation of mean total normal stress in the coupling of heterogeneous models is mandatory to satisfy energetic consistency between them. Existing methodologies are based on modifications of the Navier–Stokes variational formulation, which are undesired when dealing with fluid-structure interaction or black box codes. The presented methodology makes possible to couple one-dimensional and three-dimensional fluid-structure interaction models, enforcing the continuity of mean total normal stress while just imposing flow rate data or even the classical Neumann boundary data to the models. This is accomplished by modifying an existing iterative algorithm, which is also able to account for the continuity of the vessel area, whenever required. Comparisons are performed to assess differences in the convergence properties of the algorithms when considering the continuity of mean normal stress and the continuity of mean total normal stress for a wide range of flow regimes. Finally, examples in the physiological regime are shown to evaluate the importance, or not, of considering the continuity of mean total normal stress in hemodynamics simulations.

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Bruno Despres UPMC-LJLL, FR Uniform convergence of Asymptotic Preserving schemes on general meshes Minisymposium Session ASHO: Wednesday, 10:30 - 11:00, CO2 Diffusion Aymptotic Preserving schemes on multiD general meshes display many difficulties, some which are technical and some which are fundamental. The salient one is the structure of the mesh which generates a distortion of the A.P. properties of 1D scheme: in practice the multiD scheme can become non A.P. This topic has been investigated recently in a joint work with Buet and Franck. In this context uniform estimates of convergence are mandatory to assess the A.P. feature: uniform means uniform with respect to the stiffness parameter and the mesh size. I will explain the problem for diffusion A.P. schemes for the hyperbolic heat equation in 2D, and show a new error estimate. This estimate is finally incorporated in the standard strategy of proof. Joint work with C.Buet (CEA), E. Franck (Max-Planck), T. LEroy (CEA, and PhD).

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Daniele Di Pietro Université de Montpellier 2, FR A generalization of the Crouzeix–Raviart and Raviart–Thomas spaces with applications in subsoil modeling Contributed Session CT3.4: Thursday, 17:30 - 18:00, CO015 In the context of industrial simulators, lowest-order methods capable of handling general polygonal or polyhedral meshes have received an increasing attention over the last few years. The use of general elements may ease the discretization of complex domains, allow the use of nonconforming h-adaptivity, and is mandatory whenever the user cannot adapt the mesh to the needs of their numerical scheme. This is the case, e.g., in computational geosciences, where the discretization of the subsoil aims at integrating data from the seismic analysis. As a result, fairly general meshes can be encountered, possibly featuring nonmatching interfaces or degenerated elements in eroded layers. Polyhedral elements may also be used in near wellbore regions to exploit (qualitative) a priori knowledge of the solution. Among the methods that have appeared in recent years, we recall the Mimetic Finite Difference method of [Brezzi et al.(2005)], the Mixed/Hybrid Finite Volume (HFV) methods of [Eymard et al. (2010)], and the cell centered Galerkin (ccG) method of [Di Pietro (2012)]. The main result of the present work is to show how ideas from HFV and ccG methods can be combined to construct a discrete space of piecewise affine functions which extends two key properties of the classical Crouzeix–Raviart space to general meshes, namely, 1. the continuity of mean values at interfaces; 2. the existence of an interpolator which preserves the mean value of the gradient inside each element and ensures optimal approximation properties. For H set of positive meshsizes having 0 as its unique accumulation point, let (Kh )h∈H denote an admissible mesh family in the sense of Di Pietro (2012). In the spirit of Cell Centered Galerkin (ccG) methods, the discrete space is constructed in three steps: 1. we fix the vector space Vh of face- and cell-centered degrees of freedom (DOFs) on Kh ; 2. we define a discrete gradient reconstruction operator Gh acting on Vh . The reconstructed gradient results from the sum of two terms: a consistent part depending on face unknowns only and a subgrid correction involving both face- and cell-centered DOFs. The continuity of mean values at interfaces is ensured by finely tuning the latter contribution; 3. we define an affine reconstruction operator Rh acting on Vh which maps every vector of DOFs on a broken affine function obtained by an affine perturbation of face unknowns based on the discrete gradient. The discrete space is then defined as CRg(Kh ) = Rh (Vh ). An important point is that all the relevant geometric information is computed on the mesh Kh , which is therefore the only one that needs to be described and 100

manipulated by the end-user. Similar ideas can be used to construct a H(div; Ω)conforming discrete space on general meshes which mimics two key properties of the lowest-order Raviart–Thomas space on matching simplicial meshes, namely the (full) continuity of normal values at interfaces and the optimal approximation of vector-valued fields. The generalized Crouzeix–Raviart space is used to construct a locking-free primal discretization of the linear elasticity equations inspired by the method of [Brenner and Sung (1992)]. In the context of linear elasticity, locking refers to the loss of accuracy of the lowestorder Lagrange finite elements when dealing with quasi-incompressible materials for which Poisson’s ratio tends to 1/2. Numerical examples showing the robustness of the proposed method with respect to numerical locking are provided based on different mesh sequences including highly distorted and general polygonal ones.

References [Brenner and Sung (1992)] S. C. Brenner and L.-Y. Sung. Linear finite element methods for planar linear elasticity. Math. Comp., 59(200):321–338, 1992. [Brezzi et al.(2005)] F. Brezzi, K. Lipnikov, and M. Shashkov. Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes. SIAM J. Numer. Anal., 43(5):1872–1896, 2005. [Di Pietro (2012)] D. A. Di Pietro. Cell centered Galerkin methods for diffusive problems. M2AN Math. Model. Numer. Anal., 46(1):111–144, 2012. [Eymard et al. (2010)] R. Eymard, T. Gallouët, and R. Herbin. Discretization of heterogeneous and anisotropic diffusion problems on general nonconforming meshes. SUSHI: a scheme using stabilization and hybrid interfaces. IMA J. Numer. Anal., 30(4):1009–1043, 2010. Joint work with Simon Lemaire.

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Gabriel Dimitriu "Grigore T. Popa" University of Medicine and Pharmacy, RO POD-DEIM Approach on Dimension Reduction of a Multi-Species Host-Parasitoid System Contributed Session CT3.5: Thursday, 17:00 - 17:30, CO016 The reduced-order approach is based on projecting the dynamical system onto subspaces consisting of basis elements that contain characteristics of the expected solution. Currently, Proper orthogonal Decomposition (POD) is probably the mostly used and most successful model reduction technique, where the basis functions contain information from the solutions of the dynamical system at pre-specified time-instances, so-called snapshots. Due to a possible linear dependence or almost linear dependence, the snapshots themselves are not appropriate as a basis. Hence a singular value decomposition is carried out and the leading generalized eigenfunctions are chosen as a basis, referred to as the POD basis. Unfortunately, for nonlinear PDEs, the efficiency in solving the reduced-order systems constructed from standard Galerkin projection with any reduced globally supported basis set, including the one from POD, is limited to the linear or bilinear part, both for finite element or finite difference schemes since nonlinear terms still require calculation on the full dimensional model. A considerable reduction in complexity is achieved by DEIM – a discrete variation of Empirical Interpolation Method (EIM), proposed by Barrault, Maday, Nguyen and Patera in: An “empirical interpolation” method: Application to efficient reducedbasis discretization of partial differential equations, C. R. Math. Acad. Sci. Paris, 339 (2004), 667–672. According to this method, the evaluation of the approximate nonlinear term does not require a prolongation of the reduced state variables back to the original high dimensional state approximation required to evaluate the nonlinearity in the POD approximation. In this study we carry out an application of DEIM combined with POD to provide dimension reduction of a model that focuses on the aggregative response of parasitoids to hosts in a coupled multi-species system comprising two parasitoid species, two host species and a chemoattractant. The model defined by a system of five reaction-diffusion-chemotaxis equations was introduced by I.G. Pearce, M.A.J. Chaplain, P.G. Schofield, A.R.A. Anderson and S.F. Hubbard in: Chemotaxisinduced spatio-temporal heterogeneity in multi-species host-parasitoid systems, J. Math. Biol., 55 (2007), 365–388. We show DEIM improves the efficiency of the POD approximation and achieves a complexity reduction of the nonlinear term. Numerical results are presented. Joint work with Ionel Michael Navon, and Razvan Stefanescu.

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Sergey Dolgov Max Planck Institute for Mathematics in the Sciences, DE Alternating minimal energy methods for linear systems in higher dimensions. Part II: implementation hints and application to nonsymmetric systems Contributed Session CT2.8: Tuesday, 15:00 - 15:30, CO123 In this talk we further develop and investigate the rank-adaptive alternating methods for high-dimensional tensor-structured linear systems. The ALS method is reformulated in a recurrent variant, which performs a subsequent linear system reduction, and the basis enrichment is derived in terms of the reduced system. This algorithm appears to be more robust than the method based on a global steepest descent correction, and additional heuristics allow to speedup the computations. Furthermore, the very same method is applied to nonsymmetric systems as well. Though its theoretical justification is based on the FOM method, and is more difficult than in the SPD case, the practical performance is still very satisfactory, which is demonstrated on several examples of the Fokker-Planck and chemical master equations. Joint work with Dmitry Savostyanov.

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Marco Donatelli Department of Science and High Technology - University of Insubria, IT Multigrid preconditioning for nonlinear (degenerate) parabolic equations with application to monument degradation Minisymposium Session CTNL: Wednesday, 11:00 - 11:30, CO015 We consider linear systems of large dimensions, (locally) structured, resulting from the linearization of systems of nonlinear equations obtained by discretizing nonlinear parabolic equations (possibly degenerate) by means of finite differences in space and implicit schemes in time. In the first part, we consider a uniform discretization in space. Using the theory of sequences of (locally) Toeplitz matrices for studying the spectrum of the Jacobian matrix of Newton’s method, we prove the convergence and we derive optimal preconditioners based on multigrid techniques. The numerical tests are conducted on the equation of porous media and on a particular nonlinear parabolic equation that models the sulfation of marble by polluting agents [1,2]. Subsequently, driven by the presence of a boundary layer in the model of sulfation, we extend the previous results to the case of not uniform grids in space, using preconditioners based on algebraic multigrid [3]. References: [1] M. Donatelli, M. Semplice, S. Serra-Capizzano - Analysis of multigrid preconditioning for implicit PDE solvers for degenerate parabolic equazions - SIAM J. Matrix Anal. Appl. 32 (2011) 1125–1148. [2] M. Semplice - Preconditioned implicit solvers for nonlinear PDEs in monument conservation - SIAM J. Sci. Comp. 32 (2010) 3071-3091. [3] M. Donatelli, M. Semplice, S. Serra-Capizzano - AMG preconditioning for nonlinear degenerate parabolic equations on nonuniform grids with application to monument degradation - Appl. Numer. Math., in press. Joint work with M. Semplice, and S. Serra-Capizzano.

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Martin Ehler Helmholtz Zentrum Muenchen, DE Signal reconstruction from magnitude measurements via semidefinite programming Minisymposium Session ACDA: Monday, 11:10 - 11:40, CO122 Inspired by high-dimensional data analysis and multi-spectral imaging, we aim to reconstruct a finite dimensional vector from a set of magnitudes of its subspace components. First, we develop closed formulas for signal reconstruction. Second, we use semi-definite programming and random subspaces to reduce the number of required subspace components. We also address the optimal choice of the subspace dimension. Motivated by applications in physics, we also discuss the reconstruction of a finitedimensional signal from the absolute values of its Fourier coefficients. In many optical experiments the signal magnitude in time is also available. We combine time and frequency magnitude measurements to obtain closed reconstruction formulas. A hybrid scheme of random Fourier and deterministic time measurements are discussed to reduce the number of required frequency magnitudes. Joint work with Christine Bachoc.

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Virginie Ehrlacher Ecole des Ponts Paristech/INRIA, FR Greedy algorithms for high-dimensional eigenvalue problems Minisymposium Session LRTT: Tuesday, 10:30 - 11:00, CO3 In this talk, some new greedy algorithms in order to compute the lowest eigenvalue and an associated eigenvector of a high-dimensional eigenvalue problem will be presented. The principle of these numerical methods consists in expanding a tentative eigenvector associated to this eigenvalue as a sum of so-called tensor product functions and compute each of these tensor product function iteratively as the best possible, in a sense which will be made clear in the talk. The advantage of this family of methods relies in the fact that the resolution of the original high-dimensional problem is replaced with the resolution of several lowdimensional problems, which are more easily implementable. The convergence results we proved for our algorithms will be detailed, along with some convergence rates in finite dimension. Joint work with Eric Cancès, and Tony Lelièvre.

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Virginie Ehrlacher Ecole des Ponts Paristech/INRIA, FR Optimization of a structurally graded microstructured material Minisymposium Session MSMA: Monday, 15:30 - 16:00, CO3 An approach for the optimization of non-periodic microstructured material through the homogenization method will be presented. The central idea, simsilar to the one used by Pantz and Trabelsi [1], consists in modeling the material as a macroscopic deformation of an initially periodic material. Following the path of Bensoussan, Lions and Papanicolaou [2], homogenization fomulas can be derived to obtain the expression of the effective stiffness elasticity tensor in the limit when the size of the microcells composing the material tends to zero. Using reduced-order models obtained via greedy algorithms, the optimization procedure is performed either using the homogenization method. Numerical results obtained on a two-dimensional material will be presented. Joint work with Claude Le Bris, Frederic Legoll, Günter Leugering, Michael Stingl, and Fabian Wein.

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Lukas Einkemmer University of Innsbruck, Austria A discontinuous Galerkin approximation for Vlasov equations Minisymposium Session TIME: Thursday, 11:30 - 12:00, CO015 In astro- and plasma physics the behavior of a collisionless plasma is modeled by the Vlasov equation ∂t f (t, x, v) + v · ∇f (t, x, v) + F · ∇v f (t, x, v) = 0, a kinetic model that in certain applications is also called the collisionless Boltzmann equation. It is posed in a 3 + 3 dimensional phase space, where x denotes the position and v the velocity. The density function f is the sought-after particleprobability distribution, and the (force) term F describes the interaction of the plasma with the electromagnetic field. Discontinuous Galerkin methods have received considerable attention in recent years; they have been used and analyzed for various kinds of applications. In this talk we will consider a discontinuous Galerkin based Strang splitting method for solving the Vlasov equation coupled to an appropriate model of the electromagnetic field. Due to the Strang splitting scheme, the problem is essentially reduced to solving two (Vlasov–Poisson) or three (Vlasov–Maxwell) advection equations per step. High order approximations in space, which are easy to achieve in this context, can provide a significant advantage due to the up to six dimensional phase space employed in such simulations. A rigorous convergence analysis of this Strang splitting algorithm with a discontinuous Galerkin approximation in space can be conducted, for example, for the 1+1 dimensional Vlasov–Poisson equations. It is shown that for f0 ∈ C max{`+1,3} , i.e.
if the initial value is sufficiently regular, the error is of order O τ 2 + h` + h` /τ , where τ is the size of a time step, h is the cell size, and ` the order of the discontinuous Galerkin approximation. It is well known that piecewise constant approximations in velocity space lead to a recurrence phenomenon that is purely numerical in origin. We will present a number of numerical simulations which show that a recurrence-like effect, originating from the finite cell size, is still visible even for higher order approximations. To confirm the stability properties of the method investigated, a numerical simulation of the Molenkamp–Crowley test has been conducted. It is shown that the scheme investigated does not suffer from the instabilities described in [3]. This talk is based on [1, 2].

References [1] L. E INKEMMER AND A. O STERMANN, Convergence analysis of Strang splitting for Vlasov-type equations. Preprint (arXiv:1207.2090), 2012. [2] L. E INKEMMER AND A. O STERMANN, Convergence analysis of a discontinuous Galerkin/Strang splitting approximation for the Vlasov–Poisson equations. Preprint (arXiv:1211.2353), 2012. [3] K.W. M ORTON , A. P RIESTLEY, AND E. S ÜLI, Stability of the Lagrange-Galerkin method with non-exact integration, Modél. Math. Anal. Numér., 22 (1988), pp. 625–653. 108

Joint work with Alexander Ostermann.

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Daniel Elfverson Uppsala University, SE Discontinuous Galerkin method for convection-diffusion-reaction problems Minisymposium Session SMAP: Monday, 12:10 - 12:40, CO015 In this work we study the solution of second order convection-diffusion-reaction problems. The coefficients are heterogeneous and highly varying without any assumptions on scale separation of periodicity. This type of problems arise in many branches of scientific computing and is often impossible to simulate using standard (one scale) methods, since the variations in the coefficients need to be resolved to reach an acceptable tolerance. Instead, we use a different approach with a corrected basis which takes the variations into account without resolving it globally on a single mesh. We are interested in finding a solution u ∈ H01 such that a(u, v) = (A∇u, ∇v)L2 (Ω) + (β · ∇u + γu, v)L2 (Ω) = (f, v)L2 (Ω) := F (v), for all v ∈ H01 . Here, Ω ∈ Rd for d = 2, 3, is a Lipschitz domain with polygonal ∞ d boundary, A ∈ L∞ (Ω, Rd×d sym ) is uniformly elliptic, β ∈ [L (Ω)] and divergence ∞ 2 free, 0 ≤ γ ∈ L (Ω), and f ∈ L (Ω). This is approximated using the discontinuous Galerkin multiscale method [1, 2]. To this end let us first introduce a fine and a coarse shape-regular mesh, Th and TH with mesh functions h < H, and theirs respective discontinuous Galerkin space Vh and VH ⊂ Vh for tetrahedral and quadrilateral elements. Also, let λT,j denote the coarse basis function that spans VH , i.e., VH = span{λT,j | T ∈ TH , j = 1, . . . , r} where r is the number of degrees of freedom on element T . The multiscale splitting is defined by VH := ΠH Vh and V f := (1 − ΠH )Vh , where ΠH is the (orthogonal) L2 -projection onto the coarse space VH . The multiscale method uses a space spanned by corrected basis ms,L L functions VH = span{λT,j − φL T,j | T ∈ TH , j = 1, . . . , r}, where each φT,j is calculated on patches/subgrids and has local support, and L indicates the size of f L the patches. That is for all T ∈ TH and j = 1, . . . , r, we seek φL T,j = V (ωT ) = f {v ∈ V | v|Ω\ωTL = 0} such that ah (φL T,j , v) = ah (λT , v),

for all v ∈ V f (ωTL ),

where the bilinear form ah (·, ·) is associated with the fine mesh Th . The disconms,L tinuous Galerkin multiscale method reads: find ums,L ∈ VH such that, H , v) = F (v), ah (ums,L H

ms,L . for all v ∈ VH

ms,L Note that dim(VH ) = dim(VH ). The following result holds under moderate assumptions on the magnitude of β,

|||u − ums,L ||| ≤ |||u − uh ||| + CkH(f − ΠH f )kL2 H choosing the size of the patches proportional to H log(H −1 ). The constant C is independent of the variation in the coefficients but may depend on the ratio of the their minimum and maximum bounds and uh ∈ Vh is the (one scale) discontinuous Galerkin solution on the fine scale. This result holds independent of the regularity of the solution. For the method to make sense we assume that uh resolves the variation in the coefficients. However, note that uh is not computed in practice. [1] D. Elfverson, G. H. Georgoulis and A. Målqvist, An Adaptive Discontinuous Galerkin Multiscale Method for Elliptic Problems, Submitted for publication, 2011. 110

[2] D. Elfverson, G. H. Georgoulis, A. Målqvist and D. Peterseim, Convergence of a discontinuous Galerkin multiscale method, Submitted for publication, available as preprint arXiv:1211.5524, 2012. Joint work with Axel Målqvist.

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Stefan Engblom Uppsala University, SE Sensitivity estimation and inverse problems in spatial stochastic models of chemical kinetics Contributed Session CT3.2: Thursday, 18:00 - 18:30, CO2 In this talk I will consider computational modeling of diffusion-controlled reactions with applications mainly in molecular cell biology. I will give a brief overview of the modeling involved, in the non-spatial as well as in the fully spatial setting, and I will consider practical means by which perturbations can be propagated through the simulations. This is relevant as experimental data is often not known with a high degree of accuracy, but also because inverse formulations generally relies on being able to effectively and accurately estimate the effects of small perturbations. For this purpose I will present our implementation of an “all events method” and give two concrete examples of its use. Spatial stochastic chemical kinetics In the classical case of non-spatial stochastic modeling of chemical kinetics, the reaction rates are understood as transition intensities in a Markov chain. When spatial considerations are important, space is discretized in voxels. Between voxels diffusion rates become transition intensities in a Markov chain which now takes place in a state space which is much larger. As before reactions take place within each voxel, but scaled appropriately to take into account the voxel volume. Large such stochastic reaction-diffusion models can be simulated by resolving the geometry using two- or three-dimensional unstructured meshes as in our modular software framework URDME (www.urdme.org). Thanks to its flexible structure and well-defined interfaces, new solvers may be developed in an independent manner and connected directly to the underlying layers of the simulation environment. A viable “All Events Method”-implementation Within the URDME framework we have developed a solver for stochastic sensitivity analysis which allows for path-wise control of all discrete events occurring during the simulation of kinetic networks. This allows us to compare single trajectories under arbitrary perturbations of input data and opens up for accurate estimation of model parameters as well as optimizing models under different configurations. Refer to Figure 1 for a very intuitive usage. To demonstrate the use of this method, in the first setup we consider a spatial model of the following enzymatic law, k c·e

C + E −−−→ E,

(1)

where we think of E as an enzyme and C an intermediate complex which matures into a product not explicitly modeled here. The model is completed by adding the laws αC

αE

∅ C,

∅ E,

βC c

βE e

(2)

where a certain part of these rates actually cover transport events. The effects of the perturbation αE → αE (1 − δ) are studied, where δ depends on the position. See Figure 2 for results on this simulation. 112

In a second setup (not detailed here) a simple concept of ‘optimality’ will be defined for a certain biochemical network, and then tentative solutions for the control signal which achieves this optimality will be presented. Here one is particularly interested in the differences between the stochastic regime and the deterministic one.

60 50 40 30 20 10 0 0

50

100

Figure 1: Comparing single trajectories from a birth-death model.

40

30

20

10 −1

0

1

1

0

−1

Figure 2: Spatial stochastic focusing. Here the perturbation depends on the spatial coordinate and is increasing towards the center of the circle. We study here a measure of the effect of increasing the secretion of the enzyme E and discover an almost fourfold increase of the production rate in the critical region where the perturbation is the highest. This is a stochastic nonlinear phenomenon and cannot be explained by linear analysis nor by deterministic approximations.

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Christian Engwer University of Münster, DE Mini-symposium keynote: Bridging software design and performance tuning for parallel numerical codes Minisymposium Session PARA: Monday, 11:10 - 11:40, CO016 Motivation As applications grow in complexity, the need for sustainable development of software for partial differential equations (PDEs) is increasing rapidly. Modern numerical ingredients such as unstructured grids, adaptivity, high-order discretiations and fast and robust multilevel solvers are required to achieve high numerical efficiency, and several physical models must be combined in challenging applications. Reusable software components are crucial to maintain flexibility in this situation, this applies particularly to well-designed interfaces between different components. Numerical and implementational details should ideally be hidden from users of the software, especially application scientists, and at the same time the interfaces must be designed so that efficiency is not lost when different building blocks are combined, e.g., discretisations and solvers. At the same time a dramatic change in the underlying hardware can be observed: The memory and power wall problems are becoming hard limitations, and further performance improvements are only achieved by locality, multiple levels of parallelism, heterogeneity and specialisation. It is no longer feasible to neglect tuning techniques designed for these aspects of the hardware; rather, current increasingly parallel and heterogeneous systems require uniform tuning and code specialisation of all components. Typical workstations and cluster nodes now comprise at least two multicore CPUs capable of executing tens of concurrent threads simultaneously. Accelerator technologies such as GPUs or Xeon Phi are included as coprocessors in workstations and bigger machines, their performance stems from a much more fine-grained execution of hundreds to thousands of hardware threads. Furthermore, carefully arranged data structures and memory access patterns are crucial to extract reasonable performance. Challenges It is important to note that this change in hardware is not just a momentary aspect, but a definite trend that will not be reverted, since it stems from underlying physical principles in electrical engineering, mostly from energy considerations and associated issues like leaking voltage and heat dissipation. Consequently, these changes results in substantial challenges for designers and implementers of numerical software packages: • Modularity, maintainability, reusability and flexibility of software packages must be maintained. • Hardware details must be hidden as much as possible from application scientists, and to a certain degree also from numerical analysts. • Generic implementations with maximum flexibility in mind must be balanced with specialisations for certain hardware architectures. • Careful compromises must be made when choosing the level of specialisation, honouring rather generic trends of hardware architectures (e.g., implications of coarse- and fine-grained parallelism) than utmost performance extraction for one particular processor instance. 114

• Most importantly, the numerical methodology must be revisited to substantially improve their locality, fine-grained parallelism, communication vs. computation ratios, arithmetic intensities etc. (hardware-oriented numerics). Summary of this talk The goal of the mini-symposium is to bridge these gaps, by bringing together experts from all involved areas. The mini-symposium has a very practical focus, preferring algorithmic and implementation aspects over advances in application domains. This introductory talk will summarise the state of the art in terms of hardware and software engineering, survey generic design and implementation techniques, present tips and tricks associated with higher-level abstractions, and set the stage for fruitful discussions based on the contributed talks. Joint work with Dominik Goeddeke.

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Alexandre Ern University Paris-Est, CERMICS, FR Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs Minisymposium Session STOP: Thursday, 14:00 - 14:30, CO1 We consider nonlinear algebraic systems resulting from numerical discretizations of nonlinear partial differential equations of diffusion type. To solve these systems, some iterative nonlinear solver, and, on each step of this solver, some iterative linear solver are used. We derive adaptive stopping criteria for both iterative solvers. Our criteria are based on an a posteriori error estimate which distinguishes the different error components, namely the discretization error, the linearization error, and the algebraic error. We stop the iterations whenever the corresponding error does no longer affect the overall error significantly. Our estimates also yield a guaranteed upper bound on the overall error at each step of the nonlinear and linear solvers. We prove the (local) efficiency and robustness of the estimates with respect to the size of the nonlinearity owing, in particular, to the error measure involving the dual norm of the residual. Our developments hinge on equilibrated flux reconstructions and yield a general framework. We show how to apply this framework to various discretization schemes like finite elements, nonconforming finite elements, discontinuous Galerkin, finite volumes, and mixed finite elements; to different linearizations like fixed point and Newton; and to arbitrary iterative linear solvers. Numerical experiments for the p-Laplacian illustrate the tight overall error control and important computational savings achieved in our approach. More details on the overall approach, analysis, and results can be found in [1].

References [1] A. E RN AND M. V OHRALÍK, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs. SIAM J. Sci. Comput., to appear (2013), HAL Preprint 00681422 v2, 2012. Joint work with Martin Vohralik.

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Antonio Falcó Unversidad CEU Cardenal Herrera, ES Proper Generalized Decomposition for Dynamical Systems Minisymposium Session SMAP: Monday, 14:30 - 15:00, CO015 Tensor-based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces, such as partial differential equations arising from stochastic calculus (e.g. Fokker-Planck equations) or quantum mechanics (e.g. Schrödinger equation), stochastic parametric partial differential equations in uncertainty quantification with functional approaches, and many mechanical or physical models involving extra parameters (for parametric analyses) among others. For such problems, classical approximation methods based on the a priori selection of approximation bases suffer from the so called “curse of dimensionality” associated with the exponential (or factorial) increase in the dimension of approximation spaces. In [1] the authors give a mathematical analysis of a family of progressive and updated Proper Generalized Decompositions for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor Banach space. In this talk we discuss the approach for continuous time dynamical systems. To this end we revise the Dirac-Frenkel variational principle to justify the Proper Generaliced Decomposition in this time–dependent framework.

References [1] A. Falcó and A. Nouy: Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces. Numer. Math. 121 (2012), 503–530.

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Miloslav Feistauer Charles University Prague, Faculty of Mathematics and Physics, CZ Space-time DGFEM for the solution of nonstationary nonlinear convection-diffusion problems and compressible flow Contributed Session CT3.4: Thursday, 18:00 - 18:30, CO015 The paper will be concerned with the numerical solution of nonstationary problems with nonlinear convection and diffusion by the space-time discontinuous Galerkin finite element method (DGFEM) and with applications to the simulation of compressible flow. The first part will be devoted to some theoretical aspects of the space-time DGFEM. The time interval is split into subintervals and on each time level a different space mesh with hanging nodes may be used in general. In the discontinuous Galerkin formulation we use the nonsymmetric, symmetric or incomplete version of the discretization of the diffusion terms and interior and boundary penalty (i.e., NIPG, SIPG or IIPG versions). For the space and time discretization, piecewise polynomial approximations of different degrees p and q, respectively, are used. The question of optimal error estimates will be treated under various assumptions on the boundary conditions and nonlinearities in the convection and diffusion. Special attention will be paid to the question of the stability of the method. It is an important question, because in works [1], [2] and [3], in the case of a general form of the boundary condition (when the boundary data do not behave in time as a polynomial of degree ≤ q), the error estimates were derived under the CFLlike condition τ ≤ Ch. The goal is to prove the stability without this condition. Theoretical results will be demonstrated by numerical experiments. In the second part, the space-time DGFEM will be applied to the solution of the compressible Navier-Stokes equations. Our goal is to develop sufficiently accurate, efficient and robust numerical schemes allowing the solution of compressible flow for a wide range of Reynolds and Mach numbers. The main attention will be paid to the analysis of the low Mach number flows close to incompressible limit.

References [1] M. Feistauer, V. Kučera, K. Najzar and J. Prokopová, Analysis of space-time discontinuous Galerkin method for nonlinear convection-diffusion problems, Numer. Math. 117 (2011), pp. 251–288. [2] J. Česenek, M. Feistauer: Theory of the space-time discontinuous Galerkin method for nonstationary parabolic problems with nonlinear convection and diffusion. SIAM J. Numer. Anal. 30 (No.3) (2012) 1181–1206. [3] M. Vlasák, V. Dolejší, J. Hájek: A priori error estimates of an extrapolated space-time discontinuous Galerkin method for nonlinear convection-diffusion problems. Numer. Methods Partial Differential Eq. 27 (2011), 1456–1482. Joint work with M. Balazsova, M. Hadrava, and A. Kosik.

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Dalia Fishelov Afeka Tel-Aviv Academic College of Engineering, IL Convergence analysis of a high-order compact scheme for time-dependent fourthorder differential equations Contributed Session CT3.8: Thursday, 16:30 - 17:00, CO123 In [1] we established the convergence of a fourth-order compact scheme to the time-independent one-dimensional biharmonic problem  (4) u (x) = f (x), 0 < x < 1, (1) u(0) = 0, u(1) = 0, u0 (0) = 0, u0 (1) = 0. It approximate solution satisfies  δx4 vj = f (xj ) 1 ≤ j ≤ N − 1,   (a) 2 1 1 vx,j−1 + vx,j + vx,j+1 = δx vj , 1 ≤ j ≤ N − 1, (b)  6 3 6  (c) v0 = 0, vN = 0, vx,0 = 0, vx,N = 0. Here, δx4 is the three-point compact operators defined by   12 vx,j+1 − vx,j−1 vj+1 + vj−1 − 2vj δx4 vj = 2 − , 1 ≤ j ≤ N − 1, h 2h h2

(2)

(3)

and vx,j is the Padé approximation of the derivative of v at point xj , 1 2 1 vj+1 − vj−1 vx,j−1 + vx,j + vx,j+1 = , 1 ≤ j ≤ N − 1. 6 3 6 2h

(4)

This scheme invokes values of the unknown function as well as Padé approximations of its first-order derivative. The truncation error of the scheme is of fourthorder at interior points and of first order at near boundary points. Although the truncation error of the discrete biharmonic scheme is of fourth-order at interior point, the truncation error drops to first-order at near-boundary points. Nonetheless, we prove that the scheme retains its fourth-order (optimal) accuracy. We have proved the following theorem. Theorem 1. Let u be the exact solution of (1) and assume that u has continuous derivatives up to order eight on [0, 1]. Let v be the approximation to u, given by the (2). Then, the error e = v − u satisfies max

1≤j≤N −1

|ej | ≤ C(f )h4 .

(5)

where C depends only on f . A number of numerical examples corroborate this effect. We extend our study to time-dependent problems. We present a proof for the convergence of the scheme for the problem ut = −uxxxx and show that the error is bounded by Ch4− for arbitrary  > 0. Then, we consider a more-general time-dependent problem  ut = −uxxxx + b uxx + c ux + d u + f (x, t), 0 < x < 1, (6) u(0) = 0, u(1) = 0, u0 (0) = 0, u0 (1) = 0. 119

It is approximated by the compact scheme  d 4 ˜2 (a) dt vj = −δx vj + b δx v + c vx,j + d vj + f (xj , t), (b) v0 = 0, vN = 0, vx,0 = 0, vx,N = 0, where δ˜x2 vj = 2δx2 vj − δx vx,j = δx2 vj − We prove the following proposition.

h2 4 12 δx vj .

1 ≤ j ≤ N − 1, (7)

Proposition 2. Let u(x, t) be the exact solution of (6) and assume that u has continuous derivatives with respect to x up to order eight on [0, 1] and up to order 1 with respect to t. Let v(t) be the approximation to u, given by (7). The, the error ej (t) = vj (t) − u(xj , t) satisfies max |e(t)|h ≤ C(T )h3.5 ,

0≤t≤T

(8)

where C(T ) depends only on f , g and T . In addition, we have also proved convergence of the compact scheme for the timedependent equation uxxt = uxxxx + buxx + dux + cu + f (x, t)

(9)

on an interval, and by its approximation by the semi-discrete finite-difference scheme d ˜2 δ vj = δx4 vj + bδ˜x2 vj + cvx,j + dvj + f (xj , t). (10) dt x In this case too we prove that the error is bounded by Ch3.5 . Proposition 3. Let u(x, t) be the exact solution of (9) and assume that u has continuous derivatives with respect to x up to order eight on [0, 1] and up to order 1 with respect to t. Let v(t) be the approximation to u, given by (10). The, the error ej (t) = vj (t) − u(xj , t) satisfies max |e(t)|h ≤ max |δx+ e(t)|h ≤ C(T )h3.5 ,

0≤t≤T

0≤t≤T

(11)

where C(T ) depends only on f , g and T . In addition, we study of the eigenvalue problem uxxxx = νuxx . This is related to the stability of the linear time-dependent equation uxxt = νuxxxx . We derive the full set of continuous eigenvalues and eigenfunctions. In addition, the discrete set of eigenvalues and eigenvectors are computed. The latter are displayed graphically and compared with the continuous eigenfunctions and eigenvalues.

References [1] D. Fishelov and M. Ben-Artzi and J-P. Croisille Recent advances in the study of a fourth-order compact scheme for the one-dimensional biharmonic equation, J. Sci. Comput., , Vol. 53, pp. 55–70, (2012). Joint work with M. Ben-Artzi, and J.-P. Croisille.

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Michel Flueck EPFL, CH Domain decomposition for computing ferromagnetic effects Minisymposium Session MMHD: Thursday, 14:30 - 15:00, CO017 We consider a physical model for simulating the screen effect of ferromagnetic steel plates in presence of very strong direct currents. There is no electric current in the plates, and we assume in this model that the plates have no impact on the surrounding currents. First we theoretically study the mathematical model which is expressed for one unknown scalar field defined on the whole tridimensional space. Then we give a Dirichlet-Dirichlet domain decomposition method using Poisson’s formula to solve this problem. Finaly we present a standard finite element approximation of that problem with some numerical results. First we show the academic situation of an electric conductor on one side of a rectangular plate and we study the induction field on the other side, showing a screening effect. Then we turn to an industrial example of an aluminum electrolysis cell which is built in a big steel shell protecting the interior of the cell from surrounding currents feeding that cell. Joint work with J. Rappaz, and A. Janka.

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Melina Freitag University of Bath, GB Computing Jordan blocks in parameter-dependent eigenproblems Minisymposium Session NEIG: Thursday, 14:30 - 15:00, CO2 We introduce a general method for computing a 2-dimensional Jordan block in a parameter-dependent matrix eigenvalue problem. The requirement to compute Jordan blocks arises in a number of physical problems, for example panel flutter problems aerodynamical stability, the stability of electrical power systems, and in quantum mechanics. The algorithm we suggest is based on the Implicit Determinant Method and requires the solution of a small nonlinear system instead of solving a large eigenvalue problem. We provide theory and convergence properties for this method and give numerical results for a number of problems arising in practice. Joint work with Alastair Spence.

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Maxim Frolov Saint-Petersburg State Polytechnical University, RU Reliable a posteriori error estimation for plane problems in Cosserat elasticity Contributed Session CT2.4: Tuesday, 15:30 - 16:00, CO015 Functional type a posteriori error estimates are proposed for approximate solutions to plane problems arising in the Cosserat theory of elasticity. In comparison to the classical elasticity such type of models possesses an advanced spectrum of properties - they can more adequately describe materials with microstructure. The growing interest to generalizations of the classical elasticity theory arose from the beginning of 60s and, nowadays, it is important to provide an efficient procedure of reliable error estimation for numerical solutions. The implemented approach is based on functional grounds (in particular, on the duality theory in the Calculus of Variations). Estimates, which we present for the Cosserat model, are reliable under quite general assumptions and are explicitly applicable not only to approximations possessing the Galerkin orthogonality property. This work is based on previous investigations of the authors (Probl. Mat. Anal., 2011; J. Math. Sci., 2012) in which we only dealt with the case of isotropic media with displacements and independent rotation given on the boundary of a computational domain. For numerical justification of the approach, we use approximation of the nonsymmetric stress tensor that is based on the lowest-order element suggested by D.N. Arnold, D. Boffi, and R.S. Falk (SIAM J. Numer. Anal., 2005). According to recent numerical results for the functional type a posteriori error estimate for linear elasticity problems (S.I. Repin. Radon Series on Computational and Applied Mathematics, 4. Berlin: de Gruyter, 2008), this approach allowing a non-symmetric stress approximation is promising. It provides a significant improvement of the efficiency index in comparison with continuous approximations by the standard finite element procedure based on bilinear shape functions. Joint work with Prof. Sergey I. Repin.

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Petr Furmanek Faculty of Mechanical Engineering, CTU in Prague, CZ Numerical Simulation of Flow Induced Vibrations with Two Degrees of Freedom Contributed Session CT4.5: Friday, 08:50 - 09:20, CO016 Aeroelastic effects that appear in real flows around wings and profiles have usually a huge influence on both the flow field and the profile itself. Possibilities of numerical simulation of these effects (as is buffeting or flutter) in commercial CFD codes are still limited and are often solved by a problem-tailored software. The aim of this contribution is to show and investigate one of such approaches. The so called Modified Causon’s scheme of the 2nd order in space and time (based on TVD form of the classical MacCormack scheme for finite volume method) is enhanced with the use of the ALE method for simulations of unsteady flows, namely flow over the NACA0012 profile. The profile is considered with two degrees of freedom oscillations around a given reference point and vibration along the vertical axis. Motion in given directions is induced by the flow itself and is described by a set of ordinary differential equations. More values of initial velocities are considered and the fluid is simulated both as incompressible and compressible. The final results are compared in-between and with computational data from NASTRAN and in-house codes by P. Svacek and R. Honzatko from the Department of Technical Mathematics, Faculty of Mechanical Engineering, CTU in Prague. Based on the obtained results, the following conclusions can be drawn: there is a significant difference in computational demands of compressible and incompressible model, which allows for computations with much smaller velocities; critical velocities for instability are in both cases in the same range. The future steps intended are implementation of turbulence model and extension into three dimensions. Joint work with Karel Kozel.

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Lucia Gastaldi University of Brescia, IT Fictitious Domain Formulation for Immersed Boundary Method Minisymposium Session ANMF: Tuesday, 10:30 - 11:00, CO1 The aim of this talk is to present a new variational formulation of the Immersed Boundary Method (IBM) which can present improved stability properties. The Immersed Boundary Method was proposed by Peskin (see [7] for a review) in order to simulate the blood dynamics in the heart. It was then applied to several fluidstructure interaction problems. The main feature of the IBM is that the structure is considered as a part of the fluid where additional forces are located. Hence the Navier–Stokes equations have to be solved all over the domain, with a source term which is localized on the structure by means of the Dirac delta function. Moreover, the movement of the structure is imposed by constraining the velocity of the structure to be equal to that of the fluid at the points where the structure is located. The original discretization was based on finite differences which require the approximation of the Dirac delta function. In [2, 3, 4, 1], we have used a variational formulation of the Navier–Stokes equations, which allows to deal with the Dirac delta function in a natural way, so that the finite element method can be applied for the space discretization of the the IBM. On the contrary the position of the structure is determined pointwise. The stability analysis of the space-time discretization of the resulting scheme shows that a priori estimates for the energy can be obtained provided a CFL condition involving the time step and the mesh parameters of the fluid and in the structure domains is satisfied. Here we present a new approach based on a totally variational formulation of the problem. In this approach the equation describing the movement of the structure is seen as a constraint which links the equations for the fluid and for the structure. Therefore we introduce a distributed Lagrange multiplier, so that the final formulation can be interpreted as a Fictitious Domain approach to the fluidstructure interaction problems. We refer for the fictitious domain method to the works [6, 5]. The space-time scheme based on this formulation provide improved stability condition with weak restrictions on the discretization parameters.

References [1] Daniele Boffi, Nicola Cavallini, and Lucia Gastaldi. Finite element approach to immersed boundary method with different fluid and solid densities. Math. Models Methods Appl. Sci., 21(12):2523–2550, 2011. [2] Daniele Boffi and Lucia Gastaldi. A finite element approach for the immersed boundary method. Comput. & Structures, 81(8-11):491–501, 2003. In honour of Klaus-Jürgen Bathe. [3] Daniele Boffi, Lucia Gastaldi, and Luca Heltai. On the CFL condition for the finite element immersed boundary method. Comput. & Structures, 85(1114):775–783, 2007. [4] Daniele Boffi, Lucia Gastaldi, Luca Heltai, and Charles S. Peskin. On the hyper-elastic formulation of the immersed boundary method. Comput. Methods Appl. Mech. Engrg., 197(25-28):2210–2231, 2008.

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[5] Vivette Girault, Roland Glowinski, and T. W. Pan. A fictitious-domain method with distributed multiplier for the Stokes problem. In Applied nonlinear analysis, pages 159–174. Kluwer/Plenum, New York, 1999. [6] R. Glowinski and Yu. Kuznetsov. Distributed Lagrange multipliers based on fictitious domain method for second order elliptic problems. Comput. Methods Appl. Mech. Engrg., 196(8):1498–1506, 2007. [7] Charles S. Peskin. The immersed boundary method. Acta Numer., 11:479–517, 2002.

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Ludwig Gauckler TU Berlin, DE Plane wave stability of the split-step Fourier method for the nonlinear Schrödinger equation Minisymposium Session TIME: Thursday, 14:30 - 15:00, CO015 The cubic nonlinear Schrödinger equation i∂t ψ = −∆ψ + λ|ψ|2 ψ,

ψ = ψ(x, t),

(1)

with periodic boundary conditions in space (x ∈ Rd /(2πZ)d ) has solutions that are plane waves: ψ(x, t) = ρei(m·x−ωt) with m ∈ Zd solves (1) for ω = |m|2 + λρ2 . In the talk we will discuss the stability of these solutions and the stability of their numerical approximation. We first study the stability of plane waves in the exact solution. We show orbital stability of plane waves over long times. In the second part of the talk we study a very popular method for the numerical discretization of the nonlinear Schrödinger equation, the split-step Fourier method. This method combines a Fourier spectral method in space with a splitting integrator in time. We will pursue the question whether the stability of plane waves in the exact solution transfers to this numerical discretization. Joint work with Erwan Faou and Christian Lubich.

127

Jean-Frédéric Gerbeau INRIA, FR Luenberger observers for fluid-structure problems Minisymposium Session NFSI: Thursday, 14:00 - 14:30, CO122 This talk is devoted to inverse problems in fluid-structure interaction problems, in particular for blood flow in arteries. Our strategy is based on two kinds of methods: Luenberger observers for the state variables and Unscented Kalman Filter for the parameters. This presentation will mainly focused on Luenberger observers. We analyze the performances of two types of observers, the Direct Velocity Feedback and the Schur Displacement Feedback algorithms, originally devised for elastodynamics. The measurements are assumed to be restricted to displacements or velocities in the solid. We first assess the observers using hemodynamics-inspired test problems with the complete model, including the Navier-Stokes equations in Arbitrary Lagrangian-Eulerian formulation, in particular. Then, in order to obtain more detailed insight we consider several well-chosen simplified models, each of which allowing a thorough analysis – emphasizing spectral considerations – while illustrating a major phenomenon of interest for the observer performance, namely, the added mass effect for the structure, the coupling with a lumped-parameter boundary condition model for the fluid flow, and the fluid dynamics effect per se. Whereas improvements can be sought when additional measurements are available in the fluid domain in order to more effectively deal with strong uncertainties in the fluid state, in the present framework this establishes Luenberger observer methods as very attractive strategies – compared, e.g. to classical variational techniques – to perform state estimation, and more generally for uncertainty estimation since other observer procedures, like nonlinear filtering, can be conveniently combined to estimate uncertain parameters. Joint work with C. Bertoglio, D. Chapelle, M.A. Fernández, and P. Moireau.

128

Tomas Gergelits Faculty of Mathematics and Physics, Charles University in Prague, CZ Composite polynomial convergence bounds in finite precision CG computations Contributed Session CT3.3: Thursday, 16:30 - 17:00, CO3 The convergence rate of the method of conjugate gradients (CG) used for solving linear algebraic system Ax = b (1) with large and sparse Hermitian and positive definite (HPD) matrix A ∈ CN ×N with eigenvalues 0 < λ1 < . . . < λN is commonly associated with linear convergence bounds derived using shifted and scaled Chebyshev polynomials. However, the CG method is nonlinear and its convergence tends to accelerate during the iteration process (it exhibits the so-called superlinear convergence) and thus the linear bounds are typically highly pessimistic. In order to describe the superlinear convergence, Axelsson [1] and Jennings [5] considered in presence of m large outlying eigenvalues the composite polynomial qm (λ)χk−m (λ)/χk−m (0),

(2)

where χk−m (λ) denotes the Chebyshev polynomial of degree k−m shifted to the interval [λ1 , λN −m ] and qm (λ) has the roots at the outlying eigenvalues λN −m+1 , . . . , λN , which resulted in the bound !k−m p κm (A) − 1 kx − xk kA ≤2 p , k = m, m + 1, . . . , (3) kx − x0 kA κm (A) + 1 where κm (A) ≡ λN −m /λ1 is the so-called effective condition number. This quantity is typically substantially smaller than the condition number κ(A) ≡ λN /λ1 which indicates a possibly faster convergence after m initial iterations. All this assumes, however, exact arithmetic. In finite precision computations the CG convergence can be significantly delayed due to rounding errors. Such delays are pronounced, in particular, in the presence of large outlying eigenvalues, and they can make the composite convergence bounds practically useless; see [5], [3], [7], [2], [6, Chapter 5]. Despite the early experimental warnings and theoretical arguments, misleading conclusions and inaccurate statements keep reappearing in literature. This contribution emphasizes that due to the requirement of keeping short recurrences, in practical CG computations effects of rounding errors must always be taken into consideration. Acknowledgement: This work has been supported by the ERC-CZ project LL1202, by the GACR grant 201/09/0917 and by the GAUK grant 695612.

References [1] O. Axelsson: A class of iterative methods for finite element equations. Comput. Methods Appl. Mech. Engrg., 9, 2, pp. 123–127, 1976. [2] T. Gergelits, Z. Strakoš: Composite convergence bounds based on Chebyshev polynomials and finite precision conjugate gradient computations. accepted for publication in Numerical Algorithms, April, 2013,

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[3] A. Greenbaum: Behaviour of slightly perturbed Lanczos and conjugategradient recurrences. Linear Algebra Appl., 113, pp. 7–63, 1989. [4] M. R. Hestenes, E. Stiefel: Methods of conjugate gradients for solving linear systems. J. Research Nat. Bur. Standards, 49, pp. 409–436, 1952. [5] A. Jennings: Influence of the eigenvalue spectrum on the convergence rate of the conjugate gradient method. J. Inst. Math. Appl., 20, 1, pp. 61–72, 1977. [6] J. Liesen, Z. Strakoš: Krylov subspace methods: principles and analysis. Numerical Mathematics and Scientific Computation, Oxford University Press, 2012. [7] Y. Notay: On the convergence rate of the conjugate gradients in presence of rounding errors. Numer. Math., 65, 3, pp. 301–317, 1993. Joint work with Zdenek Strakos.

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Omar Ghattas The University of Texas at Austin, US Stochastic Newton MCMC Methods for Bayesian Inverse Problems, with Application to Ice Sheet Dynamics Plenary Session: Thursday, 08:20 - 09:10, CO1 We address the problem of quantifying uncertainties in the solution of ill-posed inverse problems governed by expensive forward models (e.g., PDEs) and characterized by high-dimensional parameter spaces (e.g., discretized heterogeneous parameter fields). The problem is formulated in the framework of Bayesian inference, leading to a solution in the form of a posterior probability density. To explore this posterior density, we propose several variants of so-called Stochastic Newton Markov chain Monte Carlo (MCMC) methods, which employ, as MCMC proposals, a local Gaussian approximation whose covariance is the inverse of a local Hessian of the negative log posterior, made tractable via randomized low rank approximations and adjoint-based matrix-vector products. The stochastic Newton variants are applied to an inverse ice sheet flow problem governed by creeping, viscous, incompressible, non-Newtonian flow. The inverse problem is to infer the coefficient field of the basal boundary condition from surface velocity observations. We assess the performance of the methods and interpret the resulting parameter uncertainties with respect to the information content of both the prior and the data. This is joint work with Tan Bui-Thanh, Carsten Burstedde, Tobin Isaac, James Martin, Noemi Petra, and Georg Stadler.

131

Luc Giraud Inria, FR Recovery policies for Krylov solver resiliency Minisymposium Session CTNL: Wednesday, 10:30 - 11:00, CO015 The advent of exascale machines will require the use of parallel resources at an unprecedented scale with potentially billions of computing units leading to a high rate of hardware faults. High Performance Computing applications that aim at exploiting all these resources will thus need to be resilient, i.e., being able to eventually compute a correct output in presence of core faults. Contrary to checkpointing techniques or Algorithm Based Fault Tolerant (ABFT) mechanisms, strategies based on interpolation for recovering lost data do not require extra work or memory when no fault occurs. We apply this latter strategy to Krylov iterative solvers for systems of linear equations, which are often the most computational intensive kernels in HPC simulation codes. We propose and discuss several variants able to possibly handle multiple simultaneous faults. We study the impact and the overhead of the recovery methods, the fault rate and the number of processors on the resilience of the most popular solvers that are CG, GMRES and BiCGStab solvers. Joint work with E. Agullo, A. Guermouche, J. Roman, and M. Zounon.

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Mohammad Golbabaee CNRS, CEREMADE (Applied Math Research Centre), Universite Paris 9-Dauphine, FR Model Selection with Piecewise Regular Gauges Minisymposium Session ACDA: Monday, 11:40 - 12:10, CO122 In this talk, we investigate in a unified way the structural properties of a large class of convex regularizers for linear inverse problems. We consider regularizations with convex positively 1-homogenous functionals (so-called gauges) which obey a weak decomposability property. Weak decomposability promotes solutions of the inverse problem conforming to some notion of simplicity/low complexity by living on a low dimensional sub-space. This family of priors encompasses many special instances routinely used in regularized inverse problems such as l1, l1l2 (group sparsity), Trace norm, or the l-inf norm. The weak decomposability requirement is flexible enough to cope with analysis-type priors that include a pre-composition with a linear operator, such as for instance the total variation (TV). Weak decomposability is also stable under summation of regularizers, thus enabling to handle mixed regularizations (e.g. Trace+l1l2). We discuss the theoretical recovery performance of this class of regularizers. We provide sufficient conditions that allow to provably controlling the deviation of the recovered solution from the true underlying object, as a function of the noise level. More precisely we show that the solution to the inverse problem is unique and lives on the same low dimensional subspace as the true vector to recover, with the proviso that the minimal signal to noise ratio is large enough. This extends previous results well-known for the l1 norm, analysis l1 semi-norm, and the Trace norm to the general class of weakly decomposable gauges. Joint work with Samuel Vaiter, Jalal Fadili, and Gabriel Peyre.

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Maria Gonzalez Universidad de A Coruña, ES A new a posteriori error estimator of low computational cost for an augmented mixed FEM in linear elasticity Contributed Session CT2.4: Tuesday, 14:30 - 15:00, CO015 We consider the augmented mixed finite element method introduced in [4, 5] for the linear elasticity problem in the plane and extended in [6] to the threedimensional case. When Dirichlet boundary conditions are prescribed, the corresponding Galerkin scheme is well-posed and free of locking for any choice of finite element subspaces. This fact turns out to be the main advantage of this method. The use of adaptive algorithms based on a posteriori error estimates guarantees good convergence behavior of the finite element solution of a boundary value problem. Several a posteriori error estimators are already available in the literature for the usual mixed finite element method in linear elasticity. Concerning the a posteriori error analysis of the augmented scheme presented in [4], an a posteriori error estimator of residual type was introduced in [2] in the case of pure homogeneous Dirichlet boundary conditions. That analysis was extended recently to the cases of pure non-homogeneous Dirichlet boundary conditions and mixed boundary conditions with non-homogeneous Neumann data; cf. [3]. The a posteriori error estimators derived in [2] and [3] are both reliable and efficient, but involve the computation of eleven residuals per element in the case of homegeneous Dirichlet boundary conditions, and thirteen residuals per element in the case of non-homogeneous Dirichlet boundary conditions, including in both cases normal and tangential jumps. In this work, we derive a new a posteriori error estimator for the augmented dualmixed method proposed in [4]-[6] in the case of Dirichlet boundary conditions. The analysis is based on the use of a projection of the error and allows to derive an a posteriori error estimator that only requires the computation of four residuals per element in the case of homogeneous boundary conditions, and six residuals per element in the case of non-homogeneous boundary conditions. In both cases, the derived a posteriori error indicators do not require the computation of normal nor tangential jumps across the edges or faces of the mesh, which simplifies the numerical implementation, specially in the 3d case. Besides, we prove that the new a posteriori error estimator is both reliable and locally efficient in the case of homogeneous Dirichlet boundary conditions. When non-homogeneous boundary conditions are imposed, it is reliable and locally efficient only in those elements that does not touch the boundary. Finally, we provide numerical experiments that illustrate the performance of the corresponding adaptive algorithms and support the theoretical results.

References [1] T.P. Barrios and G.N. Gatica. An augmented mixed finite element method with Lagrange multipliers: A priori and a posteriori error analyses. J. Comput. Appl. Math. 200, 653–676 (2007). [2] T.P. Barrios, G.N. Gatica, M. González and N. Heuer. A residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity. M2AN Math. Model. Numer. Anal. 40, 843–869 (2006). 134

[3] T.P. Barrios, E.M. Behrens and M. González. A posteriori error analysis of an augmented mixed formulation in linear elasticity with mixed and Dirichlet boundary conditions. Comput. Methods Appl. Mech. Engrg. 200, 101-113 (2011). [4] G.N. Gatica. Analysis of a new augmented mixed finite element method for linear elasticity allowing RT0 -P1 -P0 approximations. M2AN Math. Model. Numer. Anal. 40, 1–28 (2006). [5] G.N. Gatica. An augmented mixed finite element method for linear elasticity with non-homogeneous Dirichlet conditions. Electronic Transactions on Numerical Analysis, vol. 26, pp. 421-438, (2007). [6] G.N. Gatica, A. Márquez and S. Meddahi. An augmented mixed finite element method for 3D linear elasticity problems. J. Comput. Appl. Math. 231, 2, 526–540 (2009). Joint work with T.P. Barrios, and E.M. Behrens.

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Simsek Gorkem Eindhoven University of Technology, NL Error Estimation for The Convective Cahn – Hilliard Equation Contributed Session CT2.4: Tuesday, 15:00 - 15:30, CO015 The Cahn–Hilliard phase-field (or diffuse-interface) model has a wide range of applications where the interest is the modelling of phase segregation and evolution of multiphase flow systems. In order to capture the physics of these systems, diffuse-interface models presume a nonzero interface thickness between immiscible constituents, see [1]. The multiscale nature inherent in these models (interface thickness and domain size of interest) urges the use of space-adaptivity in discretization. In this contribution we consider the a-posteriori error analysis of the convective Cahn–Hilliard [4] model for varying Péclet number and interfacethickness (diffusivity) parameter. The adaptive discretization strategy uses mixed finite elements, a stable time-stepping algorithm and residual-based a-posteriori error estimation [2, 5]. Let Ω ⊂ Rd be a bounded domain with d = 1, 2, 3 and ∂Ω be the boundary which has an outward unit normal n. The convective Cahn-Hilliard equation can be written as follows: Find the real valued functions (c, µ) : Ω × [0, T ] → R for T > 0 such that ∂t c −

1 4µ + ∇ · (uc) = 0 Pe µ = φ0 (c) − 2 ∇c c(·, 0) = c0

∂n c = ∂n µ = 0

in

ΩT := Ω × (0, T ]

in

ΩT

in

Ω

on ∂ΩT := ∂Ω × (0, T ],

where ∂t (·) = ∂(·)/∂t, ∂n (·) = n · ∇(·) is the normal derivative, φ is the real-valued free energy function, u is a given function such that ∇ · u = 0 in Ω and u · n = 0 on ∂Ω, P e is the P e´clet number and  is the interface thickness. The nonlinear energy function φ(c) is of the double well form and we consider the following C 2 -continuous function :  2 (c + 1) c < −1,     
2 1 2 φ(c) := c ∈ [−1, 1] , 4 c −1      2 (c − 1) c > 1. In order to obtain the weak formulation, we consider the following function space and the corresponding norm as a suitable space for µ: Z T 2 1 2 V := L (0, T; H (Ω)), kvkV := kv(t)k2H1 dt (Ω)

0

and the space suitable for the phase variable c is W := {v ∈ V : vt ∈ V 0 }, where V 0 := L2 (0, T; [H1 (Ω)]0 ) is the dual space of V with the norm Z T kvt k2W := kvk2V + kvt k2V 0 and kvt k2V 0 := kvt (t)k2[H1 (Ω)]0 dt. 0

Then the weak form of the problem becomes: Find (c, µ) ∈ Wc0 × V : 136

1 (∇µ, ∇w) = 0 Pe 2 (µ, v) − (φ(c), v) +  (∇c, ∇v) = 0

∀w ∈ H1 (Ω)

hct , wi + (u∇c, w) +

∀v ∈ H1 (Ω),

for t ∈ [0, T], where Wc0 is the subspace of W of which the trace at t = 0 coincide with c0 . To derive an a-posteriori error representation, we will employ the mean-valuelinearized adjoint problem. The dual problem can be defined in terms of dual variables (p, χ) where the dual variable p is a function in the space W q¯ := {v ∈ W : v(T ) = q¯} . Then the dual problem can be written as follows: Find (p, χ) ∈ W q¯ × V : −∂t p + u∇p + 2 4χ − φ0 (c, cˆ)χ = q1 1 4p = q2 χ− Pe p = q¯ ∂n p = ∂n χ = 0

in

Ω × [0, T )

in

Ω × [0, T )

on Ω × {t = T } on ∂Ω × [0, T ],

where the nonlinear function φ0 (c, cˆ) is a mean-value-linearized function 0

Z

φ (c, cˆ) = 0

1

φ00 (sc + (1 − s)ˆ c) ds.

This analysis for the convective model forms a basic step in our research and will be helpful for the coupled Cahn–Hilliard/Navier–Stokes system [3] which is the desired model for future research.

References [1]

Anderson, D.M., McFadden, G.B. and Wheeler, A.A. Diffuse-Interface Methods in Fluid Mechanics. Annu. Rev. Fluid Mech. 30:139–65, 1998

[2]

Bartels, S., Müller, R. A-posteriori error controlled local resolution of evolving interfaces for generalized Cahn–Hilliard equations. Interfaces and Free Boundaries, 12:45–73, 2010

[3]

Boyer, F., Lapuerta, C., Minjeaud, S., Piar, B. and Quintard, M. Cahn– Hilliard Navier-Stokes Model for the Simulation of Three-Phase Flows. Transport in Porous Media, 82:463 – 483, 2010

[4]

Kay, D., Styles, V. and Süli, E. Discontinuous Galerkin Finite Element Approximation of the Cahn–Hilliard Equation with Convection. SIAM J. Numer. Anal., 47:2660–2685, 2009

[5]

Van der Zee, K. G., Oden, J. T., Prudhomme, S. and Hawkins-Daarud, A. Goal-oriented error estimation for Cahn–Hilliard models of binary phase transition. Numer. Methods Partial Differ. Equationsm, 27:160–196, 2011

Joint work with Kris G. van der Zee, and E. Harald van Brummelen.

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Alexandre Grandchamp LCVMM-EPFL, CH Multi-scale DNA Modelling and Birod Mechanics Contributed Session CT1.4: Monday, 17:00 - 17:30, CO015 A standard description of DNA fragments is provided by the Kratky-Porod model of bending and twisting, which corresponds in continuum mechanics language to an inextensible and unshearable elastic rod [1,2]. Classical rod theory, which is useful in applications over a wide range of length scales, e.g. from polymers to plant tendrils and wire ropes, allows one to describe more complex mechanical behaviours involving shear and extension which are also pertinent to DNA [3,4]. However, at short scales, strand separation is central to the function of DNA, which requires a continuum mechanics theory of interacting double-stranded filaments called birods [5]. We show that birod equilibrium configurations are solutions of non-canonical Hamiltonian evolution in arc-length. For multi-scale DNA modeling we parametrize the birod Hamiltonian in a sequence-dependent way starting from coarse grained rigid base models which are in turn parameterized by finer grain molecular dynamic simulations [6]. Finally, we show how birod two-point boundary value problems can be solved with parameter continuation in order to investigate features of DNA equilibrium probability distributions that can be probed experimentally. As time allows, we will present the analogous non-canonical Hamiltonian system of N coupled Cosserat rods, as arises for example in collagen filaments (N=3) or in bacterial flagellum (N=20). [1] C. J. Benham, S. P. Mielke, DNA Mechanics, Ann. rev. Biomed. Eng. 2005. 7:21-53 [2] M. Doi, S. F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986 [3] S. S. Antman , Nonlinear Problem of Elasticity, SpringerVerlag, New-York, 1995. [4] D. J. Dichmann, Y. Li and J. H. Maddocks, Hamiltonian Formulations and Symmetry in Rod Mechanics, Math. Appr. to Biomol. Struct. and Dyn., 82 (1996), Springer, New-York [5] M. Moakher, J.H. Maddocks, A Double-Strand Elastic Rod Theory, Arch. Rational Mech. Anal. 177 (2005) 53 - 91. [6] O. Gonzalez, D. Petkeviciute, J.H. Maddocks, A Sequence-Dependent Rigid Base Model of DNA, J. Chem. Phys. 138, 055102 (2013) Joint work with Prof. J. H. Maddocks and Jarosław Głowacki.

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Gwenol Grandperrin EPFL - SB - MATHICSE - CMCS, CH Multiphysics Preconditioners for Fluid–Structure Interaction Problems Minisymposium Session PSPP: Thursday, 15:00 - 15:30, CO3 We are interested in preconditioning problems arising in blood–flow simulations. These are characterized by Fluid–Structure Interaction (FSI) in three dimensional geometries. The nonlinearity of the equations is solved by using the Newton method and the Jacobian system by preconditioned GMRES iterations. The preconditioner is based on an inexact factorization derived from a block Gauss-Seidel method. Each factor represents a specific physics: the fluid, the structure, and the harmonic extension. This allows for the selection of physics–specific preconditioners. For example, we take advantage of approximate versions of state of the art preconditioners for the fluid part of the FSI model (modeled with the Navier–Stokes equations), namely the Pressure Convection–Diffusion (PCD) preconditioner, and SIMPLE. We focus on the important factors to measure parallel performances of a preconditioner: the independence on the number of iterations, in terms of CPU time (scalability of the preconditioner), on the mesh size (optimality), and on the physical parameters (robustness), as well as the strong and weak scalability. We use these metrics to demonstrate the efficiency of our preconditioners in typical situations for blood–flow simulations. All the computations are carried out using LifeV (http://www.lifev.org), an open source finite element library based on Trilinos.

Joint work with Dr. Paolo Crosetto, Dr. Simone Deparis, and Prof. Alfio Quarteroni.

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Isabelle Greff Laboratoire de Mathématiques et de leurs Applications - Pau, FR Conservation of Lagrangian and Hamiltonian structure for discrete schemes Contributed Session CT1.4: Monday, 18:00 - 18:30, CO015 Many problems arising in various fields (such as physics, mechanics, fluid mechanics or finance) are described using partial differential equations (PDEs). Although explicit solutions are not available in general, important classes of PDEs do present strong structural properties: classical examples are symmetry properties, maximum principle or conservation properties. It is quite essential for the numerical methods to provide a translation of these structural properties from the continuous level to the discrete level so enforcing the numerical solutions to obey qualitative behaviours in agreement with the underlying physics of the problem. Two fundamental notions arising in classical mechanics are Lagrangian and Hamiltonian structures. Lagrangian systems are made of one functional, called the Lagrangian functional, and a variational principle called the least action principle. From the least action principle is derived a second order differential equation called the Euler-Lagrange equation, see e.g. [1]. The Lagrangian structure is much more fundamental than its associated Euler-Lagrange equation: it contains information that the Euler-Lagrange equation does not. A range of numerical methods forget about the Lagrangian to focus on the Euler-Lagrange equation itself. Let us consider the following question: consider a PDE deriving from a Lagrangian/Hamiltonian and a least action principle. When discretising this PDE, how is embedded the attached Lagrangian/Hamiltonian structure at the discrete level ? More precisely we ask whether the discretised PDE can be seen as deriving from a discrete least action principle associated with a discrete Lagrangian/Hamiltonian structure. Basically, in case the Lagrangian structure is embedded at the discrete level, then the variational property of the original equation (at the continuous level) may be preserved by the discrete problem. Our purpose is to study the conservation of variational properties for a given problem when discretising it. This can be seen as an extension to PDEs of the works on variational integrators (as [2, 3]). Precisely we are interested with Lagrangian or Hamiltonian structures and thus with variational problems attached to a least action principle. Considering a partial differential equation (PDE) deriving from such a variational principle, a natural question is to know whether this structure at the continuous level is preserved at the discrete level when discretising the PDE. To address this question a concept of coherence is introduced. Both the differential equation (the PDE translating the least action principle) and the variational structure can be embedded at the discrete level. This provides two discrete embeddings for the original problem. In case these procedures finally provide the same discrete problem we will say that the discretisation is coherent. Our purpose is illustrated with the Poisson problem. Coherence for discrete embeddings of Lagrangian structures is studied for various classical discretisations (finite elements, finite differences and finite volumes). Hamiltonian structures are shown to provide coherence between a discrete Hamiltonian structure and the discretisation of the mixed formulation of the PDE, both for mixed finite elements and mimetic finite differences methods. Nevertheless, many PDEs do not derive from a variational formulation in the classical sense. However it is possible to bypass the obstruction to the existence of a Lagrangian formulation, as an example, we will consider the convection-diffusion

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equation.

References 1 Vladimir I. Arnold. Mathematical methods of classical mechanics, volume 60 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1989. Translated from the Russian by K. Vogtmann and A. Weinstein. 2 Ernst Hairer, Christian Lubich, and Gerhard Wanner. Geometric numerical integration, volume 31 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 2006. Structure-preserving algorithms for ordinary differential equations. 3 Jerrold E. Marsden and Matthew West. Discrete mechanics and variational integrators. Acta Numer., 10:357–514, 2001.

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Sven Gross Chair of Numerical Mathematics, RWTH Aachen University, DE XFEM for pressure and velocity singularities in 3D two-phase flows Minisymposium Session FREE: Tuesday, 10:30 - 11:00, CO2 Two-phase systems play an important role in chemical engineering, for example mass transport between droplets and a surrounding liquid in extraction columns (liquid-liquid system) or heat transfer in falling films (liquid-gas system). The velocity and pressure field are smooth in the interior of each phase, but undergo certain singularities at the interface Γ between the phases. Surface tension induces a pressure jump across Γ, and a large viscosity ratio leads to a kink of the velocity field at Γ, especially for liquid-gas systems. If interface capturing methods (like VOF or level set techniques) are applied, the finite element grid is usually not aligned with the interface. Then for standard √ FEM the approximation of functions with such singularities leads to poor O( h) convergence. The application of suitable extended finite element methods (XFEM) provides optimal approximation properties, essentially reducing spurious currents at the interface. Figure 1 shows the pressure jump of a static bubble induced by surface tension using a standard and an extended finite element space. In this talk we consider 3D flow simulations of such two-phase systems on adaptive multilevel tetrahedral grids. We present a Heaviside enrichment of the pressure space [1] yielding second order convergence of the L2 (Ω) pressure error [4] and a ridge enrichment [3] of the velocity space leading to first order convergence of the H 1 (Ω1 ∪ Ω2 ) velocity error. At the end of the talk, we present application examples of 3D droplet and falling film simulations obtained by our 3D two-phase flow solver DROPS [2, 5].

References [1] S. Groß and A. Reusken. An extended pressure finite element space for twophase incompressible flows with surface tension. J. Comput. Phys., 224:40–58, 2007. [2] S. Groß and A. Reusken. Numerical Methods for Two-phase Incompressible Flows, volume 40 of Springer Series in Computational Mathematics. Springer, 2011. [3] N. Moës, M. Cloirec, P. Cartraud, and J.-F. Remacle. A computational approach to handle complex microstructure geometries. Comput. Methods Appl. Mech. Engrg., 192:3163–3177, 2003. [4] A. Reusken. Analysis of an extended pressure finite element space for two-phase incompressible flows. Comput. Visual. Sci., 11:293–305, 2008. [5] DROPS package for simulation of two-phase flows. http://www.igpm.rwth-aachen.de/DROPS/.

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Figure 1: Pressure jump of a static bubble using piecewise linear FEM (left) and suitable XFEM (right). Joint work with Arnold Reusken.

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Marcus Grote University of Basel, CH Runge-Kutta based explicit local time-stepping methods for wave propagation Minisymposium Session TIME: Thursday, 11:00 - 11:30, CO015 The efficient simulation of time-dependent wave phenomena is of fundamental importance in a wide variety of applications from acoustics, electromagnetics and elasticity. For acoustic wave propagation, the scalar damped wave equation utt + σut − ∇ · (c2 ∇u)

= f in Ω × (0, T )

σ ≥ 0,

(1)

often serves as a model problem. Next, we discretize (1) in space by using standard continuous (H 1 -conforming) finite elements with mass lumping or a nodal DG discretization, while leaving time continuous. Either discretization leads to a system of ordinary differential equations with an essentially diagonal mass matrix, which can written as a first order system y 0 (t) = By(t) + F (t)

(2)

Locally refined meshes impose severe stability constraints on explicit time-stepping methods for the numerical solution of (1). Local time-stepping (LTS) methods overcome that bottleneck by using smaller time-steps precisely where the smallest elements in the mesh are located. In [1, 2], explicit second-order LTS integrators for transient wave motion were developed, which are based on the standard leapfrog scheme. In the absence of damping, i.e. σ = 0, these time-stepping schemes, when combined with the modified equation approach, yield methods of arbitrarily high (even) order. To achieve arbitrarily high accuracy in the presence of damping, while remaining fully explicit, explicit LTS methods based on Adams-Bashforth multi-step schemes were derived in [3]. We now propose explicit LTS methods of high accuracy based on both explicit classical and low-storage Runge-Kutta schemes. In contrast to Adams-Bashforth methods, Runge-Kutta methods are one-step methods; hence, they do not require a starting procedure and easily accommodate adaptive time-step selection. Although, Runge-Kutta methods do require several further evaluations per timestep, that additional work is compensated by a less stringent stability restriction on the time-step. The resulting LTS-RK schemes have the same high rate of convergence as the original classical or low-storage RK methods. To illustrate the versatility of our approach, we consider a computational rectangular domain of size [0, 2] × [0, 1] with two rectangular barriers inside forming a narrow gap. We use continuous P 2 elements on a triangular mesh, which is highly refined in the vicinity of the gap, as shown in Fig. 1 (left). For the time discretization, we choose the third-order low-storage Runge-Kutta based LTS. Since the typical mesh size inside the refined region is about p = 7 times smaller than that in the surrounding coarser region, we take p local time steps of size ∆τ = ∆t/p for every time step ∆t. Thus, the numerical method is third-order accurate both in space and time with respect to the L2 -norm. As shown in Fig. 2 (right), a vertical Gaussian pulse initiates two plane waves propagating in opposite directions. The right-moving wave propagates until it impinges on the obstacle. A fraction of the wave then penetrates the gap and generates a circular wave.

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References [1] J. Diaz, M. Grote: Energy conserving explicit local time-stepping for secondorder wave equations. SIAM Journal on Scientific Computing, 31 (2009), 1985–2014. [2] M. Grote, T. Mitkova: Explicit local time-stepping for Maxwell’s equations. Journal of Computational and Applied Mathematics, 234 (2010), 3283–3302. [3] M. Grote, T. Mitkova: High-order explicit local time-stepping methods for damped wave equations. Journal of Computational and Applied Mathematics, 239 (2013), pp. 270–289.

Figure 1: The initial triangular mesh.

Figure 2: The numerical solution at time t = 0.6 Joint work with Michaela Mehlin, and Teodora Mitkova.

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Nicola Guglielmi University of L’Aquila, IT Computing the distance to defectivity Minisymposium Session NEIG: Thursday, 15:30 - 16:00, CO2 Let A be a complex either real matrix with distinct eigenvalues. We are interested to compute the distance of A from the set of complex/real defective matrices. In order to compute such a distance we propose a novel method which is based on a gradient system in a low-rank manifold of matrices. We provide the main theoretical results and several illustrative examples. This is a joint work with Paolo Butta’ and Silvia Noschese (Roma) and Manuela Manetta (L’Aquila).

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Elie Hachem MINES ParisTech, FR Unified variational multiscale method for compressible and incompressible flows using anisotropic adaptive mesh Minisymposium Session ADFE: Tuesday, 10:30 - 11:00, CO016 We propose in this work a unified numerical method to address easily the coupling between compressible and incompressible multiphase flows. The same set of primitive unknowns and equations is described everywhere in the flow. A levelset function enables the precise position of the interfaces and provides homogeneous physical properties for each subdomain. The coupling between the pressure and the flow velocity is ensured by introducing mass conservation terms in the momentum and energy equations. The system is then solved using a new derived Variational Multiscale stabilized finite element method [1]. Combined with anisotropic mesh adaptation, we show that the proposed method provides an accurate modeling framework for two-phase compressible isothermal flows and for fluid-structure interaction problems. Therefore, a new a posteriori estimate based on the length distribution tensor approach and the associated edge based error analysis is presented to ensure an accurate capturing of the discontinuities at the interfaces [2]. It enables to calculate a stretching factor providing a new edge length distribution, its associated tensor and the corresponding metric. The optimal stretching factor field is obtained by solving an optimization problem under the constraint of a fixed number of edges in the mesh. With such an advantage, we can now provide a useful tool for doing accurate numerical simulations [3]. We assess the behaviour and accuracy of the proposed formulation in the simulation of 2D and 3D time-dependent numerical examples. [1] E. Hachem, B. Rivaux, T. Kloczko, H. Digonnet, T. Coupez, Stabilized finite element method for incompressible flows with high Reynolds number, Journal of Computational Physics, Vol. 229 (23), 8643-8665, 2010 [2] T. Coupez, G. Jannoun, N. Nassif, H.C. Nguyen, H. Digonnet, E. Hachem, Adaptive Time-step with Anisotropic Meshing for Incompressible Flows, accepted in Journal of Computational Physics, http://dx.doi.org/10.1016/j.jcp.2012.12.010, 2013 [3] E. Hachem, S. Feghali, R. Codina and T. Coupez, Anisotropic Adaptive Meshing and Monolithic Variational Multiscale Method for Fluid-Structure Interaction, Computer and Structures, http://dx.doi.org/10.1016/j.compstruc.2012.12.004, 2013

Joint work with Thierry Coupez.

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Martin Hadrava Charles University in Prague, Faculty of Mathematics and Physics, CZ Space-time Discontinuous Galerkin Method for the Problem of Linear Elasticity Contributed Session CT2.5: Tuesday, 14:00 - 14:30, CO016 The paper will be concerned with the numerical solution of the problem of dynamic linear elasticity by several time-discretization techniques based on the application of the discontinuous Galerkin method (DGM) in space. The DGM is a class of numerical methods for solving partial differential equations. It combines features of the finite volume method (the ability to capture discontinuities) and the finite element method (arbitrary polynomial degree yielding accurate high-order schemes). The method was initially introduced in 1970s by Reed and Hill as a technique to solve neutron transport problems [3]. During the subsequent decades, the DGM has been applied to various problems arising from physics, biology and economics and in particular became a popular method for solving problems in computational fluid dynamics, electrodynamics and plasma physics. A detailed survey about the evolution of the DGM can be found in, e.g., [1]. The first part of the paper will be devoted to the description of the problem and the derivation of the discretization schemes under investigation. We will present several discretizations based on finite-difference approximations of the time derivative terms and the discretization based on the space-time discontinuous Galerkin method (STDGM). In contrast to the standard applications of the DGM to nonstationary problems, in the STDGM the main concept of the discontinuous Galerkin method - discontinuous piecewise polynomial approximation - is applied both in space and in time and hence a more robust and accurate scheme is obtained. The application of the STDGM to solve the nonlinear convection-diffusion equation was presented in [2]. A general presentation of the theory and applications of DG methods can be found in the manuscript of Rivière [4], where the STDGM is applied to the parabolic equation. A detailed description of the application of the STDGM to the problem of dynamic linear elasticity will be given in the paper. In the second part we will investigate the numerical properties of the presented methods by comparing the results obtained by numerical experiments. We will show the estimated rate of convergence of all methods under investigation, while keeping focus on showing the difference between the accuracy of the STDGM and the discretizations based on finite-difference approximations. We will demonstrate the efficiency of the STDGM - although it consumes more computational time, it rewards us with a significantly more accurate approximate solution.

References [1] B. Cockburn, G. E. Karniadakis and C. W. Shu, The development of discontinuous Galerkin methods, (1999). [2] M. Feistauer, V. Kučera, K. Najzar and J. Prokopová, Analysis of space-time discontinuous Galerkin method for nonlinear convection-diffusion problems, Numer. Math. 117 (2011), pp. 251–258. [3] W. Reed and T. Hill, Triangular Mesh Methods for the Neutron Transport Equation, Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory (1973). 148

[4] B. M. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Frontiers in Applied Mathematics (2008). Joint work with Miloslav Feistauer, Adam Kosík, and Jaromír Horáček.

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Ernst Hairer Université de Genève, CH Long-term analysis of numerical and analytical oscillations Plenary Session: Monday, 09:50 - 10:40, Rolex Learning Center Auditorium Two completely different topics will be adressed in this talk: - the numerical solution of Hamiltonian systems with linear multistep methods over long times, - adiabatic invariants in highly oscillatory Hamiltonian differential equations. In both situations, high oscillations are present and their influence to the longtime dynamics of solutions is of interest to us. Whereas in the second situation oscillatory solutions arise from the special form of the differential equation, they are due to the multistep character and the presence of parasitic solution components in the first situation. We show that certain symmetric multistep methods, when applied to second order Hamiltonian systems, behave very similar to symplectic one-step methods (excellent long-time energy-preservation, near-preservation of angular momentum, linear error growth for nearly integrable systems). On the other hand, for multiscale systems where harmonic oscillators with several high frequencies are coupled to a slow system, near-preservation of the oscillatory energy over long times is shown without any non-resonance condition. For the proof of these results the technique of modulated Fourier expansions is used. The surprising fact is that the same ideas that permit to prove the nearpreservation of the oscillatory energy in multiscale Hamiltonian systems, can also be applied to get insight into the long-time behavior of numerical solutions obtained by symmetric linear multistep methods. The presented results have been obtained in collaboration with Christian Lubich, David Cohen, Ludwig Gauckler, and Paola Console. References [1] E. Hairer, Ch. Lubich: Long-time energy conservation of numerical methods for oscillatory differential equations. SIAM J. Numer. Anal. 38 (2001), 414–441. [2] E. Hairer, Ch. Lubich, G. Wanner: Geometric Numerical Integration. StructurePreserving Algorithms for Ordinary Differential Equations. 2nd edition. Springer Verlag, Berlin, Heidelberg, 2006. [3] L. Gauckler, E. Hairer, Ch. Lubich: Energy separation in oscillatory Hamiltonian systems without any non-resonance condition. To appear in Comm. Math. Phys. (2013). [4] P. Console, E. Hairer, Ch. Lubich: Symmetric multistep methods for constrained Hamiltonian systems. To appear in Numerische Mathematik (2013).

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Abdul-Lateef Haji-Ali King Abdullah University of Science and Technology - KAUST, SA Optimization of mesh hierarchies for Multilevel Monte Carlo Contributed Session CT4.8: Friday, 08:50 - 09:20, CO123 We consider the Multilevel Monte Carlo (MLMC) method in applications involving differential equations with random data where the underlying approximation method of individual samples is based on uniform spatial discretizations of arbitrary approximation order and cost. We perform a general optimization of the parameters defining the MLMC hierarchy in such cases. We recall the MLMC estimator A of the quantity of interest g, M` M0 L X 1 X 1 X g0 (·; ω0,m ) + (g` (·; ω`,m ) − g`−1 (·; ω`,m )) , A= M0 m=1 M` m=1 `=1

where gl (·; ωl,m ) is a sample realization of g using mesh size hl in the underlying discretization method. We assume the strong and weak errors of gl are well approximated by E[g − g` ] ' hq` 1 QW ,

2 Var[g` − g`−1 ] ' hq`−1 QS ,

for problem and method specific constants QW , QS , q1 , q2 . For a given error tolerance, TOL, and confidence parameter, Cα , we solve the optimization problem Work =

minimize ( subject to

hqL1 QW V ar(g0 ) M0

+

q

2 h`−1 `=1 M` QS

PL

PL

M` `=0 hdγ , `

≤ (1 − θ) TOL,  2 , ≤ θ TOL Cα

[bias], [statistical error],

where γ is the order of solver cost, typically 1 ≤ γ ≤ 3 for linear systems, and d is the spatial dimension of the discretization domain. The optimization parameL ters are the mesh sizes {hl }L l=0 , the numbers of samples {Ml }l=0 , the number of levels L + 1, and the optimal splitting between bias and statistical errors modeled by θ. Note that the numbers of samples {Ml }L l=0 are constrained to integers. Moreover, {hl }L are constrained to discrete sets by possible uniform discretizal=0 tions. However, in order to reduce to complexity of the optimization we do not initially enforce these constraints on the parameters. We then use the optimal unconstrained parameters as an initial guess to do a limited brute force search of the best hierarchy that takes these constraints into account. Similarly, we do a brute force search for the best integer L that minimizes the work. The resulting hierarchies are different from typical MLMC hierarchies in that they do not have a fixed ratio between successive mesh sizes. Moreover, our analysis shows that θ, which determines the best splitting of tolerance between bias and statistical errors, can be drastically different from value 1/2 traditionally used in MLMC. We will also present numerical results which highlight the functionality of the optimization by applying our method to an elliptic PDE with stochastic coefficients. We will emphasize how the optimal hierarchies change from the standard MLMC method as you include the effects of real problem parameters, such as the solver

151

cost exponent. Joint work with Nathan Collier, Abdul-Lateef Haji-Ali, Fabio Nobile, Erik von Schwerin, and Raul Tempone.

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Helmut Harbrecht University of Basel, CH On multilevel quadrature for elliptic stochastic partial differential equations Minisymposium Session UQPD: Wednesday, 12:00 - 12:30, CO1 In this talk we show that the multilevel Monte Carlo method for elliptic stochastic partial differential equations can be interpreted as a sparse grid approximation. By using this interpretation, the method can straightforwardly be generalized to any given quadrature rule for high dimensional integrals like the quasi Monte Carlo method or the polynomial chaos approach. Besides the multilevel quadrature for approximating the solution’s expectation, a simple and efficient modification of the approach is proposed to compute the stochastic solution’s variance. Numerical results are provided to demonstrate and quantify the approach. Joint work with Michael Peters, and Markus Siebenmorgen.

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Markus Hegland The Australian National University, AU Solving the chemical master equations for biological signalling cascades using tensor factorisation Minisymposium Session LRTT: Wednesday, 10:30 - 11:00, CO3 Signalling cascades are essential parts of biological organisms. In particular they can amplify and also denoise signals. As they are based on chemical reactions with relatively few copy numbers they exhibit noise and their probability functions are described as solution of a chemical master equation. Cascades have a particularly simple structure. The matrix of their chemical master equation is shown to have tensor train ranks of 3 for practically important cases. For discrete time one can then establish a graphical model for the components of the states. We discuss how to exploit this structure computationally and how to use well-known algorithms from statistics for the determination of marginal distributions. We discuss various tensor fractorisations for cascades and their application. In a practical example the Arnoldi method and the singular value decomposition are used to solve the stationary chemical master equation for simple cascades. Probability distributions with up to 10 stages have been solved. We discuss these solutions. Joint work with Jochen Garcke.

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Claus-Justus Heine Universität Stuttgart, IANS, DE Mean-Curvature Reconstruction with Linear Finite Elements Contributed Session CT1.2: Monday, 18:00 - 18:30, CO2 We present a numerical method to construct an approximative curvature-vector field from piece-wise linearly approximated co-ordinate data of a given n-dimensional hypersurface Γ ⊂ Rn+1 (n = 2, 3). The reconstruction uses an L2 -projection with an additional Laplacian diffusion term and works with piece-wise linear Lagrangian finite elements. It can be shown that the curvature reconstruction converges with h2/3 in the L2 -sense where h denotes the discretisation parameter of the underlying finite element mesh. The method can be applied to triangulated surfaces as well as to discrete surfaces defined by level-sets of piece-wise linear finite element functions.

155

Claus-Justus Heine Universität Stuttgart, IANS, DE Mean-Curvature Reconstruction with Linear Finite Elements Minisymposium Session GEOP: Tuesday, 10:30 - 11:00, CO122 We present a numerical method to construct an approximative curvature-vector field from piece-wise linearly approximated co-ordinate data of a given n-dimensional hypersurface Γ ⊂ Rn+1 (n = 2, 3). The reconstruction uses an L2 -projection with an additional Laplacian diffusion term and works with piece-wise linear Lagrangian finite elements. It can be shown that the curvature reconstruction converges with h2/3 in the L2 -sense where h denotes the discretisation parameter of the underlying finite element mesh. The method can be applied to triangulated surfaces as well as to discrete surfaces defined by level-sets of piece-wise linear finite element functions.

156

Patrick Henning Uppsala University, SE Error control for a Multiscale Finite Element Method Minisymposium Session ADFE: Tuesday, 11:00 - 11:30, CO016 In this presentation, we introduce an adaptive mesh refinement strategy for the multiscale finite element method (MsFEM) for solving elliptic problems with rapidly oscillating coefficients. Starting from a general version of the MsFEM with oversampling, we present an a posteriori estimate for the H 1 -error between the exact solution of the problem and a corresponding MsFEM approximation. Our estimate holds without any assumptions on scale separation or on the type of the heterogeneity. The estimator splits into different contributions which account for the coarse grid error, the fine grid error and even the oversampling error. Based on the error estimate we construct an adaptive algorithm that is validated in numerical experiments. Joint work with Mario Ohlberger, and Ben Schweizer.

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Henar Herrero Universidad de Castilla-La Mancha, ES The reduced basis approximation applied to a Rayleigh-Bénard problem Contributed Session CT3.5: Thursday, 17:30 - 18:00, CO016 titleThe reduced basis approximation applied to a Rayleigh-Bénard problem The reduced basis approximation is a discretization method that can be implemented for solving parameter-dependent problems P(φ(µ), µ) = 0 with parameter µ in cases of many queries. This method consists of approximating the solution φ(µ) of P(φ(µ), µ) = 0 by a linear combination of appropriate preliminary computed solutions φ(µi ) with i = 1, 2, ...N such that µi are parameters chosen by an iterative procedure using a strong greedy algorithm [2, 4]. In this work [1] it is applied to a two dimensional Rayleigh-Bénard problem with constant viscosity that depends on the Rayleigh number R, P (φ(R), R) = ~0, as follows: 0 = ∇ · ~v , in Ω, 1 (∂t~v + ~v · ∇~v ) = R θe~3 − ∇P + ν∆~v , in Ω, Pr ∂t θ + ~v · ∇θ = ∆θ, in Ω,

(1) (2) (3)

with boundary conditions defined in Ref. [1], where Ω = [0, Γ]×[0, 1], φ = (~v , θ, P ) and ~v is the velocity vector field, θ is the temperature field, P is the pressure, ~e3 is the unitary vector in the vertical direction and P r the Prandtl number. The classical approximation scheme used here to solve the stationary version of equations (1-3) with the corresponding boundary conditions for different values of the Rayleigh number R is a Legendre spectral collocation method. A linear stability analysis of these solutions has been performed in [3]. The value of the aspect ratio Γ = 3.495 has been chosen and R varies in two intervals: [1, 150; 3, 000] associated to the upper branch of stationary solutions after the primary Pitchfork bifurcation and [1, 560; 3, 000] associated to the upper branch of stationary solutions after the secondary Pitchfork bifurcation. We apply the reduced basis method within this framework to compute the stable solutions corresponding to many values of R on these two branches. For each branch, the reduced basis has been obtained by considering a greedy approach on the corresponding branch. In figure 1 the projection error of stationary solutions on the space generated by the reduced basis is represented. From this figure we check the maximum error is upper bounded by O(10−6 ), the projection on the reduced basis space is a good approximation to a stationary solution. The problem is numerically solved by the Galerkin variational formulation using the Legendre Gauss-Lobatto quadratureP formulas together with the reduced basis N {φ(Ri ), i = 1, 2, ...N } such that φ(R) ∼ i=1 λi φ(Ri ). The difference between the solution obtained with the reduced basis method and the solution obtained with Legendre collocation in case R ∈ [1, 150; 3, 000] on the upper branch is O(10−4 ). A rather simple post-processing allows to recover the same accuracy as the projection O(10−6 ) from the Reduced Basis Galerkin approximation. The reduced basis method permits to speed up the computations of these solutions at any value of the Rayleigh number chosen in a fixed interval associated with a single bifurcation branch while maintaining accuracy.

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References [1] H. Herrero, Y. Maday and F. Pla. RB (Reduced basis) for RB (Rayleigh-Bénard). Computer Methods in Applied Mechanics and Engineering (to appear) (2013). [2] Y. Maday, A.T. Patera and G. Turinici. Convergence theory for reduced-basis approximations of single-parameter elliptic partial differential equations, J. Sci. Comput. 17, no. 1-4, 437-446 (2002). [3] F. Pla, A.M. Mancho and H.Herrero. Bifurcation phenomena in a convection problem with temperature dependent viscosity at low aspect ratio. Physica D 238 , 572-580 (2009). [4] C. Prud’homme, D.V. Rovas, K. Veroy, L. Machiels, Y. Maday, A.T. Patera and G. Turinici. Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. Journal of Fluids Engineering, 124 (1), 70-80 (2002).

Figure 1: Errors of the projections of the global stationary solutions on the reduced basis. Blue lines correspond to the branches of the primary bifurcation and red lines to the branches of the secondary bifurcation. Joint work with Yvon Maday, and Francisco Pla.

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Martin Hess Max Planck Institute Magdeburg, DE Reduced Basis Methods for Maxwell’s Equations with Stochastic Coefficients Contributed Session CT3.5: Thursday, 18:00 - 18:30, CO016 Parametrized partial differential equations (PDEs) in many-query and real-time settings require the solution of high-dimensional systems for a large number of parameter configurations. We discuss the Reduced Basis Method (RBM) for timeharmonic Maxwell’s equations under deterministic and stochastic parameters. We consider the time-harmonic Maxwell’s equation ∇ × µ−1 ∇ × E + iωσE − ω 2 E = iωJ

in Ω,

(1)

in the electric field E with permeability µ, conductivity σ and permittivity . The considered frequency is ω, the source current density J and i the imaginary unit. The equation (1) is considered in the computational domain Ω. To study material uncertainties, we introduce a stochastic coefficient (x; ω) with x ∈ Ω and ω the stochastic inputs. The RBM model reduction significantly reduces the system size while preserving a certified accuracy by employing rigorous error estimators, see [1]. The uncertainty in the coefficients is considered in the Karhunen-Loève expansion.

(x; ω) =

∞ p X λk ξk (ω)k (x),

(2)

k=1

with λk and k (x) the eigenvalues and eigenfunctions of the covariance operator and ξk (ω) uncorrelated random variables with zero mean and variance 1. To quantify the statistical outputs like mean and variance for Maxwell’s equations under stochastic uncertainties, we combine the techniques from [1] with sparse grid approaches as presented in [3]. As an alternative, we review the method introduced in [2] and show how to apply it to (1). Numerical experiments are performed on large-scale 3D models of microwave semiconductor devices. [1] B. Haasdonk, K. Urban, B. Wieland, Reduced Basis Methods for Parametrized Partial Differential Equations with Stochastic Influences using the KarhunenLoève Expansion, SIAM Journal on Uncertainty Quantification, (2013). Accepted [2] P. Nair, A. Keane, Stochastic Reduced Basis Methods, American Institute Aeronautics and Astronautics Journal (2002) 40, (8), 1653-1664. [3] B. Peherstorfer, S. Zimmer, H.-J. Bungartz, Model Reduction with the Reduced Basis Method and Sparse Grids, Publisher: In Sparse Grids and Applications, Volume 88 of Lecture Notes in Computational Science and Engineering, (2013), Springer. Joint work with Peter Benner.

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Jan Hesthaven Brown University, USA High-order accurate reduced basis multiscale finite element methods Plenary Session: Thursday, 09:10 - 10:00, CO1 The development of numerical methods for problems with highly oscillating coefficients remains an active and important field of research. To overcome the computational cost of resolving the fine scale(s), multiscale finite element methods (MsFEM) have been proposed and developed by several authors - see [1] for an overview and introduction. In such methods, accuracy is achieved by solving a local fine scale problem to build the multiscale finite element basis functions needed to capture the small scale information of the leading order differential operator. Alternative approaches to this particular technique are several, e.g., the multiscale variational method [2] and the heterogeneous multiscale method(HMM) [3], although they all share some aspects. In this presentation we focus on techniques most naturally formlated as multiscale finite element methods and develop a new multiscale finite element method for problems with highly oscillating coefficients. The method, discussed first in the context of elliptic problems, is inspired by the multiscale finite element method developed in [4]. Howeverm rather than using a composition rule, a more explicit nonconforming multiscale finite element space is constructed. Accuracy is ensured by using a Petrov-Galerkin formulation and oversampling techniques to reduced the impact of the resonance error. We show that the method is natural for highorder finite element methods used with advantage to solve the coarse grained problem and discuss optimal error estimates. Following related past work [5,6,7], we consider the use of a reduced model to accurately and efficiently represent the multiscale basis. For uniform rectangular meshes, the local oscillating test functions are most naturally parametrized by the centers of the elements. For triangular meshes, inspired by the idea that oversampled oscillating test functions yield a better approximation of the global map, we propose to first build the reduced basis set on uniform rectangular elements containing the original triangular elements and then restrict the oscillating test function on triangular elements. This approach allows for the development of efficient and accurate multiscale methods on general unstructured grids and can also be generalized to the case where coefficients dependent on other independent parameters. Time permitting, we shall discuss the extension to include the development of efficient high-order accurate multiscale methods for wave problems where the highorder accuracy of the coarse solver is of particular value. Throughout the presentation we shall illustrate the behavior and results with computational examples [1] Y. Efendiev and T. Y. Hou, Multiscale Finite Element Methods: Theory and Applications, Springer, 2009. [2] T. J. R. Hughes, G. R. Feijoo, L. Mazzei, and J-B. Quincy The variational multiscale methoda paradigm for computational mechanics, Computer Methods in Applied Mechanics and Engineering 166(12), 3–24,1998. [3] A. Abdulle, W. E, B. Engquist and E. Vanden-Eijnden, The heterogeneous multiscale method, Acta Numerica, 2012, 1–87. [4] G. Allaire and R. Brizzi, A multiscale finite element method for numerical homogenization, SIAM MMS 4, (2005) 790-812.

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[5] N.C. Nguyen, A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales. J. Comput. Phys., 227 (2007) 9807–9822. [6] S. Boyaval, Reduced-Basis approach for homogenization beyond the periodic setting, Multiscale Model. Simul. 7 (1) (2008) 466–494. [7] A. Abdulle and Y. Bai, Reduced basis finite element heterogeneous multiscale method for high-order dis- cretizations of elliptic homogenization problems, J. Comput. Physics, 231, 21, 2012, 7014–7036. Joint work with S. Zhang, and X. Zhu.

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Holger Heumann Department of Mathematics, Rutgers University, New Brunswick, NJ, US Stabilized Galerkin for Linear Advection of Differential Forms Minisymposium Session MMHD: Thursday, 15:30 - 16:00, CO017 The spaces H(curl) and H(div) are the natural spaces of the various vector fields in Maxwell’s equations and magnetohydrodynamics. In the language of exterior calculus, the vector fields in these two spaces correspond either to 1-forms or 2-forms; that means, that we differentiate between vector fields that have a welldefined action on lines (e.g. the electric field E) and well-defined action on surfaces (e.g. the magnetic induction B). The so-called Lie-derivative is the natural linear advection operator for differential forms. It is the generalization of the directional derivative of scalar functions to differential forms and measures the rate of change of the action of a differential form on advected manifolds. In this talk we will exploit such structural properties to formulate and analyze stabilized Galerkin methods for linear advection problems of vector fields. We will pay particular attention to stabilized Galerkin methods with H(curl)- and H(div)-conforming approximation spaces [1]. Vector fields of H(curl)- and H(div)conforming approximation spaces are globally continuous only in certain components. Hence, stabilized Galerkin with these spaces is beyond the existing terminology of stabilized methods for globally continuous or discontinuous approximation spaces. [1 ] H. Heumann and R. Hiptmair, Stabilized Galerkin methods for magnetic advection, SAM-report, 2012-26, submitted to M2AN. Joint work with Ralf Hiptmair.

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Christian Himpe WWU Muenster - Institute for Computational and Applied Mathematics, DE Combined State and Parameter Reduction of Large-Scale Hierarchical Systems Minisymposium Session ROMY: Thursday, 14:00 - 14:30, CO016 Hierarchical systems have widespread use in computer science and applied mathematics. One example is a hierarchical network, distributing information from a single source. Such a system, with L levels and a maximum of M children per L+1 node can be treated as a N × N linear system, with N = M M −1 . In large scale settings, with many levels or many nodes per level, such that N  1, the issue of reducibility arises to cap computational cost. By treating this system as a linear control system with a single input and unit output of the leaf nodes, the proven model reduction concept of balanced truncation commends oneself. Here random but stable, rooted M -ary trees with parametrized edges are explored in terms of state reduction, parameter reduction and combined reduction by the use of empirical gramians, from the emgr framework. Joint work with Mario Ohlberger.

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Michael Hintermueller Humboldt-University of Berlin, DE An adaptive finite element method for variational inequalities of second kind with applications in L2-TV-based image denoising and Bingham fluids Minisymposium Session ADFE: Wednesday, 12:00 - 12:30, CO016 Adaptive finite element methods for a class of variational inequalities of second kind are studied. In particular problems related to the first order optimality system of a total variation regularization based variational model with L2-data-fitting in image denoising (L2-TV problem) as well as to Bingham fluids are highlighted. For a finite element discretization of the variational inequality problem, an a posteriori error residual based error estimator is derived and its reliability and (partial) efficiency are established. The results are applied to solve the L2-TV problem by means of the adaptive finite element method. The adaptive mesh refinement relies on the newly derived a posteriori error estimator and, in the case of image processing, on an additional heuristic providing a local variance estimator to cope with noisy data. The numerical solution of the discrete problem on each level of refinement is obtained by a superlinearly convergent algorithm based on Fenchel-duality and inexact semismooth Newton techniques and which is stable with respect to noise in the data. Numerical results justifying the advantage of adaptive finite elements solutions are presented.

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Michael Hintermueller Humboldt-University of Berlin, DE Optimal shape design subject to elliptic variational inequalities Minisymposium Session GEOP: Tuesday, 11:30 - 12:00, CO122 The shape of the free boundary arising from the solution of a variational inequality is controlled by the shape of the domain where the variational inequality is defined. Shape and topological sensitivity analysis is performed for the obstacle problem and for a regularized version of its primal-dual formulation. The shape derivative for the regularized problem can be defined and converges to the solution of a linear problem. These results are applied to an inverse problem and to the electrochemical machining problem. Joint work with A. Laurain.

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Marlis Hochbruck Karlsruhe Institute of Technology, Germany Error analysis of implicit Runge-Kutta methods for discontinuous Galerkin discretizations of linear Maxwell’s equations Minisymposium Session TIME: Thursday, 10:30 - 11:00, CO015 In this talk we present an error analysis for implicit Runge–Kutta methods for linear Maxwell’s equations. We start with the time discretization and write consider Maxwell’s equations as an abstract initial value problem u0 (t) = Au(t) + f (t),

u(0) = u0

on a suitable Hilbert space H. Here, A is an unbounded operator with domain D(A) ⊂ H. Since A is a skew-symmetric operator on its domain, we consider Gauß collocation methods with constant time step size τ . These methods are unconditionally stable and have nice geometric properties. Our error analysis is based on energy technique discussed in [1]. For s-stage Gauss collocation methods we obtain an order reduction to order s + 1 instead of the classical order 2s of Gauß collocation methods. Next, the we consider the full discretization error. We discretize Maxwell’s equations in space using the discontinuous Galerkin finite element method. This yields the semidiscrete problem u0h (t) = Ah uh (t) + πh f (t),

uh (0) = πh u0 .

Here h denotes maximum diameter of the finite elements, Ah denotes the discrete operator that approximates A, and πh denotes the L2 -projection onto the finite element space. It is well known that the spatial error for general simplicial meshes is of size O(hp+1/2 ), where p denotes the degree of the polynomials used in the finite elements. As in the continuous case we apply Gauß collocation methods to discretize in time. We can prove that the full discretization error is of size O(hp+1/2 + τ s+1 ). Finally, we illustrate our theoretical results by numerical experiments. References [1] C. Lubich, A. Ostermann, Runge-Kutta approximation of quasi-linear parabolic equations, Mathematics of Computation, Vol. 64, No. 210, pp 601-628, 1995.

Joint work with Tomislav Pazur.

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Haakon Hoel KAUST university, SA On non-asymptotic optimal stopping criteria in Monte Carlo simulations Contributed Session CT4.8: Friday, 09:20 - 09:50, CO123 We consider the setting of estimating the mean of a random variable Xi by a sequential stopping rule Monte Carlo (MC) method: Given small, fixed constants T OL > 0 and δ > 0, and the estimation goal ! M X Xi − µ > T OL ≤ δ, (1) P M i=1

our objective is to construct a sequential stopping rule method for determining the number of samples M (T OL, δ) that is required to ensure that (1) is met. The performance of a typical second moment based sequential stopping rule MC method, determining M (T OL, δ) by means of sequential samples of the sample variance, is shown to be unreliable in such settings both by numerical examples and through analysis. By analysis and approximations, we construct a higher moment based stopping rule which is shown in numerical examples to perform more reliably and only slightly less efficiently than the second moment based stopping rule.

Figure 1: A Pareto distributed r.v. Xi with parameters α = 3.1 and p xm = (α − 1) (α − 2)/α is sampled. Plots of the function f (T OL, δ) = PM (T OL,δ) δ −1 P (|M (T OL, δ)−1 i=1 Xi − µ| > T OL) where M (T OL, δ) is generated by a second moment based stopping rule in the top plot and by our higher moment based stopping rule in the lower plot.

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Joint work with Christian Bayer, Erik von Schwerin, and Raul Tempone.

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Johan Hoffman KTH Royal Institute of Technolgy, SE Adaptive finite element methods for turbulent flow and fluid-structure interaction: theory, implementation and applications Minisymposium Session ADFE: Wednesday, 10:30 - 11:00, CO016 We present recent advanced in the area of adaptive finite element methods for turbulent flow and fluid-structure interaction, including high performance parallel algorithms and software implementation in the open source software project FEniCS. Basic theory is presented, as well as applications to aerodynamics, aeroacoustics and biomedicine, including modeling of the turbulent flow past a full aircraft, the blood flow in the human heart, and phonation by simulation of the fluid-structure interaction of air and the vocal folds. Joint work with Johan Jansson, Aurélien Larcher, Niclas Jansson, Rodrigo Vilela De Abreu, Jeannette Hiromi Spühler, Kaspar Müller, and Cem Niyanzi Degirmenci.

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Jiří Holman CTU in Prague, Faculty of Mechanical Engineering, Dept. of Technical Mathematics, CZ Numerical Simulation of Compressible Turbulent Flows Using Modified Earsm Model Contributed Session CT4.5: Friday, 09:20 - 09:50, CO016 This work deals with the numerical solution of compressible turbulent flows. Turbulent flows are modeled by the system of averaged Navier-Stokes equations [6] closed by the Explicit Algebraic Reynolds Stress Model (EARSM) of turbulence [5]. The EARSM model used in this work is based on the Kok’s TNT model equations [4]. New set of model constants which is more suitable for conjunction with EARSM model has been derived. The most crucial part is calibration of the diffusion constants. Their values are determined from the simplified model behavior near the outer edges of the shear layer. Kok derived requirements in form of the set of inequations for the diffusion constants. Hellsten shows [2] that this set of inequations si not valid for the nonlinear constitutive relations and proposes new set of inequations which respects nonlinear behavior of the EARSM model. We are using this inequations to obtain new diffusion constants for the closure TNT model equations. Recalibrated model of turbulence together with the system of averaged NavierStokes equations is then solved by the in-house made software based on the finite volume method [1]. Inviscid numerical fluxes are approximated by the HLLC Riemann solver with the piecewise linear MUSCL or WENO reconstruction [3]. Viscous numerical fluxes are approximated by the central differencing with aid of the dual mesh [3]. The resulting system of ordinary differential equations is then solved by the explicit two-stage TVD Runge-Kutta method with local time-step and point implicit treatment of the source terms [3]. For validation purposes the subsonic flow over a flat plate was solved at first. From Figure 1 one can see that velocity profile obtained with EARSM model with original Kok’s model constants has qualitatively wrong shape. On the other hand, recalibrated model is in good argeement with Hellsten model. Example of application in external aerodynamics is flow around the RAE 2822 airfoil (AGARD case 10: M∞ = 0.754, α∞ = 2.57◦ , Re = 6.2 · 106 ). This problem is transonic flow with small separation of the boundary layer due to interaction with shockwave. From Figure 2 we can see very good agreement of the modified EARSM model with experiment.

References [1] Ferziger J. H., Peric M.: Computational Methods for Fluid Dynamics. Springer, 1999. [2] Hellsten, A.: New two-equation turbulence model for aerodynamics applications, Report A-21, Helsinki University of Technology, 2004 [3] Holman, J.: Numerical solution of compressible turbulent flows in external and internal aerodynamics, Diploma thesis CTU in Prague, 2007 (in czech) [4] Kok, J. C.: Resolving the Dependence on Freestream Values for k − ω Turbulence Model, AIAA Journal, Vol. 38, No. 7, 2000 171

[5] Wallin, S.: Engineering turbulence modeling for CFD with focus on explicit algebraic Reynoldes stress models, Ph.D. thesis, Royal Institutte of Technology, 2000 [6] Wilcox, D. C.: Turbulence Modeling for CFD, Second Edition, DWC Industries, 1994

Figure 1: Velocity profile

Figure 2: Comparison with experimental data Joint work with Jiří Fürst.

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Thomas Huckle Technische Universitaet München, DE Tensor representations of sparse or structured vectors and matrices Minisymposium Session LRTT: Wednesday, 12:00 - 12:30, CO3 In this talk we derive tensor train representations for vectors/matrices with special symmetries that often appear in applications. Typical symmetries are persymmetry, centrosymmetry, translation invariance. These representations can be helpful to derive faster and more accurate solutions esp. for Matrix Product States in Quantum simulation. Furthermore, we discuss tensor train forms of sparse vectors that might be useful for applications like high-dimensional compressed sensing or ill-posed inverse problems with sparse solution data.

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Matteo Icardi KAUST, SA Bayesian parameter estimation of a porous media flow model Contributed Session CT2.7: Tuesday, 15:30 - 16:00, CO122 Porous materials appear in a number of important industrial applications, in particular related to subsurface flows, membranes and filters, materials modeling, structural mechanics, etc. The exact micro-scale description of these materials is often impractical, therefore, average models are usually used to describe, for example, the flow and transport in such porous media. These models are, however, often affected by high uncertainty (and therefore unreliability) because of the high heterogeneity and multi-scale structure of the media. Homogenization and spatial averaging methods can be applied to the fluid flow equations in the pore space, under the assumption that the scales are well separated, obtaining the well-known Darcy equation (and Forchheimer extension for non-linear regime). In this case, the parameters appearing in the Darcy equation (porosity and permeabilities) are well defined and in general can be represented at the field scale with anisotropic and inhomogeneous scalar fields. The same holds for other upscaled equations such as the Advection-Dispersion-Reaction (ADR) equation for solute transport and its parameters (reaction rates and diffusivity tensor). These parameters are usually more affected by uncertainty and variability than the parameters of the flow (Darcy) equation. In this work, we assume an equation for macro-scale scalar transport (ADR equation) with unknown parameters (effective porosity and diffusivity) and we setup a statistical method, based on the Bayesian framework, to validate the model and estimate a distribution for the unknown parameters. The data used to solve the inverse problem are the results of accurate pore-scale computational fluid dynamics (CFD) simulations. Three-dimensional geometries, representing actual sand samples, have been reconstructed and the flow field and scalar transport are solved with a finite volume code on an unstructured octree-based mesh. The micro-scale results are then averaged and used as data for solving the inverse problem at the macro-scale. The one-dimensional macro-scale forward problem is solved numerically with standard finite difference techniques. A Bayesian method has been implemented to estimate the overall hydrodynamic dispersion under different flow rates and bulk (molecular) diffusivities. In this problem, in addition to the standard measurement normal-distributed error, also the possible modeling error in the selection of the macro-scale model should also be taken into account. The results are in agreement with previous works on dispersion in porous media but highlight the inaccuracy of the simple model assumed at the macro-scale for realistic irregular three-dimensional geometry. Possible modifications and extension are then discussed to take into account the stagnant zone and the effect of the boundary conditions.

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Figure 1: Posterior distribution of porosity and dispersion parameters.

Figure 2: Solute concentration data, least square fitting and maximum a posteriori estimation.

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Reijer Idema Delft University of Technology, NL On the Convergence of Inexact Newton Methods Contributed Session CT3.3: Thursday, 17:30 - 18:00, CO3 Assume an iterative method that, given current iterate xi , has a unique new iterate x ˆi+1 . If instead an approximation xi+1 of the exact iterate x ˆi+1 is used to continue the process, we speak of an inexact iterative method. Inexact Newton methods are examples of inexact iterative methods. Let δ c and εc be the distance of the current iterate xi to the exact iterate x ˆi+1 and the solution x∗ respectively. Likewise, let δ n and εn be the distance of the new (inexact) iterate xi+1 to the exact iterate x ˆi+1 and the solution x∗ . The superscript c denotes “current”, while the superscript n denotes “new”. Let further εˆ be the distance of the exact iterate x ˆi+1 to the solution x∗ . For a graphical representation, see Figure 1. n The ratio εεc is a measure for the improvement of the inexact iterate xi+1 over the current iterate xi , in terms of the distance to the solution x∗ . Likewise, the ratio δn δ c is a measure for the improvement of the inexact iterate xi+1 , in terms of the n distance to the exact iterate x ˆi+1 . As the solution is unknown, so is the ratio εεc . n Assume, however, that some measure for the ratio δδc is available and controllable. For example, for an inexact Newton method the relative linear residual norm krk k δn kF (xi )k can be used as a measure for δ c . The aim is to have an improvement in the controllable error translate into a similar improvement in the distance to the solution, i.e., to have εn δn ≤ (1 + α) c c ε δ

(1)

for some reasonably small α > 0. We show that the minimum α for which Equation (1) is guaranteed to hold can be written as: "  # −1 γ δn αmin = +1 , (2) 1−γ δc where γ = δεˆc is a measure for the quality of the exact iterate x ˆi+1 . This means δn that the smaller γ is, the smaller δc can be made without compromising αmin . We combine the above ideas with inexact Newton convergence theory to proof the following theorem, where J(x) denotes the Jacobian of the nonlinear problem, and ηi are the forcing terms. The linearized equations are solved up to an accuracy kri k kF (xi )k ≤ ηi . Theorem: Let ηi ∈ (0, 1) and choose α > 0 such that (1 + α) ηi < 1. Then there exists an ε > 0 such that, if kx0 − x∗ k < ε, the sequence of inexact Newton iterates xi converges to x∗ , with kJ(x∗ ) (xi+1 − x∗ ) k < (1 + α) ηi kJ(x∗ ) (xi − x∗ ) k.

(3)

This theorem implies that, if the initial iterate is close enough to the solution, it is always possible to choose forcing terms ηi such that the method converges without oversolving. With proper choice of the forcing terms, if the initial iterate is close enough to the solution, it is therefore possible to solve a nonlinear problem with 176

an inexact Newton method in such a way that the nonlinear residual converges as fast as the linear residuals of the linearized equations. Numerical experiments on power flow problems [1,2] are presented that illustrate the practical value of these results. [1] R. Idema, D. J. P. Lahaye, C. Vuik, and L. van der Sluis. Scalable NewtonKrylov solver for very large power flow problems. IEEE Transactions on Power Systems, 27(1):390–396, February 2012. [2] R. Idema. Newton-Krylov Methods in Power Flow and Contingency Analysis. PhD Thesis, Delft University of Technology, November 2012.

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Figure 1: Inexact iterative step Joint work with D.J.P. Lahaye, and C. Vuik.

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Hiroki Ishizuka Keio University, JP Simulating information propagation by near-field P2P wireless communication Contributed Session CT1.1: Monday, 18:00 - 18:30, CO1 The main target of this research is information propagation by near-field P2P wireless communication. More specifically, this research focuses on information propagation in an ad-hoc network of P2P communication by bluetooth on smartphone carried by each pedestrian in a city. This phenomenon is similar to that of the spread of disease, in that these are both caused by person-to-person contact. There are a lot of researches about spread of disease. For example, Draief [1] carried out research into the spread of a virus by proposing a simple model of infection by use of graph theory and Markov chain. In that model, there are two walkers, one being infectious and the other healthy. They do random walk on a regular graph and if these two random walkers encounter at the same node by accident, the healthy walker will be infected with a certain probability. In this research, using Markov chain, Draief [1] derived the equation to figure out the time that elapses before the healthy walker is infected. Learning from the above study, we used graph theory to analyze information propagation with bluetooth. Specifically, as fig.1 shows, we treated the field as “almost 8-regular graph” and made pedestrians walk on this graph. The black pedestrian has information and the gray pedestrians don’t. The circle, the center of which is the black pedestrian, shows transmission distance of bluetooth. All smartphones which the pedestrians hold try to connect to other smartphones in certain intervals. If any pedestrian whose smartphone is in the “try connect” mode, moves into that circle when the condition of the black pedestrian’s smartphone is also in the “try connect” mode, he will get information. The most difficult point of this research was for the purpose of making the pedestrians walk realistic, we couldn’t use random walk method, which also meant, we couldn’t use Markov chains. For this reason, it was difficult to express the phenomenon of information propagation with bluetooth in a mathematical form. Therefore, we developed “directed walk algorithm” reflecting real pedestrian movement, and used multi-agent simulation(MAS) to simulate this phenomenon. Changing the number of pedestrian in the field and interval of “try connect”, we found percolation transition of the number of pedestrian who has information. [1] M. Draief, A. Ganesh, “A random walk model for infection on graphs: spread of epidemics & rumours with mobile agent”, Discrete Event Dynamical Systems, Vol.21, pp. 41-61, (2011).

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Figure 1: Pedestrians walk on the graph Joint work with Kenji Oguni.

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Alessandra Jannelli Department of Mathematics and Computer Science, University of Messina, IT Quasi-uniform Grids and ad hoc Finite Difference Schemes for BVPs on Infinite Intervals Contributed Session CT2.6: Tuesday, 15:00 - 15:30, CO017 We consider finite differences schemes on quasi-uniform grids applied to the numerical solution of BVPs defined on infinite intervals. Quasi-uniform grids have successfully applied to the numerical solution of partial differential equations on unbounded domains. We apply the proposed approach to the Falkner-Skan model of boundary layer theory, to a problem of interest in foundation engineering and to a nonlinear problem arising in physical oceanography. Let us consider the smooth strict monotone quasi-uniform map x = x(ξ), the so-called grid generating function, x = −c · ln(1 − ξ) ,

(1)

where ξ ∈ [0, 1], x ∈ [0, ∞], and c > 0 is a control parameter. We notice that xN −1 = c ln N for (1). The problem under consideration can be discretized by introducing a uniform grid ξn of N +1 nodes in [0, 1] with ξ0 = 0 and ξn+1 = ξn +h with h = 1/N , so that xn is a quasi-uniform grid in [0, ∞]. The last interval in (1), namely [xN −1 , xN ], is infinite but the point xN −1/2 is finite, because the non integer nodes are defined by   n+α , (2) xn+α = x ξ = N with n ∈ {0, 1, . . . , N − 1} and 0 < α < 1. This map allows us to describe the infinite domain by a finite number of intervals. The last node of such grid is placed on infinity so right boundary condition is taken into account correctly. Top frame of figure shows the quasi-uniform mesh defined by (1) with c = 5 and N = 20. We can define the values of u(x) on the mid-points of the grid un+1/2 ≈

xn+1 − xn+1/2 xn+1/2 − xn un + un+1 . xn+1 − xn xn+1 − xn

(3)

As far as the first derivative is concerned we apply the following approximation du un+1 − un
. ≈ (4) dx n+1/2 2 xn+3/4 − xn+1/4 This formulae uses the value uN = u∞ , but not xN = ∞. In order to justify finite difference formula (4), by considering u = u(ξ(x)), we can write
du du dξ un+1 − un 2 ξn+3/4 − ξn+1/4
. (5) = ≈ dx n+1/2 dξ n+1/2 dx n+1/2 ξn+1 − ξn 2 xn+3/4 − xn+1/4 The last formula on the right hand side of equation (5) reduces to the right hand side of equation (4)
because we are using a uniform grid for ξ and therefore 2 ξn+3/4 − ξn+1/4 = ξn+1 − ξn . The two finite difference approximations (3) and (4) have order of accuracy O(N −2 ). Figure shows the numerical solution of Falkner-Skan model with β = 1 obtained by (1) with c = 5 for N = 80. 180

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Bärbel Janssen Universität Bern, CH The hp-adaptive Galerkin time stepping method for nonlinear differential equations with finite time blow-up Contributed Session CT1.5: Monday, 18:30 - 19:00, CO016 We consider hp-adaptive Galerkin time stepping methods for nonlinear ordinary differential equations. The occuring nonlinearity is assumed to be bounded by a constant times the solution to a power β which is larger than one. We prove dual based a posteriori error estimates. Existence of discrete solutions is shown using reconstruction techniques. By means of numerical examples we show that the blow-up is well preserved. Joint work with Thomas P. Wihler.

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Manuel Jaraczewski Helmut Schmidt University, DE On the asymptotics of discrete Riesz energy with external fields Contributed Session CT3.8: Thursday, 17:00 - 17:30, CO123 Potential theory has been intensively studied for a long time due to its intrinsic relations to many other fields both in physics and in mathematics [2]. Particular for plane sets, the close connection of logarithmic potentials and complex analysis offers an extremely rich theory, see, e.g., [6]. During the last 20 years an increasing interest in algorithmic and computational aspects of potential theory has arisen. This among other aspects motivated research in the discrete counterparts of potentials and minimal energy. Many techniques related to discrete minimal logarithmic energy or minimal Newton energy can be transferred to the more general setting of s-Riesz energy (s ≥ 0) in Rd (d ≥ 1). It is well know that the discrete minimal s-Riesz energy of a system of n points on a compact set Ω ⊆ Rd converges to its continuous counterpart as n tends to infinity, if the latter exists, i.e., if 0 ≤ s < d. The corresponding asymptotic behavior and discrete minimal configurations have been intensively studied. However, apart from many results in the complex plain (see, e.g., [6]) most of the higher dimensional investigations focus on the sphere or on a torus: For the sphere in Rd , e.g., explicit error bounds for the asymptotic approximation of the continuous s-Riesz energy by discrete energy follow from results due to Wagner [7], and due to Kuijlaars and Saff [1]. This work deals with two extensions of the theory of discrete minimal energy problems: First, an estimate on the asymptotic behavior of the discrete s-Riesz energy in the cases 0 < s ≤ d − 2 for a large class of sets is derived. It turned out that Ahlfors-David regularity (see, e.g., [3]) is a suited notion of regularity to derive an estimate on the asymptotic behavior of the discrete minimal s-Riesz energy. This is a very mild hypothesis, which is fulfilled by a large class of sets, including images of a ball under Bilipschitz maps. The s-Riesz potentials with 0 ≤ s ≤ d − 2 are (super) harmonic and, hence, the equilibrium measure of the potentials concentrates on the outer boundary of the considered set Ω ⊆ Rd . As a consequence, results for d-Ahlfors-David regular sets carry over to sets that bound an Ahlfors-David regular set. The second extension is related to minimal s-Riesz energy in the presence of an external field. This has been intensively studied by, e.g., Saff and Totik in the case of logarithmic potentials [5], where it leads to the notion of weighted extremal points. We, hence, examine the asymptotic behavior of discrete minimal s-Riesz energy under an external field. Finally, it is discussed, in how far known relations can be transferred to the more general setting, such as, e.g., the connection of extremal points and good quadrature formulae. References: [1] A. B. J. Kuijlaars and E. B. Saff, Asymptotics for minimal discrete Energy on the Sphere, Transactions of the American Mathematical Society, Volume 350, Number 2, 523-538 (1998) [2] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag Berlin Heidelberg New York (1972) [3] P. Mattila and P. Saaranen, Ahlfors-David regular sets and Bilipschitz maps, Annales Academiæ Scientiarum Fennicæ Mathematica, Volume 34, 487-502 (2009)

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[4] E. B. Saff and A. B. J. Kuijlaars, Distributing Many Points on a Sphere, The Mathematical Intelligencer, Springer-Verlag New York, Volume 19, Number 1 , 5-11 (1997) [5] E. B. Saff and V. Totik, Logarithmic Potentials with external Fields, Grundlehren der mathematischen Wissenschaften 316, Springer-Verlag Berlin Heidelberg (1997) [6] M. Tsuji, Potential theory in modern function theory, 2nd edition, Chelsea Publishing Company, New York (1975) [7] G. Wagner, On Means of Distances on the Surface of a Sphere (Lower Bounds), Pacific Journal of Mathematics, Volume 144, No. 2, 389-398 (1990) Joint work with M. Stiemer.

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Elias Jarlebring KTH Royal Institute of Technology, SE An iterative block algorithm for eigenvalue problems with eigenvector nonlinearities Minisymposium Session NEIG: Thursday, 11:30 - 12:00, CO2 Let A(V ) ∈ Rn×n be a symmetric matrix depending on V ∈ Rn×k which is a basis of a vector space, and suppose A(V ) is independent of the choice of basis of the vector space. We here consider the problem of computing V such that (Λ, V ) is an invariant pair of the matrix A(V ), i.e., A(V )V = V Λ. We present a block algorithm for this problem, where every step involves solving one or several linear systems of equations. We show that the algorithm is a generalization of (shift-andinvert) simultaneous iteration for the standard eigenvalue problem and that the generalization inherits many of its properties. The algorithm is illustrated with the application to a model problem in quantum chemistry.

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Pavel Jiranek CERFACS, FR A general framework for algebraic multigrid methods Minisymposium Session CTNL: Tuesday, 11:00 - 11:30, CO015 Algebraic multigrid methods (AMG) form a popular class of solvers and preconditioners for large sparse systems of linear algebraic equations arising mainly in the context of discretised partial differential equations due to their scalability properties inherited from their geometric counterpart. Unlike in geometric multigrid, AMG constructs the hierarchy of levels using solely the algebraic information contained in the system to be solved and thus can be easily applied in the “black-box” manner in practice. Various AMG algorithms and software packages implementing them exist nowadays and differ essentially in the way how the coarsening on the fine levels is realised and how the transfer operators are constructed on the given coarsening. Main representatives of different coarsening approaches are classical AMG methods (where the coarse grid is identified with certain independent subset of the fine-grid variables) and aggregation-based methods (where the coarse grid is associated with some disjoint subsets of the fine-grid variables). The basic AMG approaches for solving scalar problems can be also usually extended to more general problems including systems of partial differential equations and indefinite saddle point problems. One of the drawbacks of the most of the existing AMG implementations is the focus on a particular AMG scheme and to some extent to a fixed problem type while there are certainly various multigrid components which are common to any AMG implementation. Our attempt is to create a general object-oriented environment for AMG methods which would cover this gap and allow to realise essentially any kind of AMG method for a large variety of problem types in a single framework. The setup of the multigrid hierarchy is realised by a set of interconnected components implementing certain elementary part of the coarsening algorithm and because of their hierarchical object structure they can be easily modified and extended with new features. The general design of the setup process also allows to reuse these elementary algorithms for more general types of problems including structured saddle point systems arising, e.g., in the mixed finite element method. We illustrate the use of the framework and its parallel performance on some academic test problems including practical problems arising in reservoir simulations.

Joint work with S. Gratton, X. Vasseur, and P. Henon.

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Lorenz John Institute of Computational Mathematics, Graz University of Technology, AT A multilevel preconditioner for the biharmonic equation Contributed Session CT4.2: Friday, 08:20 - 08:50, CO2 We present a multilevel preconditioner for the mixed finite element discretization of the biharmonic equation of first kind. While for the interior degrees of freedom a standard multigrid method can be applied, a different approach is required on the boundary. The construction of the preconditioner is based on a BPX type multilevel representation in fractional Sobolev spaces. Numerical examples illustrate the obtained theoretical results. Joint work with Olaf Steinbach.

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Pierre Jolivet Laboratoire Jacques-Louis Lions, FR How to easily solve PDE with FreeFem++ ? Minisymposium Session PARA: Monday, 14:30 - 15:00, CO016 Implementing a finite element software that can support arbitrary meshes and arbitrary finite elements spaces can be highly time-consuming. In this talk, FreeFem++ will be presented. It is a simple Domain Specific Language that can be used to quickly solve partial differential equations given their variational formulation. In the first part of the talk, the inner workings of the language will be explained (lexical and syntactical analysis and code generation). Then, we will move on to the second part of the talk which explains how FreeFem++ can be used in conjuction with a simple framework for domain decomposition methods to solve problems on large scale architectures. Joint work with Frédéric Hecht, Frédéric Nataf, and Christophe Prud’homme.

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Mika Juntunen Aalto University Department of Mathematics and Systems Analysis, FI A posteriori estimate of Nitsche’s method for discontinuous material parameters Contributed Session CT1.9: Monday, 17:30 - 18:00, CO124 One of the advantages of the Nitsche’s method is the simplicity of joining subdomains with non-matching meshes. If the division follows material boundaries, the parameters are discontinuous over the subdomain interfaces. If the jump in the material parameters is moderate, the straightforward extension of the method as it was described in, e.g., [3] readily applies but large discontinuities may lead to poor results [1,4]. Some of the problems are avoided with material parameter weighted mean flux [1,5] but to fully avoid the problems the stabilizing terms need to be modified too [2]. In this work we propose Nitsche’s method for discontinuous material parameters and derive a residual based a posteriori estimate for the method. Both the method and the a posteriori estimate take the discontinuity in the material parameter carefully into account. If the material parameters are continuous, the method reduces to the straightforward method. In the case of extreme discontinuity, the method reduces to setting Dirichlet boundary conditions with Nitsche’s method. The straightforward a posteriori estimate tends to over-refine the mesh near the interface if the material parameters have large discontinuity. The proposed a posteriori estimate inherits the good properties of the method and avoids the overrefinement even in the case of extreme discontinuity of the material parameters. The derived method and a posteriori estimate are tested numerically for a Poisson model problem. [1] Chandrasekhar Annavarapu, Martin Hautefeuille, and John E. Dolbow. A robust Nitsche’s formulation for interface problems. Computer Methods in Applied Mechanics and Engineering, 225-228:44–54, 2012. [2] Nelly Barrau, Roland Becker, Eric Dubach, and robert Luce. A robust variant of NXFEM for the interface problem. C. R. Math. Acad. Sci. Paris, 350(15–16):789–792, 2012. [3] Roland Becker, Peter Hansbo, and Rolf Stenberg. A finite element method for domain decomposition with non-matching grids. M2AN Math. Model. Numer. Anal., 37(2):209–225, 2003. [4] Tod A. Laursen, Michael A. Puso, and Jessica Sanders. Mortar contact formulations for deformable-deformable contact: past contributions and new extensions for enriched and embedded interface formulations. Comput. Methods Appl. Mech. Engrg., 205/208:3–15, 2012. [5] Rolf Stenberg. Mortaring by a method of J. A. Nitsche. In Computational mechanics (Buenos Aires, 1998), pages CD–ROM file. Centro Internac. Métodos Numér. Ing., Barcelona, 1998.

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Ashraful Kadir Royal Institute of Technology, SE How accurate is molecular dynamics for crossings of potential surfaces? Part II: numerical tests Contributed Session CT4.9: Friday, 08:50 - 09:20, CO124 I will present numerical examples related to the talk given by Prof. Szepessy on ‘How accurate is molecular dynamics for crossings of potential surfaces? Part I: error estimates’. The numerical tests show that the Schrödinger observables are approximated with the error estimate O(pe + M −1/2 ) by molecular dynamics observables, where pe is the probability for an electron to be in an excited state and M is the nuclei-electron mass ratio. A numerical algorithm is developed to approximate pe based on Ehrenfest molecular dynamics simulations, which enables the practical use of the error estimate. I will compare the approximated pe with the solutions obtained from the discrete time-independent Schrödinger eigenvalue problems for crossings and near avoided crossings of potential surfaces, see Figure 1. Based on numerical tests the talk will explain the approximation results: namely the discretization error, the sampling error and the modeling error. The time discretization error comes from approximating the differential equation for molecular dynamics with a numerical method, based on replacing time derivatives with difference quotients for a positive step size ∆t. The sampling error is due to truncating the infinite T in an ergodic limit and using a finite value of T . The modeling error originates from eliminating the electrons in the Schrödinger nuclei-electron system and replacing the nuclei dynamics with their classical paths; this approximation error was first analyzed by Born and Oppenheimer.

Figure 1: Plots showing pe + M −1/2 with conical intersections at (a1 , 0). Joint work with Håkon Hoel (KAUST), Petr Plechac (Univ. Delaware), Mattias Sandberg (KTH), and Anders Szepessy (KTH).

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Dante Kalise Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austria An accelerated semi-Lagrangian/policy iteration scheme for the solution of dynamic programming equations Minisymposium Session NMFN: Monday, 11:40 - 12:10, CO2 We present some recent results concerning the efficient numerical approximation of static Hamilton-Jacobi-Bellman equations of the form λu(x) + sup {−f (x, a) · Du(x) − g(x, a)} = 0, a∈A

x ∈ Rn ,

characterizing the value function u(x) of an optimal control problem in Rn . One of the main challenges in the solution of this equation relates to its high-dimensionality, and therefore the design of efficient methods turns to be a fundamental task. In this talk we present a scheme based on a semi-Lagragian/finite differences discretization [2] combined with an iterative scheme in the space of policies [1, 3]. Moreover, we exploit the idea that a reasonable initialization of the policy iteration procedure yields a faster numerical convergence to the optimal solution. For such purpose, the scheme features a pre-processing step with value iterations in a coarse grid. A series of numerical tests, spanning a wide variety of applications, assess the robust and efficient performance of the method.

References [1] O. Bokanowski, S. Maroso, H. Zidani,Some convergence results for Howard’s algorithm, SIAM Journal on Numerical Analysis 47 (2009), 3001–3026. [2] E. Carlini, M. Falcone, R. Ferretti,An efficient algorithm for Hamilton-Jacobi equations in high dimension, Computing and Visualization in Science (2004), 15–29. [3] M.S. Santos and J. Rust,Convergence properties of policy iteration, SIAM J. Control Optim., 42 (2004), 2094–2115. Joint work with Alessandro Alla, and Maurizio Falcone.

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Kenichi Kamijo Graduate School of Life Sciences, Toyo University, JP Numerical Method for Fractal Analysis on Discrete Dynamical Orbit in n-Dimensional Space Using Local Fractal Dimension Contributed Session CT4.1: Friday, 08:50 - 09:20, CO1 The orbit of a discrete dynamical system in n-dimensional space can be considered to be a kind of discrete signal. The local fractal dimension (LFD) has been defined and calculated in a finite short “processing window” on the orbit. In order to evaluate the fractal structure in the orbit, a numerical method for signal processing has been proposed. Then, the moving LFD can be obtained by sliding the said window along the line on the orbit. Logistic mapping has been selected at each coordinate as an example, and a computer simulation has been carried out in this paper. It is shown that the probability distribution of the moving LFD becomes almost a normal distribution within the restricted range of the control parameter concerned with logistic time development, in which these parameters raise the socalled chaotic fluctuations up as discrete dynamical orbits. Also, the relationships between the control parameter and the mean or standard deviation of the moving LFD, or SN ratio have been investigated. The proposed method can be applied to statistical quality control or analysis for general random processes with the same procedure. Key words: discrete dynamical system, random process, logistic time development, local fractal dimension, statistical quality control

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Bulent Karasozen Institute of Applied Mathematics, TR Adaptive Discontinuous Galerkin Methods for nonlinear Diffusion-Convection-Reaction Models Contributed Session CT3.4: Thursday, 17:00 - 17:30, CO015 Many engineering problems such as chemical reaction processes, population dynamics, ground water contamination are governed by coupled diffusion-convectionreaction partial differential equations (PDEs) with nonlinear source or sink terms. In the linear case, when the system is convection dominated, stabilized finite elements and discontinuous Galerkin methods are capable of handling the nonphysical oscillations. Nonlinear reaction terms pose additional challenges. Nonlinear transport systems are typically convection and/or reaction dominated with characteristic solutions possessing sharp layers. In order to eliminate spurious localized oscillations in the numerical solutions discontinuity or shock-capturing techniques are applied in combination with the streamline upwind Petrov-Galerkin(SUPG) method. In contrast to standard Galerkin finite element methods, the discontinuous Galerkin methods produce stable solutions without need of extra stabilization techniques to overcome the spurious oscillations for convection dominated problems. In this talk we present the application of adaptive discontinuous Galerkin methods to convection dominated models containing quadratic and Monod type reaction rates. A posteriori error estimates for linear problems in space discretization are extended to PDEs with nonlinear reaction terms. Numerical results for steady state and time dependent coupled systems arising in contaminant biodegradiation process demonstrate the accuracy and efficiency of the adaptive DGFEM compared over the SUPG and shock capturing techniques Joint work with Murat Uzunca.

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Vladimir Kazeev Seminar for Applied Mathematics, ETH Zürich, CH Tensor-structured approach to the Chemical Master Equation Minisymposium Session LRTT: Wednesday, 11:00 - 11:30, CO3 The Chemical Master Equation (CME) is a cornerstone of stochastic analysis and simulation of models of biochemical reaction networks. Yet direct solutions of the CME have remained elusive. Although several approaches overcome the infinite dimensional nature of the CME through projections or other means, a common feature of proposed approaches is their susceptibility to the curse of dimensionality, i.e. the exponential growth in memory and computational requirements with respect to the number of problem dimensions. To “lift” this curse of dimensionality, we use the recently proposed Quantized Tensor Train (QTT) decomposition of high-dimensional tensors. It relies on two key ingredients. The first is quantization which consists in splitting each “physical” dimension into a few virtual levels and results in a tensor with the same entries, but more dimensions and smaller mode sizes. The second is the Tensor Train (TT) representation of high-dimensional arrays based on the separation of variables. The TT representation enjoys two crucial advantages. First, the low-rank approximation of a tensor in the TT format is related to the low-rank approximation of certain matrices related to the tensor in question. Therefore, it can be performed with the use of standard, well-established matrix algorithms. Second, the TT format has complexity and memory requirements which are linear or almost linear in the number of dimensions for many applications. The use of quantization, leading to the QTT decomposition, allows to resolve even more structure in the matrices and vectors involved and to further reduce the complexity and memory requirements. We analyze the QTT structure of the CME and apply the exponentially-converging hp-discontinuous Galerkin discretization in time to reduce it to a QTT-structured system of linear equations to be solved at each time step. As a solver of linear systems, we use an algorithm based on the Density Matrix Renormalization Group (DMRG) approach from quantum chemistry. While there is currently no estimate of the convergence rate for the DMRG algorithm, our numerical experiments show the solver to be highly efficient. We demonstrate the efficiency of our method, compared to Monte Carlo simulations, by applying it to a few examples from systems biology.

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Radka Keslerova Czech Technical University in Prague, Dep. of Tech. Mathematics, CZ Numerical Simulation of Steady and Unsteady Flows for Viscous and Viscoelastic Fluids Minisymposium Session MANT: Tuesday, 11:30 - 12:00, CO017 This work deals with the numerical solution of viscous and viscoelastic fluids flow. The governing system of equations is based on the system of balance laws for mass and momentum for incompressible laminar fluids. Different models for the stress tensor are considered. For viscous fluids flow Newtonian model is used. For the describing of the the behaviour of the mixture of viscous and viscoelastic fluids Oldroyd-B model is used. div u = 0 ∂u ρ + ρ(u.∇)u = −∇P + div Ts + div Te ∂t 2µe 1 ∂Te + (u.∇)Te = D − Te + (W Te − Te W ) + (DTe + Te D) ∂t λ1 λ1 where P is the pressure, ρ is the constant density, u is the velocity vector. The symbols Ts and Te represent the Newtonian and viscoelastic parts of the stress tensor and δTe = 2µe D Ts = 2µs D, Te + λ1 δt where D is symmetric part of the velocity gradient and W is antisymmetric part of the velocity gradient. Numerical solution of the described models is based on cell-centered finite volume method using explicit Runge–Kutta time integration. Steady state solution is achieved for t → ∞. In this case the artificial compressibility method can be applied. The flow is modelled in a bounded computational domain where a boundary is divided into three mutually disjoint parts: a solid wall, an outlet and an inlet. At the inlet Dirichlet boundary condition for velocity vector is used and for a pressure and the stress tensor Neumann boundary condition is used. At the outlet the pressure value is given and for the velocity vector and the stress tensor Neumann boundary condition is used. The homogenous Dirichlet boundary condition for the velocity vector is used on the wall. For the pressure and stress tensor Neumann boundary condition is considered. In the case of unsteady computation dual-time stepping method is considered. The principle of dual-time stepping method is following. The artificial time τ is introduced and the artificial compressibility method in the artificial time is applied. The system of Navier-Stokes equations is extended to unsteady flows by adding artificial time derivatives ∂W/∂τ to all equations. Presented mathematical models are tested in the two and three dimmensional branching channel.

Acknowledgment This work was partly supported by the grant GACR 201/09/0917 and GACR P201/11/1304. 195

References [1] T. Bodnar and A. Sequeira: Numerical Study of the Significance of the NonNewtonian Nature of Blood in Steady Flow through s Stenosed Vessel (Editor: R. Ranacher, A. Sequeira), Advances in Mathematical Fluid Mechanics (2010) 83–104. [2] A.J. Chorin, A numerical method for solving incompressible viscous flow problem, Journal of Computational Physics, 135, 118–125 (1967). [3] A. Jameson, W. Schmidt and E. Turkel, Numerical solution of the Euler equations by finite volume methods using Runge-Kutta time-stepping schemes, AIAA 14th Fluid and Plasma Dynamic Conference California (1981). [4] R. Keslerová and K. Kozel, Numerical solution of laminar incompressible generalized Newtonian fluids flow, Applied Mathematics and Computation, 217, 5125–5133 (2011). [5] R. LeVeque, Finite-Volume Methods for Hyperbolic Problems. Cambridge University Press, (2004). [6] P. Louda, K. Kozel, J. Příhoda, L. Beneš, T. Kopáček: Numerical solution of incompressible flow through branched channels, Journal of Computers & Fluids 46 (2011) 318–324.

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Radka Keslerova Czech Technical University in Prague, Dep. of Tech. Mathematics, CZ Numerical Simulation of the Atmospheric Boundary Layer Flow over coal mine in North Bohemia Contributed Session CT4.5: Friday, 08:20 - 08:50, CO016

1

Introduction

This contribution presents numerical results and wind tunnel measurement obtained for the air flow over a real orography and a scaled-down model of the coal mine in the North Bohemia. Pollution dispersion is one of the critical aspects of economy development. Therefore the accurate prediction of pollutants propagation in the environment is crucial for future industrialization as well as for natural resources exploitation. Within this context the airborne dust dispersion in complex terrain is of major interest. In order to be able to predict the dust pollution it is necessary to explore the air-flow in detail first. The terrain (in-situ) measurements are naturally the best source of information. They are however quite expensive and thus their availability is very limited while solving real-life problems. Thus it is necessary to employ various ways of physical and mathematical modeling as a complement and extension of available meteorological data.

2

Problem description

The solved case is directly based on orography of the opencast coal mine in the North Bohemia. The real area of the mine cover more then 30km2 of the landscape with forests, villages and part of mountains. The wind-tunnel model is based on orography of this mine. The model scale is 1 : 9000. The whole wind tunnel model has horizontal dimensions of 1500 × 1500 mm. The detailed experimental data were collected just in a small portion of this area. The nominal velocity is about 0.25 m/s which means that the corresponding Reynolds number is of the order 103 . More details about the experimental setup and methodology can be found e.g. in [4].

3

Mathematical model and numerical methods

The mathematical model chosen for this case is the system of Navier-Stokes equations for viscous incompressible non-stratified flow. Because of the rather low Reynolds number for the wind tunnel scale experiment, the case is considered as laminar and thus no turbulence model is used. The first numerical scheme used to solve this model is a modification of the semiimplicit finite difference method described in [2]. It uses artificial compressibility formulation to resolve pressure. The governing equations are discretized in a semiimplicit way using a combination of forward and backward differences at time levels n and n+1 which leads to a central scheme with second order of accuracy in space. Numerical stabilization is carried out using a fourth order artificial viscosity. The whole problem is solved by a time-marching technique, where the steady state solution is searched as a limit of unsteady problem solution for time t −→ ∞. 197

The second method is based on the finite volume formulation. The finite volume scheme AUSM scheme is used for the spatial semidiscretization of the inviscid fluxes. Quantities on the cell faces are computed using the MUSCL reconstruction with the Hemker-Koren limiter. The scheme is stabilized by the pressure diffusion. The viscous fluxes are discretized using the central approach on a dual mesh (diamond type scheme). The spatial discretization results in a system of ODE’s which is solved by the second-order BDF formula. Arising set of nonlinear equations is then solved by the artificial compressibility method in the dual time by the explicit 3-stage secondorder Runge-Kutta method. Numerical results obtained by both of these schemes are compared to each other and also the comparison with the experimental data is presented. Influence of the boundary conditions is studied.

References [1] T. Bodnár and L. Beneš. On some high resolution schemes for stably stratified fluid flows. In Finite Volumes for Complex Applications VI, Problems & Perspectives, volume 4 of Springer Proceedings in Mathematics, pages 145– 153. Springer Verlag, 2011. [2] T. Bodnár, L. Beneš, and K. Kozel. Numerical simulation of flow over barriers in complex terrain. Il Nuovo Cimento C, 31(5–6):619–632, 2008. [3] I. Sládek, T. Bodnár, and K. Kozel. On a numerical study of atmospheric 2D and 3D - flows over a complex topography with forest including pollution dispersion. Journal of Wind Engineering and Industrial Aerodynamics, 95(9– 11), 2007. [4] Š. Nosek, Z. Jaňour, K. Jurčáková, R. Kellnerová, and L. Kukačka. Dispersion over open-cut coal mine: wind tunnel modelling strategy. In P. Jonáš and V. Uruba, editors, Proceedings of the Colloquium FLUID DYNAMICS 2011, pages 27–28. Institute of Thermomechanics AS CR, 2011. Joint work with L. Benes, and T. Bodnar.

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Sebastian Kestler University of Ulm, DE On the adaptive tensor product wavelet Galerkin method in view of recent quantitative improvements Minisymposium Session SMAP: Monday, 15:00 - 15:30, CO015 Based on the fundamental work of Cohen, Dahmen and DeVore (see [1]), in recent years much progress and many contributions have been made in the field of adaptive wavelet methods for operator problems (see [6] and the references given therein). In particular, it was shown that this type of method allows for quasioptimal algorithms for different types of operator problems including linear elliptic and parabolic problems (for both local and non-local operators), non-linear (leastsquares) problems as well as PDEs with stochastic influences. In this talk, we first shortly repeat the basic principles behind wavelet discretizations of operator problems and adaptive wavelet (Galerkin) methods (see [2]). In the main part of the talk, we present an optimal algorithm for the fast evaluation of non-sparse stiffness matrices (see [4]) and a new efficient way of computing a reliable and effective a-posteriori error estimator within the adaptive tensor product wavelet Galerkin method applied to linear operator problems (see [3]). We shall also show how to solve operator problems on unbounded domains by adaptive wavelet methods (see [5]).

References [1] A. Cohen, W. Dahmen, and R. DeVore. Adaptive wavelet methods for elliptic operator equations – Convergence rates. Mathematics of Computations, 70(233):27–75, 2001. [2] T. Gantumur, H. Harbrecht, and R. P. Stevenson. An optimal adaptive wavelet method without coarsening of the iterands. Mathematics of Computations, 76(258):615–629, 2007. [3] S. Kestler and R. P. Stevenson. An efficient approximate residual evaluation in the adaptive tensor product wavelet method. Journal of Scientific Computing, 2013. doi: 10.1007/s10915-013-9712-1 [4] S. Kestler and R. P. Stevenson. Fast evaluation of system matrices w.r.t. multitree collections of tensor product refinable basis functions. Technical report, 2012. Submitted. [5] S. Kestler and K. Urban. Adaptive wavelet methods on unbounded domains. Journal of Scientific Computing, 53(2):342–376, 2012. [6] R. P. Stevenson. Adaptive wavelet methods for solving operator equations: An overview. In R. DeVore and A. Kunoth, editors, Multiscale, Nonlinear and Adaptive Approximation: Dedicated to Wolfgang Dahmen on the Occasion of his 60th Birthday, pages 543-598. Springer (Berlin), 2009. Joint work with R.P. Stevenson, and K. Urban.

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Venera Khoromskaia Max-Planck Institute for Mathematics in the Sciences, DE Hartree-Fock and MP2 calculations by grid-based tensor numerical methods Minisymposium Session LRTT: Monday, 12:40 - 13:10, CO1 The Hartree-Fock eigenvalue problem governed by the 3D nonlinear integro-differential operator represents the basic model in ab initio electronic structure calculations. We present a fast “black-box” Hartee-Fock solver by the tensor numerical methods based on the rank-structured calculation of the core Hamiltonian and of the twoelectron integrals tensor (TEI), using a general, well separable basis discretized on a sequence of n × n × n Cartesian grids [2,5]. The arising 3D convolution integrals are replaced by 1D algebraic operations in O(n log n) complexity, yielding high resolution at low cost [1,2], due to approximation on large spatial grids up to the size n3 ≈ 1014 . The Cholesky decomposition of TEI matrix is based on the new algorithm of multiple factorizations, which yields an almost irreducible number of product basis functions building the TEI tensor, depending on a threshold ε > 0 [4]. The factorized TEI matrix is applied in tensor calculations of MP2 energy correction [6]. We demonstrate on-line Hartree-Fock simulations for compact molecules using our prototype Matlab programs. The examples include glycine and alanine amino acids [7].

[1] B. N. Khoromskij and V. Khoromskaia. Multigrid Tensor Approximation of Function Related Arrays. SIAM J Sci. Comp., 31(4), 3002-3026 (2009). [2] V. Khoromskaia. Computation of the Hartree-Fock Exchange in the Tensorstructured Format. Comp. Meth. in Appl. Math., Vol. 10(2010), No 2, pp.204-218. [3] B. N. Khoromskij, V. Khoromskaia and H.-J. Flad. Numerical solution of the Hartree-Fock equation in Multilevel Tensor-structured Format. SIAM J Sci. Comp., 33(1), 45-65 (2011). [4] V. Khoromskaia, B.N. Khoromskij and R. Schneider. Tensor-structured Calculation of the Two-electron Integrals in a General Basis. Preprint 29/2012 MIS MPG, Leipzig, 2012. SIAM J. Sci. Comp., 2013, accepted. [5] V. Khoromskaia, D Andrae and B.N. Khoromskij. Fast and Accurate Tensor Calculation of the Fock Operator in a General Basis. Comp. Phys. Comm., 183 (2012) 2392-2404. [6] V. Khoromskaia, and B.N. Khoromskij. Møller-Plesset Energy Correction Using Tensor Factorizations of the Grid-based Two-electron Integrals. Preprint 26/2013 MIS MPG Leipzig, 2013. [7] V. Khoromskaia. 3D grid-based Hartree-Fock solver by tensor methods. in progress, 2013.

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Boris Khoromskij Max-Planck-Institute for Mathematics in the Sciences, DE Quantized tensor approximation methods for multi-dimensional PDEs Contributed Session CT2.8: Tuesday, 15:30 - 16:00, CO123 Modern numerical methods of separable approximation, combining the canonical, Tucker, and matrix product states (MPS) – tensor train (TT) formats, allow the low-parametric discretization of d-variate functions and operators on large n⊗d grids with linear complexity in the dimension, O(dn) [2]. The recent quantics-TT (QTT) approximation method [1] is proven to provide the logarithmic data-compression on a wide class of functions and operators. This opens the way to solve high-dimensional steady-state and dynamical problems using FEM approximation in quantized tensor spaces, with the log-volume complexity scaling in the full-grid size, O(d log n), instead of O(nd ). In this talk I will demonstrate how the canonical, QTT and QTT-Tucker tensor approximations apply to multi-parametric PDEs [3, 4], and to some uncertainty quantification problems for time-dependent models [5]. The efficiency of QTTbased tensor approximation is illustrated by numerical examples. http://personal-homepages.mis.mpg.de/bokh

References [1] B.N. Khoromskij. O(d log N )-Quantics Approximation of N -d Tensors in High-Dimensional Numerical Modeling. J. Constr. Approx. v. 34(2), 257-289 (2011). [2] B.N. Khoromskij. Tensors-structured Numerical Methods in Scientific Computing: Survey on Recent Advances. Chemometr. Intell. Lab. Syst. 110 (2012), 1-19. [3] B.N. Khoromskij, and Ch. Schwab, Tensor-Structured Galerkin Approximation of Parametric and Stochastic Elliptic PDEs. SIAM J. Sci. Comp., 33(1), 2011, 1-25. [4] B.N. Khoromskij, and I. Oseledets. Quantics-TT collocation approximation of parameter-dependent and stochastic elliptic PDEs. Comp. Meth. in Applied Math., 10(4):34-365, 2010. [5] S. Dolgov, and B.N. Khoromskij. Tensor-product approach to global time-space-parametric discretization of chemical master equation. Preprint 68/2012, MPI MiS, Leipzig 2012 (submitted).

201

Emil Kieri Department of Information Technology, Uppsala University, SE Accelerated convergence for Schrödinger equations with non-smooth potentials Contributed Session CT4.9: Friday, 09:20 - 09:50, CO124 When numerically solving the time-dependent Schrödinger equation (TDSE) for the electrons in an atom or molecule, the Coulomb singularity poses a challenge. The solution will have limited regularity, and high-order spatial discretisations, which are much favoured in the chemical physics community, are not performing to their full potential. By exploiting knowledge about the kinks in the solution we construct a correction, and show how this improves the convergence of Fourier collocation from second to fourth order. The new method is applied to the higher harmonic generation (HHG) from atomic hydrogen. In HHG from atomic gases, atoms are ionised by a laser beam. The detached electrons are accelerated in the electric field of the laser, and may recombine with the nucleus. A high energy photon, a higher harmonic, is then emitted. The process is sketched in Figure 1. The HHG process has extensive applications in experimental physics as the generator of short pulses in the extreme ultraviolet regime. Such pulses can be used e.g. for time-resolved spectroscopy of electron dynamics [1]. For different applications, different properties of the harmonic radiation are desirable, and by shaping the incident pulse it is possible to tailor the harmonic spectrum. Much work has gone into optimising the harmonic spectrum with respect to a certain target experimentally [3]. The aim of this work is to improve the accuracy of simulation of the HHG process. An application we have in mind is computational optimisation of the harmonic spectrum, for which efficient simulation of the process is a necessary component. For its simplicity we will consider HHG from atomic hydrogen, but the method presented is applicable to any atom. Our model is the TDSE for the hydrogen atom subject to a linearly polarised electric field. We use a two-dimensional model in cylindrical coordinates, 1 1 u − ze(t)u. iut = − ∆u − √ 2 2 r + z2 The coordinate system is centred at the nucleus with the z-axis aligned with the electric field. The wave function u contains all retrievable information about the electron. The initial wave function is taken as the atomic ground state. The Coulomb potential is singular at the origin. This poses a challenge for numerical methods, especially if high order of accuracy is desired. Fourier collocation is often used for the TDSE because of its spectral accuracy, but it is only second order accurate for this problem due to lack of regularity. We discretise the radial direction using Bessel functions. This softens the Coulomb singularity such that the potential becomes bounded and continuous. In the axial direction we use Fourier collocation with a new correction term, which is constructed in the spirit of [2]. We derive the time evolution of the kink in the solution, and use this knowledge to cancel the leading order error terms. One would then expect the order of accuracy to increase by one, but we show that it increases from two to four. This is confirmed by numerical experiments. For the time discretisation we use the Magnus midpoint rule. We conclude with a simulation for which we calculate the spectrum of emitted higher harmonics. We use a laser pulse consisting of a Gaussian carrier envelope 202

and a base frequency ω0 . The harmonic spectrum is approximated by the squared Fourier transform of the expectation value of the z-component of the dipole velocity ∂ u). The outcome is shown in Figure 2. In the harmonic spectrum, hµi ˙ = (u, i ∂z peaks show at odd multiples of ω0 , less distinct and with smaller amplitude for higher frequencies.

References [1] G. Doumy and L. F. DiMauro. Science, 322:1194–1195, 2008. [2] J.-H. Jung. J. Sci. Comput., 39:49–66, 2009. [3] C. Winterfeldt, C. Spielmann, and G. Gerber. Rev. Mod. Phys., 80:117–140, 2008.

(a)

(b)

(c)

(a) (b) (c) Figure 1: Semi-classical sketch of the HHG process. (a) The electric field of the laser beam tilts the potential. The electron may then tunnel through the potential barrier and ionise the atom. (b) The potential tilts in the other direction. The electron is then accelerated back towards the nucleus. (c) The electron recombines with the nucleus, emitting a high-energy photon.

0

10

−2

Im u

S(ω)

10

−4

10

−6

10

−8

10

z

r

0

5

10

15

20

25

30

35

40

ω/ω0 Figure 2: (left) The imaginary part of the wave function at the peak of the laser pulse. Most of the wave function is bound around the nucleus, but some have ionised and may generate higher harmonics. (right) The spectrum of emitted harmonics, normalised with respect to the base frequency ω0 .

203

Michael Kirby Colorado State University, US Flag manifolds for characterizing information in video sequences Contributed Session CT3.8: Thursday, 17:30 - 18:00, CO123 In many applications researchers are concerned with knowledge discovery from large sets of digital imagery. Of particular interest is the the problem of analyzing large amounts of data generated by video sequences. We propose to explore this problem using the mathematical framework of the flag manifold. We present a method for computing flags from raw video sequences and a metric for computing the distance between flags. A flag is a set of nested sequence of subspaces Sk of a vector space V such that S0 ⊂ S1 ⊂ S2 ⊂ S3 ⊂ · · · ⊂ SM where S0 is the empty set, SM = V and the dimension of the spaces is increasing, i.e., dim Si < dim Si+1 . To begin, we consider full flag manifolds where each flag consists of M+1 nested subspaces, i.e., dim Si = i with i = 0, . . . , M and then proceed to partial flags where dim Si 6= i. We are primarily interested in exploiting the manifold structure of the flag where the vector spaces in question are defined over the real numbers. We present a novel method for computing flag manifolds from video sequences. This involves introducing an optimization problem that computes the mean of a set of subspaces of possibly different dimensions. We observe that the Karcher mean is a special instance of such an approach but is not generally associated to a flag structure. We apply our algorithm to the problem of characterizing information in video for the purposes of classification. Joint work with Bruce Draper, Justin Marks, Tim Marrinan, and Chris Peterson.

204

Alana Kirchner Technical University of Munich, DE Efficient computation of a Tikhonov regularization parameter for nonlinear inverse problems with adaptive discretization methods Minisymposium Session FEPD: Monday, 12:40 - 13:10, CO017 Parameter and coefficient identification problems for PDEs usually lead to nonlinear inverse problems, which require regularization techniques due to their instability. We will present a combination of Tikhonov regularization, Morozov’s discrepancy principle, and adaptive finite element discretizations as a Tikhonov parameter choice rule. The discrepancy principle is implemented via an inexact Newton method, where we control the accuracy by means of mesh refinement based on a posteriori goal oriented error estimators. In order to further reduce the computational costs, we apply a generalized Gauss-Newton approach for the optimal control problem, where the stopping index for this iteration plays the part of an additional regularization parameter, also determined by the discrepancy principle. The obtained theoretical convergence results (optimal rates under usual source conditions) will be illustrated by several numerical experiments. Joint work with Barbara Kaltenbacher, and Boris Vexler.

205

Axel Klawonn Universität zu Köln, DE A deflation based coarse space in dual-primal FETI methods for almost incompressible elasticity Minisymposium Session PSPP: Thursday, 15:30 - 16:00, CO3 Domain decomposition methods of FETI-DP type have been successfully considered for mixed finite element discretizations of almost incompressible linear elasticity problems. For discretizations with discontinuous pressure elements, a zero net flux condition on each subdomain is needed to ensure a good condition number for FETI-DP or BDDC domain decomposition methods which has been shown by Li, Pavarino, Widlund, and others. Usually, this constraint is enforced for each vertex, edge, and face of each subdomain separately. Here, a coarse space is discussed where all vertex and edge constraints are treated as usual but where all faces of each subdomain contribute only a single constraint. This approach is presented within a deflation based framework for the implementation of coarse spaces into FETI-DP methods. Joint work with Sabrina Gippert, and Oliver Rheinbach.

206

Stefan Kleiss RICAM, Austrian Academy of Sciences, AT Guaranteed and Sharp a Posteriori Error Estimates in Isogeometric Analysis Contributed Session CT1.5: Monday, 17:00 - 17:30, CO016 The potential and the performance of isogeometric analysis (IGA), introduced in [1], have been well-studied for applications from many fields over the last years, see the monograph [2]. Though not a pre-requisite, most of the studies of IGA are based on non-uniform rational B-splines (NURBS). Since the straightforward implementation of NURBS leads to a tensor-product structure, local mesh refinement methods are subject of active current research. Despite the fact that adaptive mesh refinement is closely linked to the question of reliable a posteriori error estimation, the latter is still in its infancy stage in isogeometric analysis. Functional-type a posteriori error estimates, see the recent monograph [3] and the references therein, which have also been studied for a wide range of problems, provide reliable and sharp error bounds, which are fully computable and do not contain any generic, un-determined constants. We present functional-type a posteriori error estimates in isogeometric analysis. By exploiting the properties of NURBS, we present efficient computation of these error estimates. The numerical realization and the quality of the computed error distribution are addressed. The potential and the limitations of the proposed approach are illustrated using several computational examples.

References [1] T.J.R. Hughes, J. Cottrell, and Y. Bazilevs. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Engrg., 194(39-41):4135–4195, 2005. [2] T.J.R. Hughes, J. Cottrell, and Y. Bazilevs. Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Chichester, 2009. [3] S. Repin. A Posteriori Estimates for Partial Differential Equations. Walter de Gruyter, Berlin, Germany, 2008. Joint work with Satyendra K. Tomar.

207

Petr Knobloch Department of Numerical Mathematics, Faculty of Mathematics and Physics, Charles University in Pragu, CZ Finite element methods for convection dominated problems Plenary Session: Friday, 10:50 - 11:40, CO1 Many important applications involve a strong convective transport of physical quantities of interest whereas diffusion effects play a minor role only. This typically causes that the solutions of such problems contain so-called layers, which are narrow regions where the solutions change abruptly. Unless the computational mesh is sufficiently fine, approximation of these solutions using standard discretization techniques usually leads to spurious oscillations of unacceptable size that pollute the solutions in a large part of the computational domain. Therefore, many various discretization approaches have been developed during the last more than four decades of intensive research but the problem of solving convection dominated problems numerically has not been resolved in a satisfactory way yet. The talk will be devoted to various finite element techniques developed to solve convection dominated problems numerically. For clarity, most ideas will be explained for simple scalar convection–diffusion or convection–diffusion–reaction equations. Furthermore, applications to flow problems will be discussed. We shall mainly focus on linear and nonlinear stabilized methods. A basic difficulty connected with these approaches is that they involve stabilization parameters that significantly influence the quality of the computed solutions but for which the optimal choice is usually unknown. Therefore, we shall also discuss various possibilities how these parameters can be defined. The properties of the methods will be illustrated by both theoretical and numerical results.

208

Tzanio Kolev Lawrence Livermore National Laboratory, US Parallel Algebraic Multigrid for Electromagnetic Diffusion Minisymposium Session MMHD: Thursday, 11:30 - 12:00, CO017 Numerical simulation of electromagnetic phenomena is of critical importance in a number of practical applications and production codes. In many physical models, Maxwell’s equations are reduced to a second-order PDE system for one of the vector fields or for a potential. The definite Maxwell equations, for example, arise after discretization in time, while magnetostatics with a vector potential leads to the semidefinite Maxwell problem. Motivated by the practical needs of such large-scale simulations, we are developing parallel algebraic solvers for complicated systems of partial differential equations, including the definite and semidefinite Maxwell problem. One example of a typical application is the calculation of hydrodynamic stresses caused by large currents in pulsed-power experiments at Lawrence Livermore National Laboratory. The plot in Figure 1 shows the transient magnetic field and eddy currents occurring in a helical coil with two side-by-side wires. Fine resolutions of this problem are not tractable with previous solvers, such as classical algebraic multigrid (AMG) methods. Recently, there has been a significant activity in the area of auxiliary-space methods for linear systems arising in electromagnetic diffusion simulations. Motivated by a novel stable decomposition of the Nedelec finite element space due to Hiptmair and Xu, we implemented a scalable solver for second order (semi-)definite Maxwell problems, which utilizes several internal AMG V-cycles for scalar and vector nodal Poisson-like matrices. In this talk we describe this Auxiliary-space Maxwell Solver (AMS) by reviewing the underlying theory, demonstrating its parallel numerical performance, and presenting some new developments in its theory and implementation for new classes of electromagnetic problems. In particular, we will report some large-scale scalability results with the AMS implementation in the hypre library of scalable linear solvers, including the problem with 12 billion unknowns on 125,000 cores shown in Figure 2.

209

Figure 1: Simulation of electromagnetic diffusion in a bifilar helical coil using the Auxiliary-space Maxwell Solver.

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210

Igor Konshin Institution of Russian Academy of Sciences Dorodnicyn Computing Centre of RAS, RU Continuous parallel algorithm of the second order incomplete triangular factorization with dynamic decomposition and reordering Contributed Session CT1.6: Monday, 17:30 - 18:00, CO017 Modern software packages that are used for mathematical physics problems modeling, are often based on the implicit approximation schemes. It requires a high accuracy solution of ill-conditioned sparse systems of linear algebraic equations of large dimension. Another key point is the maximal parallel efficiency of the linear system solution. The paper deals with both of the above mentioned problems. Let us consider the solution of linear system Ax = b, where A ∈ RN ×N is a given nonsingular sparse matrix of large dimension N , b ∈ RN is a given right-hand side vector, x ∈ RN is a vector of unknowns. The main idea of the iterative solution is based on the second order incomplete triangular factorization [1], which follows from the relation A + E = L(I) U (I) + L(I) U (II) + L(II) U (I) ,

(1)

where L(I) and U (I) are the lower and upper triangular parts of the preconditioner, respectively, (the elements of the first order accuracy), L(II) and U (II) are the lower and upper triangular parts of the preconditioner with the elements of the second order accuracy, respectively, and E is an error matrix. In [1] there are presented some theoretical estimates of the preconditioning quality in the symmetric case. Our aim is to provide a reliable approach to the construction of the parallel preconditioner without a deterioration of convergence. The commonly used parallelization methods are based on the block Jacobi preconditioning or overlapping additive Schwarz preconditioning [2]. The idea of continuous parallel algorithm of type (1) preconditioning consists in reproducing the sequential algorithm for dynamically chosen ordering and partitioning. The above ordering is based on the sparsity of the current Schur complement and is dynamically calculated by a nested dissection (ND) ordering algorithm, after that a new decomposition to the processors and threads can be constructed. The following calculations model is introduced for the implementation of the MPI+threads continuous parallel factorization. The notion of group tree of MPI processes ‘th-tree’ is introduced in terms of processor groups (Fig. 1, left). Each vertex of th-tree is associated with a corresponding th-block of the matrix as a set of consecutive rows of the coefficient matrix. Each node of the tree which have children corresponds to the MPI processes separators between several groups of MPI processes, where the leaves are the independent sets of each MPI process. After calculating the approximate Schur complement the ND ordering is constructed for each node of th-tree by using ParMetis package, after that the binary block partitioning is constructed to provide a set of hyper blocks (h-blocks). This corresponds to the partitioning to a binary tree of MPI processes, binary ‘h-tree’ (Fig. 1, center). Similarly, after the calculation of the approximate Schur complement on each node of the MPI processes tree the ND ordering is constructed to separate the binary block partitioning to a set of blocks. The last tree is a binary tree for computational threads ‘t-tree’ (Fig. 1, right). The technique described above is used to dynamically construct the multilevel preconditioner based on decomposition and proper reordering. This method is 211

used for the parallel linear systems solution on the parallel computers of heterogeneous architectures with up to several thousands of processors (threads). There are presented the numerical results for the linear systems arising from different applications (including structural mechanics problems). [1] I.E.Kaporin. High quality preconditioning of a general symmetric positive definite matrix based on its decomposition. Numer. Linear Algebra Appl. (1998) Vol.5, 483-509. [2] I.E.Kaporin, I.N.Konshin. A parallel block overlap preconditioning with inexact submatrix inversion for linear elasticity problems. Numer. Linear Algebra Appl. (2002) Vol.9, No.2, 141-162.

Figure 1: Hierarchy of the trees: th-tree of the MPI processes groups (left); binary h-tree of MPI processes (center); and binary t-tree of computational threads (right). Joint work with Sergey Kharchenko.

212

Adam Kosík Charles University in Prague, Faculty of Mathematics and Physics, CZ The Interaction of Compressible Flow and an Elastic Structure Using Discontinuous Galerkin Method Contributed Session CT4.5: Friday, 09:50 - 10:20, CO016 In this paper we are concerned with the numerical simulation of the interaction of fluid flow and an elastic structure in a 2D domain. For each individual problem we employ the discretization by the discontinuous Galerkin finite element method (DGM). We describe the application of the DGM to the problem of compressible fluid flow in a time-dependent domain [1] and dynamic problem of the deformation of an elastic body. For the static elasticity problem, the discretization method was established in [2]. Finally, we describe our approach to the coupling of these two independent problems: both are solved separately at a given time instant, but we require the approximate solutions to satisfy certain transient conditions. These transient conditions are met through several inner iterations. In each iteration a calculation of both the elastic body deformation problem and the problem of the compressible fluid flow is performed. The application of the DGM to both problems is described. The DGM is a method for solving various kinds of partial differential equations, taking in advance some of the features of both the finite volume and the finite element methods. The DGM approach is applied for the spatial discretization of both problems. The time discretization is based either on finite-difference methods or on the spacetime discontinuous Galerkin method (STDGM). The STDGM applies the main concept of the DGM both to the time and space semi-discretizations. Flow of viscous compressible fluid is described by the Navier-Stokes equations. The time-dependent computational domain and a moving grid are taken into account employing the arbitrary Lagrangian-Eulerian (ALE) formulation of the Navier-Stokes equations. Solution of the deformation of the computational domain becomes another task which has to be dealt with to solve the problem of interaction. For the numerical solution of the dynamic 2D linear elasticity problem with mixed boundary conditions we have developed a .NET library written in C#. The library supports several time discretization techniques, built on top of the DG discretization in space with an arbitrary choice of the degree of the polynomial approximation. The time discretizations as based on the backward Euler formula, the second-order backward difference formula and the STDGM with an arbitrary choice of the degree of the polynomial approximation in time. The presented method can be applied to solve a selection of problems of biomechanics and aviation. Specifically, in this paper we are focused on the simulation of vibrations of vocal folds, which are caused by the airflow originating in human lungs. This procedure leads to the formation of voice. We consider a simplified 2D problem equipped with appropriate initial and boundary conditions. We define the properties of the flowing fluid and the material properties of the elastic body, which models the vocal folds. The geometry of the computational domain is inspired by measurements on real human vocal tract. The results are post-processed in order to get a visualization of the obtained solution. We are especially interested in the visualization of the elastic body deformation and the visualization of some chosen physical quantities.

213

References [1] M. Feistauer, J. Horáček, V. Kučera, J. Prokopová: On the numerical solution of compressible flow in time-dependent domains. Mathematica Bohemica, 137 (2012), 1–16. [2] B. M. Rivière, Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations: Theory and Implementation, Frontiers in Applied Mathematics, 2008. Joint work with Miloslav Feistauer, Martin Hadrava, and Jaromír Horáček.

214

Antti Koskela University of Innsbruck, AT A moment-matching Arnoldi method for phi-functions Minisymposium Session TIME: Thursday, 15:30 - 16:00, CO015 We consider a new Krylov subspace algorithm for computing expressions of the Pp form k=0 hk ϕk (hA)wk , where A ∈ Cn×n , wk ∈ Cn , and ϕk are matrix functions related to the exponential function. Computational problems of this form appear when applying exponential integrators to large dimensional ODEs in semilinear form u0 (t) = Au(t) + g(u(t)). Using Cauchy’s integral formula ϕk (z) =

1 2πi

Z Γ

eλ 1 dλ λl λ − z

we give a representation for the error of the approximation and derive a priori error bounds which describe well the convergence behaviour of the algorithm. In addition an efficient a posteriori estimate is derived. Numerical experiments in MATLAB illustrating the convergence behaviour are given. Joint work with Alexander Ostermann.

215

Felix Krahmer Insitute for Numerical and Applied Mathematics, University of Göttingen, DE The restricted isometry property for random convolutions Minisymposium Session ACDA: Monday, 12:40 - 13:10, CO122 The theory of compressed sensing is based on the observation that many natural signals are approximately sparse in appropriate representation systems, that is, only few entries are significant. The goal of the theory is to devise methods to recover such a signal x from linear measurements y = Φx. For example, it has been shown [1] that under the assumption of a small restricted isometry constant on the matrix Φ, approximate recovery via `1 -minimization min kzk1 z

subject to Φz = y,

(where kzkp denotes the usual `p -norm) is guaranteed even in the presence of noise. Here, for a matrix Φ ∈ Rm×n and s < n, the restricted isometry constant δs = δs (Φ) is defined as the smallest number such that (1 − δs )kxk22 ≤ kΦxk22 ≤ (1 + δs )kxk22

for all s-sparse x.

If a matrix has a small restricted isometry constant, we also say that the matrix has the restricted isometry property (RIP). A class of measurement models that is of particular relevance for sensing applications is that of subsampled convolution with a random pulse. In such a model, the convolution of a signal x ∈ Rn with a random vector  ∈ Rn given by x 7→  ∗ x, ( ∗ x)k =

n X

(k−j)

mod n

xj .

j=1

is followed by a restriction PΩ to a deterministic subset of the coefficients Ω ⊂ {1, . . . , n} and normalization of the columns. The resulting measurement map is linear; its matrix representation Φ given by 1 Φx = √  ∗ x m is called a partial random circulant matrix. In the talk, we will focus on the case that the random vector  is a Rademacher random vector, that is, its entries are independent random variables with distribution P(i = ±1) = 1/2. Note, however, that the corresponding results in [2] consider more general random vectors. In the talk, we present the following main result. Theorem 4. ([2]) Let Φ ∈ Rm×n be a draw of a partial random circulant matrix generated by a Rademacher vector . If m ≥ cδ −2 s (log2 s)(log2 n),

(1)

2

then with probability at least 1 − n−(log n)(log s) , the restricted isometry constant of Φ satisfies δs ≤ δ. The constant c > 0 is universal. This result improves the best previously known estimates for a partial random circulant matrix [3], namely that m ≥ Cδ (s log n)3/2 is a sufficient condition for achieving δs ≤ δ with high probability. In particular, Theorem 4 removes the 216

exponent 3/2 of the sparsity s, which was already conjectured in [3] to be an artefact of the proof. The proof is based on the observation that the restricted isometry constant of a partial circulant matrix Φ based on a Rademacher vector can be expressed as kVx k22 − EkVx k22 , δs (Φ) = sup x∈S n−1 | supp x|≤s

where Vx is defined through Vx y :=

√1 PΩ x m

∗ y.

As it turns out, the expression kVx k22 is a Rademacher chaos process, that is, it is of the form h, M i. This observation was already exploited in [3] to obtain their suboptimal bounds. Our result, however, incorporates the additional observation that the matrix M in the above scenario is Vx∗ Vx , hence positive semidefinite. In the talk we present a bound for suprema of chaos processes under such structural assumptions. The proof of this bound is based on decoupling and a chaining argument, see [2]. This bound then allows to establish the above theorem.

References [1] E. J. Candès, J., T. Tao, and J. Romberg. Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory, 52(2):489–509, 2006. [2] F. Krahmer, S. Mendelson, and H. Rauhut. Suprema of chaos processes and the Restricted Isometry Property. Comm. Pure Appl. Math., to appear. [3] H. Rauhut, J. K. Romberg, and J. A. Tropp. Restricted isometries for partial random circulant matrices. Appl. Comput. Harmon. Anal., 32(2):242–254, 2012. Joint work with Shahar Mendelson, and Holger Rauhut.

217

Stephan Kramer Institut f. Numerische und Angewandte Mathematik, Universität Göttingen, DE The Geometric Conservation law in Astrophysics: Discontinuous Galerkin Methods on Moving Meshes for the non-ideal Gas Dynamics in Wolf-Rayet Stars Minisymposium Session NFSI: Thursday, 15:30 - 16:00, CO122 Wolf-Rayet stars are described by the inviscid Euler equations for compressible flow enhanced by a coupling to radiation transport in the diffusion approximation and a Poisson equation for the self-gravitation. Unlike standard aerospace applications the closure is given by two equations of state, one for the pressure and one for the energy density. These equations and the opacity of the star are to a large extent only known in the form of lookup tables. To understand the details of the nonlinear dynamics in the transient states of a Wolf-Rayet star an accurate three-dimensional simulation of its atmosphere is necessary. Especially the mass losses observed require a discretization scheme which is locally conservative in space and time. To accommodate for shock waves we employ arbitrary Lagrangian-Eulerian (ALE) methods where the mesh partly moves with - or represents the motion of - a fluid particle. Due to the importance of local conservation properties of the discretization scheme we choose an DG approach. They key to a successful and consistent ALE-type DG-discretization is to respect the geometric conservation law: uniform flows should be preserved exactly for arbitrary mesh motion. We follow Mavriplis et al. [1] and discuss a DG-ALE discretization for the nonlinear gas dynamics in Wolf-Rayet stars.

[1] D. Mavriplis and C. Nastase. On the geometric conservation law for highorder discontinuous Galerkin discretizations on dynamically deforming meshes. 46th AIAA Aerospace Sciences Meeting and Exhibit, 2008. Joint work with Bartosz Kohnke, and Gert Lube.

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Stephan Kramer Insitut f. Numerische und Angewandte Mathematik, DE Converting Interface Conditions due to Excluded Volume Interactions into Boundary Conditions by FEM-BEM Methods Minisymposium Session FREE: Tuesday, 12:00 - 12:30, CO2 Recent impedance spectroscopy studies of ubiquitin in solution have revealed the influence of conformational sampling of proteins on the direct current contribution to the dielectric loss spectrum. A detailed model for this has been derived in [1]. Our contribution discusses the main numerical issues in setting up a Poisson-Nernst-Planck model for the ion dynamics and the electrostatic potential in impedance spectroscopy of globular proteins in solution: - The set of partial differential equations modeling impedance spectroscopy are derived from the continuity equation and the electro-diffusive fluxes. This is a set of convection-diffusion equations for the ion densities coupled to a Poisson equation for the electrostatic potential. - The simulation of the experiment on a generic, solvated globular protein needs appropriate boundary conditions for the impedance cell and for the protein-solvent interface. The experimental setup introduces solvent-electrode interfaces which give rise to dielectric double layers well-known from the electro-chemistry. The excluded volume interaction between protein and ions can be transformed into an integral equation for the electrostatic potential on the protein-solvent interface. This is helpful especially in the case of complicated molecular surfaces. When intramolecular dynamics are taken into account these surface might start to move. In the bulk the model is discretized by finite elements. The integral equation on the protein-solvent interface is discretized by a boundary element method. Our results show - the interface problem can be replaced by a non-local boundary condition, - how to setup the correct FEM-BEM coupling for its discretization, - curvilinear approximation of cell boundaries enhances convergence.

[1] Stephan C. Kramer PhD thesis 2012, Universität Göttingen, link: http://ediss.unigoettingen.de/handle/11858/00-1735-0000-000D-FB52-0 Joint work with Gert Lube.

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Marie Kray Universität Basel, CH A new approach to solve the inverse scattering problem for the wave equation Contributed Session CT3.1: Thursday, 18:00 - 18:30, CO1 In paper [1], we propose a new method to solve the following inverse problem: we aim at reconstructing, from boundary measurements, the location, the shape and the wave propagation speed of an unknown inclusion surrounded by a medium whose properties are known. Our strategy combines two methods recently developed by the authors: 1. the Time-Reversed Absorbing Condition method (TRAC) first introduced in [2]: It combines time reversal techniques and absorbing boundary conditions to reconstruct and regularize the signal in a truncated domain that encloses the inclusion. This enables one to reduce the size of computational domain where we solve the inverse problem, now from virtual internal measurements. 2. the Adaptive Inversion (AI) method initially proposed for the viscoelasticity equation in [3]: The originality of this method comes from the parametrization of the problem. Instead of looking for the value of the unknown parameter at each node of the mesh, it projects the parameter into a basis composed by eigenvectors of the Laplacian operator. Then, the AI method uses an iterative process to adapt the mesh and the basis of eigenfunctions from the previous approximation to improve the reconstruction. The novelty of our work is threefold. Firstly, we present a new study on the regularizing power of the TRAC method. Secondly, we adapt the Adaptive Inversion method to the case of the wave equation and we propose a new anisotropic version of the iterative process. Finally, we present numerical examples to illustrate the efficiency of the combination of both methods. In particular, our strategy allows (a) to reduce the computational cost, (b) to stabilize the inverse problem and (c) to improve the precision of the results. On Figure 1, we display our results for a penetrable pentagon. We compare the exact propagation speed (left column) to the reconstruction by using both methods, first without noise on the recorded data (center column), then with 20% level of noise (right column). We denote by 20%-noisy TRAC data, the virtual data obtained after the TRAC process from 20%-noisy boundary measurements. References: [1] M. DE B UHAN AND M. K RAY, A new approach to solve the inverse scattering problem for waves : combining the TRAC and the Adaptive Inversion methods, submitted (available on HAL), 2013. [2] F. A SSOUS , M. K RAY, F. N ATAF, AND E. T URKEL, Time Reversed Absorbing Condition : Application to inverse problem, Inverse Problems, 27(6), 065003, 2011. [3] M. DE B UHAN AND A. O SSES, Logarithmic stability in determination of a 3D viscoelastic coefficient and a numerical example, Inverse Problems, 26(9), 95006, 2010.

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Figure 1: Shape and properties reconstruction of a penetrable pentagon by using both TRAC and AI methods: (a) Propagation speed profile inside and outside the inclusion. (b) Result obtained with 0%-noisy TRAC data, relative L2 -error = 1.72%. (c) Result obtained with 20%-noisy TRAC data, relative L2 -error = 1.92%. Joint work with Dr. Maya de Buhan (CNRS-Université Paris Descartes France).

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Gunilla Kreiss Uppsala University, SE Imposing Neumann and Robin boundary conditions with added penalty term Contributed Session CT1.9: Monday, 17:00 - 17:30, CO124 In a standard finite element method model for an elliptic problem Neumann and Robin boundary conditions are imposed weakly. For smooth cases we expect the normal derivative at the boundary to converge to the prescribed value, but at a slower rate than the solution itself. This can be problematic when for example computing flow in porous media. In a typical porous media case the pressure will satisfy an elliptic equation with a Neumann boundary condition for instance where the aquifer is bounded by an impermeable rock. After solving for the pressure, the pressure gradient gives the approximate flow. At boundaries the flow approximation will only satisfy the prescribed flux approximately. From an engineering point of view a very good agreement would be desirable. In this work we modify the weak form by including a penalty term so as to decrease the error in the boundary normal derivative for the Neumann case. The same technique can be applied to Robin boundary conditions. The new bilinear form is symmetric, and the approach is inspired by Nitsche’s method for imposing Dirichlet conditions weakly. We prove that in the interior of the domain the corresponding discrete approximation converges at the same order as the solution obtained using the standard method. Numerical experiments demonstrate that the convergence rate of the normal derivative at the boundary can be improved by one order. This is true for both Neumann and Robin boundary conditions. In a second numerical example we compute streamlines based on a pressure solution on a square, with prescribed flux at the horizontal boundaries. At the right half of the upper boundary, and at the left half of the lower boundary the prescribed flux is equal to zero. Thus the streamlines of the exact solution should be parallel to the boundary there. In figures 1 and 2 we have plotted streamlines based on the standard method and on our method, respectively. Note the streamlines almost parallel to the right half of the upper boundary. In figure 2 the uppermost streamline is considerably more accurate than in figure 1, where it exits the no-flow boundary. An improved result is also found in the lower left half of the boundary. The numerical tests have been done on both Cartesian grids and quadrilateral grids with bilinear finite elements.

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Figure 1: Standard Neumann

Figure 2: Penalized Neumann Joint work with Margot Gerritsen, and Annette Stephansen.

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Wolfgang Krendl Johannes Kepler University Linz, AT Efficient preconditioning for time-harmonic control problems Contributed Session CT4.2: Friday, 08:50 - 09:20, CO2 Based on analytic results on preconditioners for time-harmonic control problems in the paper Stability Estimates and Structural Spectral Properties of Saddle Point Problems (authors: Krendl W., Simoncini V., Zulehner W.: to appear in: Numerische Mathematik), we discuss their efficient implementation. In particular, time-harmonic parabolic and time-harmonic Stokes control problems. For these problems we present practical preconditioners in combination with MINRES, which lead to robust convergence rates with respect to meshsize, frequency and cost parameters. Joint work with Valeria Simoncini, and Walter Zulehner.

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Daniel Kressner EPF Lausanne, CH Interpolation based methods for nonlinear eigenvalue problems Minisymposium Session NEIG: Thursday, 12:00 - 12:30, CO2 This talk is concerned with numerical methods for matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. In particular, we focus on eigenvalue problems for which the evaluation of the matrix-valued function is computationally expensive. Examples of such problems arise, e.g., from boundary integral formulations of elliptic PDE eigenvalue problems or coupled FEM/BEM discretizations of fluid-structure interaction problems. The cost for evaluating the matrix-valued function typically excludes the use of established nonlinear eigenvalue solvers. Instead, we propose the use of polynomial approximation combined with non-monomial linearizations. It can be shown that the obtained eigenvalue approximations converge exponentially as the degree of the polynomial decreases. In turn, a degree between 10 and 20 is often sufficient to attain excellent accuracy. Still, this means that the size of the eigenvalue problem is increased by a factor between 10 and 20, and hence the storage requirements of, e.g., Krylov subspace methods increase by this factor. For matrix polynomials in the monomial basis, the Q-Arnoldi methods and variants thereof (SOAR, TOAR) are established techniques to largely avoid this increase. If time permits, we will discuss the adaption of TOAR and deflation techniques to non-monomial bases. Parts of this work are based on collaborations with Jose Roman, Olaf Steinbach, and Gerhard Unger. Joint work with Cedric Effenberger.

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Jochen Kroll LANXESS Deutschland GmbH, DE An alternative description of the visko-elastic flow behavior of highly elastic polymer melts Minisymposium Session MANT: Wednesday, 10:30 - 11:00, CO017 The description of the visco-elastic behavior of polymer melt and solutions undergoing finite deformations is usually based on the description by generalized Maxwell processes. Achieving a sufficient approximation quality of dynamical data requires – especially in the case of commercial and thus broadly distributed polymers – the introduction of a large number of parameters with the latter being of limited physical meaning. The presented modeling approach is not only characterized by its significantly reduced number of parameters but also by its direct link to the dynamical characterization of the material. In that way a connection between the molecular information and the simulated flow behavior can be established.

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Lev Krukier Southern Federal University, Computer Center, RU Symmetric - skew-symmetric splitting and iterative methods Contributed Session CT3.3: Thursday, 17:00 - 17:30, CO3 Any matrix A can naturally be expressed as a sum of symmetric matrix A0 and skew-symmetric matrix A1 . This splitting is named symmetric - skew-symmetric spliting (SSS). Consider the linear equation system Au = f,

(1)

where A is non-symmetric matrix, u is the vector of unknown, f is the vector of the right part is considered. Iterative method based on symmetric - skew-symmetric splitting was firstly proposed for this business by Gene Golub. If A0 is a positive definite than matrix A is named positive real. We will name matrix A strongly non-symmetric if kA0 k∗ 0 are iterative parameters, u is the solution that we obtain, ek = y k − u and rk = Aek denote the error and the residual in the k-th iteration, respectively. Consider the next choice of operator B. The class of triangular skew-symmetric iterative methods is defined by (2) with the matrix B being chosen as B(ω) = Bc + ω((1 + j)KL + (1 − j)KU ), j = ±1, Bc = Bc∗ .

(3)

The class of product triangular skew-symmetric iterative methods is defined by (2) with the matrix B being chosen as −1 B = (BC + ωKU )BC (BC + ωKL ), Bc = Bc∗ ,

(4)

∗ where KL + KU = A1 , KL = −KU∗ , BC = BC . Operator BC can be chosen arbitrarily, but has to be symmetric. These methods are from class of SSIT and called as two-parameters triangular(TTM) and product triangular (TPTM) method. Convergence of TTM and TPTM has been considered and proved. We compare TTM to the conventional SOR procedure and TPTM to the conventional SSOR procedure. For the check of TPTM behavior, the standard 5-point central difference scheme on the regular mesh has been used for approximation of the convection-diffusion

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equation with Dirichlet boundary conditions and small parameter at the higher derivatives in the incompressible medium and it’s transformation by regular ordering to strongly non-symmetric linear equation systems. In the case of central difference approximation of the convective terms operator A can naturally be expressed in a sum of symmetric positive definite operator A0 , which is a difference analogue of the Laplace operator and skew-symmetric operator A1 , which is a difference analogue of the convective terms. Numerical experiments show that in considered particular cases the behavior of methods is closely related to the technique of choosing the matrix BC . Joint work with B. L. Krukier, and O.A.Pichugina.

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Vaclav Kucera Charles University in Prague, Faculty of Mathematics and Physics, Czech republic On the use of reconstruction operators in discontinuous Galerkin schemes Contributed Session CT2.5: Tuesday, 15:00 - 15:30, CO016 In this work we follow the methodology of higher order finite volume (FV) and spectral volume (SV) schemes and introduce a reconstruction operator into the discontinuous Galerkin (DG) method. In the standard FV method, such operators are used to increase the order of accuracy of the basic piecewise constant scheme by constructing higher order piecewise polynomial approximations of the exact solution from the lower order piecewise constant approximate solutions. In the DG setting, the reconstruction operators will be used to construct higher order piecewise polynomial reconstructions from the lower order DG scheme. This allows us to increase the accuracy of existing DG schemes with a problem-independent reconstruction procedure. In the talk, the technique will be presented for a nonstationary nonlinear convection equation, although the basic idea can be straightforwardly applied to any DG formulation of general evolutionary equations. Unlike the FVM, where the reconstruction stencil size must be increased in order to increase the order of accuracy, in the DG scheme the reconstruction stencil has minimal size independent of the approximation order. For example, in one spatial dimension, from a DG scheme of order n, one can reconstruct an approximate solution of order 3n + 2 using the von Neumann neighborhood only. This represents a dramatic increase in accuracy. In two spatial dimensions, an approximate solution of order n allows us to construct an approximation of order 2n + 1. One may ask, whether such a reconstruction procedure brings any advantages over using the corresponding DG scheme of higher order. However, there are several reasons why using a lower order DG scheme is more advantageous. First, test functions in the reconstructed scheme are from the lower order discrete space, therefore lower order quadrature rules are needed in the evaluation of element and boundary integrals and therefore fewer quadrature points and (numerical) flux evaluations are needed. Furthermore, the stability conditions on the time step size are inherited from the lower order scheme, therefore larger time steps can be taken, which greatly increases the efficiency of the scheme. And finally, if orthogonal bases are not used, the mass matrices resulting from the temporal discretization have smaller dimension and can therefore be inverted faster. Numerical experiments are provided to demonstrate the accuracy and efficiency of the proposed schemes. Applying reconstruction procedures in DG schemes was already proposed already in Dumbser et al. (2008) based on heuristic arguments, however we provide a more rigorous derivation, which justifies the increased order of accuracy. Then we analyze properties of the reconstruction operators form the point of view of classical finite element theory, using a generalized version of the Bramble-Hilbert lemma. Furthermore, we show the equivalence of the reconstructed DG scheme to a certain modification of the corresponding higher order DG scheme. This socalled auxiliary problem can be analyzed similarly as standard DG schemes and although a complete theory of error estimates is not yet developed, this setting gives a firm theoretical background to the reconstructed DG scheme. The author is a junior researcher in the University Center for Mathematical Modelling, Applied Analysis and Computational Mathematics (Math MAC). The research is supported by the project P201/11/P414 of the Czech Science Foundation.

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Dmitri Kuzmin University Erlangen-Nuremberg, DE Vertex-based limiters for continuous and discontinuous Galerkin methods Plenary Session: Friday, 11:40 - 12:30, CO1 This talk is concerned with the design of constrained finite element methods for convection-dominated transport equations and hyperbolic systems. We will begin with a review of algebraic flux correction schemes for enforcing the discrete maximum principle for (low-order) continuous finite elements. After formulating sufficient conditions of positivity preservation, we will present a black-box approach to limiting the antidiffusive part of the Galerkin transport operator. The limiting techniques to be discussed are based on a generalization of the fully multidimensional flux-corrected transport (FCT) algorithm. Next, we will address the aspects of slope limiting in discontinuous Galerkin (DG) methods. The representation of finite element shape functions in terms of cell averages (coarse scales) and derivatives (fine scales) makes it possible to eliminate the unresolvable fine-scale features using a vertex-based hierarchical moment limiter. The proposed limiting strategy preserves the order of accuracy at smooth extrema and may serve as a parameterfree regularity estimator. We will highlight the existing similarities to variational multiscale methods and explore the possibility of enriching a continuous (linear or bilinear) coarse-scale approximation space with discontinuous basis functions of higher order. Further topics to be discussed include the iterative treatment of nonlinear systems and the extension of scalar limiting techniques to the Euler equations of gas dynamics. The accuracy of the presented high-resolution schemes will be illustrated by numerical examples including the first use of vertex-based limiters in the context of hp adaptivity for hyperbolic conservation laws.

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Pauline Lafitte Ecole Centrale Paris, FR Projective integration schemes for kinetic equations in the hydrodynamic limit Minisymposium Session ASHO: Wednesday, 11:30 - 12:00, CO2 In order to introduce new asymptotic preserving schemes for kinetic equations in regimes leading to hyperbolic systems of conservation laws appearing e. g. in some models of radiative transfer or fluid-particle interactions, we apply the projective integration method developed by Gear and Kevrekidis in the context of large multiscale differential systems appearing in Chemistry. Joint work with A. Lejon, and G. Samaey.

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Omar Lakkis University of Sussex, GB Review of Recent Advances in Galerkin Methods for Fully Nonlinear Elliptic Equations Minisymposium Session NMFN: Monday, 12:10 - 12:40, CO2 I will make a brief overview of all numerical methods, including their analysis where available, for fully nonlinear elliptic equations based on Galerkin-type approximations, while mentioning other known methodologies, such as finite differences and related monotone schemes. In the final part, I will focus on the finite element Hessian methods introduced by Lakkis and Pryer (2010) via the nonvariational finite element method, aposteriori error estimates and their potential for convergent adaptive mesh refinement.

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Jens Lang Technische Universität Darmstadt, DE Anisotropic Finite Element Meshes for Linear Parabolic Equations Minisymposium Session TIME: Thursday, 15:00 - 15:30, CO015 In [1,2] anisotropic mesh adaptation methods for elliptic problems are studied. In a next step, we have investigated the influence of anisotropic meshes upon the time stepping and the conditioning of the linear systems arising from linear finite element approximations of linear parabolic equations. Here, we present stability results and estimates for the condition number. Both explicit and implicit time integration schemes are considered. For stabilized explicit Runge-Kutta methods, the stability condition is obtained. It is shown that the allowed maximal step size depends only on the number of the elements in the mesh and a measure of the non-uniformity of the mesh viewed in the metric specified by the inverse of the diffusion matrix. Particularly, it is independent of the mesh non-uniformity in volume measured in the Euclidean metric. For the implicit time stepping situation, bounds are obtained for the condition numbers of the coefficient matrices of the linear system and preconditioned linear system with Jacobi preconditioning. It is shown that the effects of the volume non-uniformity can be eliminated by the Jacobi preconditioning. One of our main findings is that the alignment of the mesh with the diffusion matrix plays a crucial role in the stability condition for the explicit stepping case and the condition number of the preconditioned linear system by the Jacobi preconditioning for the implicit stepping case. When the mesh is uniform with respect to metric defined by the (symmetric and uniformly positive definite) diffusion matrix, the stability condition and the condition number behaves like in the situation with constant, isotropic diffusion problems on a uniform mesh. [1] W. Huang, L. Kamenski, J. Lang, A new anisotropic mesh adaptation method based upon hierarchical a posteriori error estimates, J. Comp. Phys. 229 (2010), pp. 2179-2198. [2] W. Huang, L. Kamenski, J. Lang, Adaptive finite elements with anisotropic meshes, Numerical Mathematics and Advanced Applications 2011: Proceedings of ENUMATH 2011, the 9th European Conference on Numerical Mathematics and Advanced Applications, Leicester, September 2011, A. Cangiani et al. (eds.), pp. 33-42, Springer 2013. Joint work with Weizhang Huang, and Lennard Kamenski.

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Toni Lassila CMCS-MATHICSE, EPFL, CH Space-time model reduction for nonlinear time-periodic problems using the harmonic balance reduced basis method Minisymposium Session ROMY: Thursday, 12:00 - 12:30, CO016 In many applications of fluid dynamics, for example in simulations of turbomachinery flows or the human cardiovascular system, the behavior of the flow is such that the solution converges towards a periodic steady-state starting from an arbitrary initial state. Typically one is then only interested in computing the periodic-steady state solution. In this case, simulating the transient behavior of the unsteady flow until a periodic steady-state is reached is not an efficient approach. The harmonic balance method assumes that both the flow solution and the spatial operator of the problem are time-periodic and can be written as their Fourier series expansions. These expansions are then truncated after the first few leading terms, and the problem reduces to solving a set of fully-coupled nonlinear equations for the Fourier coefficients. In this talk, the harmonic balance method is coupled with the reduced basis method for reduction in space to construct a computationally efficient space-time reduced order model without the typical growth of error in time. It is well suited to hemodynamics applications in large arteries, where a strong pulsatile inflow drives the flow towards periodic regimes. We also discuss extending the Floquet theory of the stability of linear time-periodic systems to analyze the stability of the harmonic balance reduced basis -solutions to identify the critical Reynolds number after which the flow undergoes a bifurcation and the periodic steady-state solution becomes unstable.

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Olivier Le Maitre Duke University, US Galerkin Method for Stochastic Ordinary Differential Equations with Uncertain Parameters Minisymposium Session UQPD: Thursday, 11:30 - 12:00, CO1 We propose a Galerkin method for the resolution of a certain class of Stochastic Ordinary Differential Equations (SODE) driven by Wiener processes and involving some random parameters. The dependence of the solution with respect to the uncertain parameters is treated by Polynomial Chaos expansions, with expansion coefficients being random processes function of the Wiener processes. An hybrid Monte-Carlo Galerkin method is then proposed to compute these expansion coefficients, allowing for a complete uncertainty analysis of the solution. In particular, we show that one can retrieve the dependence on the uncertain parameters of the stochastic noise in the solution. Exemples of applications are shown for linear and non linear SODEs. Finally, the extension of the method to non-intrusive techniques and more general source of stochasticity is discussed. Joint work with Omar Knio.

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Sanghyun Lee Ph.D in Mathematics at Texas A&M University, US Numerical simulation of Kaye effects Minisymposium Session FREE: Monday, 15:00 - 15:30, CO2 The fascinating phenomenon of a leaping shampoo stream, Kaye effect, is a property of non-Newtonian fluid which was first described by Alan Kaye in 1963. It manifest itself, when a thin stream of non-Newtonian fluid is poured into a dish of the fluid. As pouring proceeds, a small stream of liquid occasinally leaps upward from the heap. Figure (1) Since there is no mathematical model or numerical simulation studied before, as a first approach, we have studied a mathematical model and algorithm to find the range of parameters to observe the Kaye effects. In this context we propose a modfied projection method for Navier-stokes equation with open boundary, level set method for free boundary and adaptivity. Also, in earlier studies it has been debated whether non-Newtonian effects are the underlying cause of this phenomenon, making the jet glide on top of a shearthinning liquid layer, or whether an entrained air layer is responsible. Here in we show that the jet slides on a lubricating air layer with numerical simulation which is the identical result that we observe from physical experiments.

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Figure 1: Kaye effect with shampoo Joint work with Andrea Bonito, and Jean-Luc Guermond.

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Jeonghun Lee Aalto University, Department of Mathematics and Systems Analysis, FI Hodge Laplacian problems with Robin boundary conditions Contributed Session CT4.6: Friday, 08:20 - 08:50, CO017 In this work, we consider mixed methods of Hodge Laplacian problems with Robin boundary conditions. Mixed methods for the Hodge Laplacian problems were studied by Arnold, Falk and Winther in [1, 2] using a framework of the de Rham complex, called the finite element exterior calculus (FEEC). In the work of Arnold, Falk and Winther, they assume the homogeneous Dirichlet or homogeneous Neumann boundary conditions. However, it is reasonable to consider more general boundary conditions in physical applications. Recently, the scalar Poisson equation with Robin boundary conditions was studied in [3]. Stenberg and his collaborators proved a priori error estimates and provided an efficient and reliable a posteriori error estimator. The author generalizes this to mixed methods of Hodge Laplacian problems with Robin boundary conditions, for general differential k forms in the FEEC framework. Robin boundary conditions for the scalar Poisson equation are well-known whereas Robin boundary conditions for Hodge Laplacian problems of differential k forms are not obvious for general k. Thus we propose appropriate Robin boundary conditions in the language of differential forms and discuss well-posedness of the problem. For discrete mixed forms of Hodge Laplacian problems, we use the Pr Λk and Pr− Λk finite element families on triangular meshes. We prove the stability of the numerical scheme, as well as discuss a priori and a posteriori error estimates.

References [1] Douglas N. Arnold and Richard S. Falk and Ragnar Winther Finite element exterior calculus, homological techniques, and applications, Acta Numer., 15 (2006), 1–155. [2] Douglas N. Arnold and Richard S. Falk and Ragnar Winther Finite element exterior calculus: from Hodge theory to numerical stability, Bull. Amer. Math. Soc., 47 (2010), no. 2, 281–354. [3] Juho Könnö and Dominik Schötzau and Rolf Stenberg Mixed finite element methods for problems with Robin boundary conditions, SIAM J. Numer. Anal., 49 (2011), no. 1, 285–308.

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Annelies Lejon Department of Computer Science, KU Leuven, BE Higher order projective integration schemes for multiscale kinetic equations in the diffusive limit Contributed Session CT1.3: Monday, 18:30 - 19:00, CO3

1

Introduction

Multiscale systems (involving multiple timescales) can be found in many real world applications, such as biological systems, traffic flow, plasma astrophysics, etc. In this talk, we consider systems that can be described by a kinetic equation that models evolution of a distribution function in position-velocity expensive to simulate this over the longer timescales we are interested in. We present a high-order projective integration scheme that is fully explicit and whose computational complexity does only depends on the macroscopic time-scales in the system. Moreover, we show an application of this technique on a semiconductor equation to illustrate the numerical performance.

2

Methods

The kinetic equation describe the evolution of the probability f (x, v, t) being at position x, moving with velocity v at time t, ρ(x, t) − f (x, v, t) + εM (ρ) v , ∂t f (x, v, t) + f (x, v, t) = ε ε2

(1)

in which ρ = hf i, and we have introduced a small-scale parameter 0 < ε  1 and a diffusive scaling. The term M (ρ) has been introduced to obtain an advectiondiffusion behaviour in the diffusion limit ε → 0, The projective integration algorithm was developed by Gear and Kevrekidis (SIAM Journal on Scientific Computing, 4:1091,1106,2003). It consists of the following steps: 1. Perform K small steps with an naive explicit integrator (with time-step δt = O(ε2 ) (this is called an inner integrator). When ε is small, this will enforce convergence of the fast modes to the slow manifold that is characterize by ρ. 2. One then performs a large time-step ∆t by extrapolation in time (this is called an outer integrator). The application of projective integrations to kinetic equations was first studied for first-order extrapolation (projective forward Euler) and a purely diffusive equation (SIAM Journal on Scientific Computing, 34:A579-A602, 2012) This work extends the method to higher order time integration, and provides a numerical analysis in a more general advection-diffusion setting. We proved that the stability condition on ∆t independent of ε for kinetic equations of type (1). Also, the required number K of steps with the inner integrator is independent of ε. We therefore constructed a stable and explicit method, with arbitrary accuracy in time and space. For the numerical results, we used a 4th order Runge–Kutta method as the outer integrator, forward Euler as the inner integrator. 239

3

Stability Regions

Furthermore, we derived analytical expressions for the stability regions for the higher order method. From figure 1, it is clear that there are two distinct regions. One part is centered around the origin and can be used to capture the fast modes of the system and the other one is located near (1, 0) and the latter will capture the slow modes.

Figure 1: In the left part of the figure the stability regions of the PRK4 method has been plotted for different values of δt and ∆t = 1 × 10−3 , K = 3 : δt = 1 × 10−6 (dashed), δt = 1 × 10−4 (dotted),δt = 1 × 1.6 × 10−5 (solid). The right part is a magnification for the region for slow eigenvalues Joint work with Pauline Lafitte, and Giovanni Samaey.

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Martin Lilienthal Graduate School of Computational Engineering / TU-Darmstadt, DE Non-Dissipative Space Time Hp-Discontinuous Galerkin Method for the TimeDependent Maxwell Equations Contributed Session CT4.6: Friday, 08:50 - 09:20, CO017 A space-time finite element method for the time-dependent Maxwell equations is presented. The method allows for local hp-refinement in space and time by employing a space-time Galerkin approach and is thus well suited for hp-adaptivity. Inspired by the continuous Galerkin methods for ODEs, nonequal test and trial spaces are employed in the temporal direction. Combined with a (centered) discontinuous Galerkin approach in the spatial directions, a stable non-dissipative method is obtained. Numerical experiments in (3+1)D indicate that the method is suitable for space-time hp-adaptivity on dynamic discretizations. The work of M. Lilienthal is supported by the ’Excellence Initiative’ of the German Federal and State Governments and the Graduate School of Computational Engineering at Technische Universität Darmstadt and the DFG under grant no. SCHN 1212/1-1. Joint work with Sascha Schnepp, and Thomas Weiland.

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Lek-Heng Lim University of Chicago, US Symmetric tensors with positive decompositions Minisymposium Session LRTT: Monday, 11:10 - 11:40, CO1 A symmetric d-tensor is positive semidefinite if, when viewed as a homogeneous form, is always nonnegative valued, or equivalently, has all eigenvalues nonnegative. The dual cone of positive semidefinite tensors is the cone of symmetric tensors that have a decomposition into rank-1 symmetric tensors with all coefficients positive. Such tensors have many nice properties: a best rank-r approximation always exist, the decomposition is unique for small values of r without any additional requirements (such as Kruskalś condition), and there are provably correct algorithms (as opposed to heuristics like alternating least squares) for finding such decompositions. We will discuss these and other properties of symmetric tensors with positive decompositions. Joint work with Greg Blekherman.

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Ping Lin University of Dundee, GB L2 projected finite element methods for Maxwell’s equations with low regularity solution Minisymposium Session MMHD: Thursday, 11:00 - 11:30, CO017 In the talk we will present an element-local L2 projected finite element method to approximate the nonsmooth solution (not in H 1 ) of the Maxwell problem on a nonconvex Lipschitz polyhedron with reentrant corners and edges. The key idea lies in that element-local L2 projectors are applied to both curl and div operators. The C 0 linear finite element (enriched with certain higher degree bubble functions) is employed to approximate the nonsmooth solution. The coercivity in L2 norm is established uniformly in the mesh size. For the solution and its curl in H r with r < 1 we obtain an error bound O(hr ) in an energy norm. Numerical examples confirm the theoretical error bound. The idea is also applied to curldiv magnetostatic problem in multiply-connected Lipschitz polyhedrons and to eigenvalue problems. Desirable error bounds are obtained as well. The talk is based on a few joint papers with H.Y. Duan and R. Tan.

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Alexander Linke Weierstrass Institute, DE Stabilizing Mixed Methods for Incompressible Flows by a New Kind of Variational Crime Contributed Session CT2.3: Tuesday, 14:00 - 14:30, CO3 In incompressible flows with vanishing normal velocities at the boundary, irrotational forces in the momentum equations should be balanced completely by the pressure gradient. Unfortunately, nearly all available discretizations for incompressible flows violate this property. The origin of the problem is that discrete velocities are usually not divergence-free. Hence, the use of divergence-free velocity reconstructions is proposed wherever an L2 scalar product appears in the discrete variational formulation - which actually means committing a new kind of variational crime. The approach is illustrated and applied to several finite volume and finite element discretizations for the incompressible Navier-Stokes equations. In a finite element context, the new variational crime makes classical grad-div stabilization unnecessary, and even delivers error estimates for the discrete velocities that are completely independent of the pressure. Several numerical examples illustrate the theoretical results demonstrating that divergence-free velocity reconstructions may indeed increase the robustness and accuracy of existing convergent flow discretizations in physically relevant situations.

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Quan Long King Abdullah University of Science and Technology, KSA A Projection Method for Under Determined Optimal Experimental Designs Contributed Session CT3.1: Thursday, 17:30 - 18:00, CO1 Shannon–type expected information gain can be used to evaluate the relevance of a proposed experiment subjected to uncertainty. The estimation of such gain, however, relies on a double-loop integration. Moreover, its numerical integration in multidimensional cases, e.g., when using Monte Carlo sampling methods, is therefore computationally intractable for realistic physical models, especially those involving the solution of partial differential equations. In this paper, we present a new methodology, based on the Laplace approximation for the integration of the posterior probability density function (pdf), to accelerate the estimation of the expected information gains in the model parameters and predictive quantities of interest for both determined and under determined models. We obtain a closed– form approximation of the inner integral and the corresponding dominant error term, such that only a single–loop integration is needed to carry out the estimation of the expected information gain. In this work, we extend that method to the general cases where the model parameters could not be determined completely by the data from the proposed experiments. We carry out the Laplace approximations in the directions orthogonal to the null space of the corresponding Jacobian matrix, so that the information gain (Kullback–Leibler divergence) can be reduced to an integration against the marginal density of the transformed parameters which are not determined by the experiments. Furthermore, the expected information gain can be approximated by an integration over the prior, where the integrand is a function of the projected posterior covariance matrix. To deal with the issue of dimensionality in a complex problem, we use Monte Carlo sampling or sparse quadratures for the integration over the prior probability density function, depending on the regularity of the integrand function. We demonstrate the accuracy, efficiency and robustness of the proposed method via several nonlinear under determined numerical examples. They include the designs of the scalar parameter in an one dimensional cubic polynomial function with two indistinguishable parameters forming a linear manifold, respectively, and the boundary source locations for impedance tomography in a square domain, considering the parameters as a piecewise linear continuous random field. Joint work with Marco Scavino, Raul Tempone, and Suojin Wang.

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Petr Louda Czech Technical University in Prague, CZ Numerical simulations of laminar and turbulent 3D flow over backward facing step Contributed Session CT4.3: Friday, 08:20 - 08:50, CO3 Numerical simulations of laminar and turbulent 3D flow over backward facing step 1

P. Louda2 , P. Sváček1 , K. Kozel1 , J. Příhoda2 , Dept. of Technical Mathematics CTU Prague, Karlovo nám. 13, CZ-121 35 Prague 2 2 Institute of Thermomechanics AS CR, Dolejškova 5, CZ-182 00 Prague 8 Corresponding author: [email protected] ABSTRACT

The work deals with 3D numerical simulations of incompressible flow in channel of rectangular cross-section with backward facing step. The flow regimes considered are laminar as well as turbulent. The mathematical model is based on NavierStokes for laminar and Reynolds averaged Navier-Stokes equations for turbulent regime. Two types of numerical methods are used: • Implicit finite volume method solving governing equations by artificial compressibility method. The approximation of convective terms is based on third order interpolation on structured grid of hexahedrons, the discretization of viscous term is second order accurate. The time discretization is backward Euler scheme. • Stabilized finite element method solving weak formulation of governing equations. The flow velocity and pressure are approximated by continuous piecewise linear functions using streamline-upwind/ Petrov-Galerkin and pressure stabilizing/ Petrov-Galerkin method together with div-div stabilization. The results of both methods are compared in the laminar case and also in turbulent cases. The turbulence is modelled by two-equation eddy-viscosity models (TNT k-ω, SST) and by an explicit algebraic Reynolds stress model (EARSM, Wallin, Hellsten) and by V2F model (Durbin). The numerical results are compared with 3D experimental data acquired using PIV technique.

References [1] P. Louda, J. Příhoda, P. Sváček, and K. Kozel. Numerical simulation of separated flows in channels. J. of Thermal Science, 21(2):145–153, 2012. [2] P. Louda, P. Sváček, K. Kozel, and J. Příhoda. Numerical simulations of separation behind backward facing steps. In D. T. Tsahalis, editor, IC-EpsMsO 4th International Conference on Experiments/Process/System/Modelling/Simulation/Optimization, pages 437–444, Laboratory of Fluid Mechanics and Energy, University of Patras, 2011. ISBN 978-960-98941-8-0.

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[3] P. Sváček. Numerical Modelling of Aeroelastic Behaviour of an Airfoil in Viscous Incompressible Flow. Applied Mathematics and Computation. vol. 217, no. 11, p. 5078-5086. ISSN 0096-3003, 2011. [4] P. Louda, K. Kozel, J. Příhoda. Numerical solution of 2D and 3D viscous incompressible steady and unsteady flows using artificial compressibility method. Int. J. for Numerical Methods in Fluids 56, pp. 1399–1407, 2008. Joint work with P. Svacek, K. Kozel, and J. Prihoda.

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Robert Luce University of Pau, FR Robust local flux reconstruction for various finite element methods Minisymposium Session ADFE: Wednesday, 11:00 - 11:30, CO016 We are interested in local reconstructions of the gradient of primal finite element approximations. We consider conforming, nonconforming and totally discontinuous (Galerkin) methods (abbreviated as CG, NC, DG in the following) of arbitrary order. Such reconstructions have many applications, such as a posteriori error estimation and numerical approximations of coupled system. Our first aim is to present a uniform approach to flux reconstruction. We start from a hybrid formulation covering all considered finite element methods. The Lagrange multipliers compensating for the different weak continuity conditions yield approximations to the normal fluxes. It turns out that they can be computed locally in all cases on patches defined by the support of the lowest-order basis functions. Then these multipliers are used to define local corrections in broken Raviart-Thomas spaces. Our second aim is to study relations between the different methods. Especially we prove that the DG-method with stabilisation parameter γ converges uniformly in h with the convergence rate 1/γ towards the CG or NC solution, depending on the employed form of stabilisation. In addition, the same convergence result holds true for the reconstructed fluxes and therefore for the error estimators. The theoretical results will be illustrated by numerical tests. Joint work with Roland Becker, and Daniela Capatina.

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Lin-Tian Luh Providence University of Taiwan, TW The Criteria of Choosing the Shape Parameter for Radial Basis Function Interpolations Contributed Session CT1.2: Monday, 17:30 - 18:00, CO2 The main purpose of this report is to present concrete and useful criteria for choosing the constant c contained in the famous radial function β β h(x) := Γ(− )(c2 + |x|2 ) 2 , β ∈ R\2N≥0 , c > 0 2

(1)

which is called multiquadric for β > 0 and inverse multiquadric for β < 0, respectively. Here Γ denotes the classical gamma function. The optimal choice of c is a longstanding question and has obsessed many experts in the field of radial basis functions(RBFs). Most time what people can do is just making experiments and try to build a model o predict the influence of c, for some special cases. Here, we make a lucid clarification for its influence on the error estimates and show it with a concrete function, denoted by M N (c). The approximated functions lie in a function space which is equivalent to Gaussians’ native space, and is denoted by Eσ . Then, |f (x) − sf (x)| ≤ M N (c) · F (δ), for all f ∈ Eσ , where sf is the frequently used interpolation function and δ is the fill distance which measures the spacing of the data points. Both M N (c) and F (δ) contribute to the error bound, but M N (c) is more influential. The constant σ describes the rate of decay for the Fourier transform of f . We find M N (c) depends on four parameters, β, σ, the dimension n, and the fill distance δ. So the optimal choice of c which minimizes the value of M N (c) also depends on the four parameters. There are three cases. Here l contained in M N (c) corresponds to the fill distance and is inversely proportional to the fill distance. Case1. β < 0, |n + β| ≥ 1 and n + β + 1 ≥ 0 Let f ∈ Eσ and h be as in (1). Then   M N (c) := c

β−n+1−4l 4

where ξ∗ =

cσ +

p

(ξ ∗ )

n+β+1 2

ecξ

∗

−

(ξ∗ )2 σ

1/2

c2 σ 2 + 4σ(n + β + 1) . 4

Case2. β = −1 and n = 1 Let f ∈ Eσ and h be as in (1). Then β

M N (c) := c 2 −l where

( M (c) :=



1/2 √ 1 + 2 3M (c) ln2

1

e1− c2 σ

√ 2 2 g( cσ+ c4 σ +4σ )

if 0 < c ≤ if

√2 3σ

√2 , 3σ

0 and n ≥ 1 Let f ∈ Eσ and h be as in (1). Then ( M N (c) := c

1+β−n−4l 4

(ξ ∗ )

1+β+n 2

e 249

(ξ∗ )2 σ

ecξ

∗

)1/2

,

where ∗

ξ =

cσ +

p

c2 σ 2 + 4σ(1 + β + n) . 4

Example of Case1: In the figure, b0 controls the domain size of the approximated functions and is roughly speaking the diameter of the domain. It’s obvious that the optimal c in this sitution is around 9. Since F (δ), which is independent of c, also contributes to the error bound, the actual error is much smaller than 10−16 .

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Vladimir Lukin Keldysh Institute of Applied Mathematics, RAS, RU Mathematical modelling of radiatively accelerated canalized magnetic jets Contributed Session CT1.3: Monday, 18:00 - 18:30, CO3 One of the most interesting classes of astrophysical processes is the formation of jet outflows in active galaxy nuclei (e.g, galaxy M87), microquasars and many other objects. The jet consists of high energetic magnetized matter bullets. It propagates inside the cone with 6◦ angle. Matter velocity in M87 jet reaches the value of 0.8c (c — velocity of light). We construct and investigate the mathematical model of the jet matter radiation acceleration inside the canal over the hot gravitating object with thin accretion disk. The model is based on [1, 2] models and includes 2D axisymmetric radiative magnetohydrodynamic (RMHD) equation system: ∂B ∂ρ + ∇ρv = 0, = ∇ × (v × B) , ∂t ∂t   ∂ ˆ +T ˆ = 1 (∇ × B) × B + Fg , (ρv + G) + ∇ · Π ∂t 4π ∂ 1 (e + U ) + ∇ · (v (e + p) + W) = ((∇ × B) × B) · v + Fg · v, ∂t 4πZ

Γ(t, x, ω, ω 0 )I(t, x, ω 0 ) dω 0 .

ω · ∇I(t, x, ω) + β(t, x)I(t, x, ω) = β(t, x)

(1) (2) (3) (4)

Ω

ˆ — impulse flow density, Here ρ is the matter density, v — velocity vector, Π Πij = pδij + ρvi vj , p — gas pressure, e — gasR energy, B — magnetic field, Fg — gravitation force, I — radiation intensity, Γ(t, x, ω, ω 0 )I(t, x, ω 0 ) dω 0 — Ω

scattering integral, β(t, x) — matter scattering coefficient. We consider the matter 2 p is the ideal gas, so e = ρ|v| 2 + γ−1 . We use Rayleigh scattering function and the scattering cross section is σT = 6.652 × 10−29 sm2 . The gravitating body of mass M is situated in the origin of coordinates. Figure 1 shows model scheme. Numerical method for unstructured triangular grids based on splitting into physical processes is used to solve the system of equations. Calculation of the unknowns at each time step consists of the following phases: solution of the gas dynamic equations system by the HLLC method; divergence-free approximation of Faraday’s law on the staggered difference cell; integration of radiation transfer equation (RTE) by the discrete directions method; approximation of the geometrical and power sources at the system right side. The parallel numerical code for this method is developed. For the algorithm of RTE (4) numerical integration we used following parallel strategies: the RTE integration along the traced on the grid beams corresponding to a given node of the spatial grid is independent for each node, so the shared memory OpenMP technology is usefull; following the nVidia CUDA technology computation of each of the Nscat elementary integrals for the given spatial node is implemented using thread block inside one graphic multiprocessor and every elementary integral is computed by one thread inside the block. Obtained calculations acceleration efficiency is 10.8 times for the OpenMP alongthe-beam integration using two 6-core Intel Xeon X5670 2.93 Ghz processors and 251

82.3 times for scattering integral calculation using nVidia Tesla GPU. The calculations were performed on the K-100 cluster, KIAM RAS. At the begin of calculations there is magnetized channel inside the computation domain. The impact of radiation field leads to uprising of radiative accelerating force acting on the rarefied matter. Driven by the radiation force the matter is rapidly accelerating up to velocity of 1/5c. The jet is well collimated, magnetic field inside the channel preserves its structure, the canal is surrounded by optically thick magnetized walls (see Figure 2). The jet flow contains the velocity discontinuities of shock wave type. Dense matter bullets are formed on the front of every discontinuity. The bullets release period is 13 days. The work has been partly supported by the Russian Foundation for Basic Research (projects 12-01-00109, 12-02-00687, 12-01-31193) and by the Science School 1434.2012.2.

References [1] M.P. Galanin, Yu.M. Toropin, and V.M. Chechetkin, Astron. Rep. 43, 119 (1999). [2] M.P. Galanin, V.V. Lukin, and V.M. Chechetkin, Math. Mod. and Comp. Sim., 4, 3 (2012).

Figure 1: Scheme of the jet launching system model

Figure 2: Density and velocity modulus distributions Joint work with M.P. Galanin, and V.M. Chechetkin.

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Francisco Macedo EPFL, CH A low-rank tensor method for large-scale Markov Chains Contributed Session CT1.1: Monday, 17:00 - 17:30, CO1 A number of practical applications lead to Markov Chains with extremely large state spaces. Such an instance arises from models for calcium channels, which are structures in the body that allow cells to transmit electrical charges to each other. These charges are carried on a calcium ion which can travel freely back and forth through the calcium channel. The state space of a Markov process describing these interactions typically grows exponentially with the number of cells. More generally, Stochastic Automata Networks (SAN s) are networks of interacting stochastic automata. The dimension of the resulting state space grows exponentially with the number of involved automata. Several techniques have been established to arrive at a formulation such that the transition matrix has Kronecker product structure. This allows, for example, for efficient matrix-vector multiplications. However, the number of possible automata is still severely limited by the need of representing a single vector (e.g., the stationary vector) explicitly. We propose the use of lowrank tensor techniques to avoid this barrier. More specifically, an algorithm will be presented that allows to approximate the solution of certain SAN s very efficiently in a low-rank tensor format. Joint work with Prof. Daniel Kressner.

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Pravin Madhavan University of Warwick, GB On a Discontinuous Galerkin Method for Surface PDEs Contributed Session CT2.5: Tuesday, 14:30 - 15:00, CO016 Partial differential equations on manifolds have become an active area of research in recent years due to the fact that, in many applications, models have to be formulated not on a flat Euclidean domain but on a curved surface. For example, they arise naturally in fluid dynamics (e.g. surface active agents on the interface between two fluids ) and material science (e.g. diffusion of species along grain boundaries) but have also emerged in areas as diverse as image processing and cell biology (e.g. cell motility involving processes on the cell membrane, or phase separation on biomembranes). Finite element methods (FEM) for elliptic problems and their error analysis have been successfully applied to problems on surfaces via the intrinsic approach in Dziuk (1988) based on interpolating the surface by a triangulated one. However, as in the planar case there are a number of situations where FEM may not be the appropriate numerical method, for instance, advection dominated problems which lead to steep gradients or even discontinuities in the solution. DG methods are a class of numerical methods that have been successfully applied to hyperbolic, elliptic and parabolic PDEs arising from a wide range of applications. Some of its main advantages compared to ‘standard’ finite element methods include the ability of capturing discontinuities as arising in advection dominated problems, and less restriction on grid structure and refinement as well as on the choice of basis functions. The main idea of DG methods is not to require continuity of the solution between elements. Instead, inter-element behaviour has to be prescribed carefully in such a way that the resulting scheme has adequate consistency, stability and accuracy properties. In my presentation I will investigate the issues arising when attempting to apply DG methods to problems on surfaces. We restrict our analysis to a linear secondorder elliptic PDE on a compact smooth connected and oriented surface. An interior penalty (IP) method is introduced on a discrete surface and we derive a-priori error estimates by relating the latter to the original surface via the lift introduced in Dziuk (1988). The estimates suggest that the geometric error terms arising from the surface discretisation do not affect the overall convergence rate of the IP method when using linear ansatz functions. This is then verified numerically for a number of test problems. Joint work with Andreas Dedner, and Bjorn Stinner.

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Immanuel Maier University of Stuttgart, DE A reduced basis method for domain decomposition problems Contributed Session CT4.4: Friday, 08:20 - 08:50, CO015 Reduced basis (RB) methods allow efficient model reduction of parametric partial differential equations. We propose a new approach for combining model reduction methods with domain decomposition techniques. Important components of the RB technique, as the decomposition into parameter-independent and parameterdependent computations (offline/online - decomposition), the greedy-algorithm for generating the basis and a-posteriori error estimation are maintained. Some related RB methods for coercive homogeneous domain decomposition problems already have been developed. Starting from the RB element method (RBEM) [2], the scRBEM [3] and the RDF method [4] represent efforts to accommodate to the decomposed nature of problems. We point out the relationship to these methods. In particular, they mainly address network-type problems. In contrast, our approach [1] treats coercive problems, where the system’s topology is known a-priori. Expensive solutions of the full system can be computed offline and used as snapshots for several RB spaces on the subdomains. The snapshots are chosen offline by a greedy procedure. For the construction of RB spaces a framework separating intra-domain and interface-associated functions is established. Online an iterative RB solution method can be formulated; convergence is proven theoretically. The overall method is investigated numerically with respect to accuracy and efficiency. We present the abstraction in a general framework and consider the extension of our method to heterogeneous domain decomposition problems. Possible problem instantiations are the coupling of free flow with porous media flow modelled by the Stokes and Darcy equations or the flow around an obstacle, modelled by Stokes in an inner region and by Laplace’s equation in an outer region (due to negligence of viscous effects) [5].

References [1] I. Maier and B. Haasdonk. An Iterative Domain Decomposition Procedure for the Reduced Basis Method. SimTech Preprint, University of Stuttgart, 2012. [2] Y. Maday and E.M. Rønquist. The Reduced Basis Element Method: Application to a Thermal Fin Problem. Journal of Scientific Computing, 26 (2004), 240–258. [3] D.B.P. Huynh, D.J. Knezevic and A.T. Patera. A Static Condensation Reduced Basis Element Method: Approximation and A Posteriori Error Estimation. Submitted to M2AN, 2011. [4] L. Iapichino. Reduced Basis Methods for the Solution of Parametrized PDEs in Repetitive and Complex Networks with Application to CFD. PhD thesis, EPFL, 2012. [5] A. Quarteroni and A. Valli. Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, 1999. 255

Joint work with Bernard Haasdonk.

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Charalambos Makridakis University of Sussex, GB Consistent Atomistic / Continuum approximations to atomistic models. Minisymposium Session MSMA: Monday, 16:00 - 16:30, CO3 We discuss recent results related to the problem of the atomistic-to-continuum passage and the design of corresponding coupled methods for crystalline materials. In particular we will comment on issues related to the analysis of Cauchy– Born/nonlinear elasticity approximations to atomistic models in two and three space dimensions. We will present new coupled atomistic/continuum methods which are consistent.

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Olli Mali University of Jyväskylä, FI Estimates of Effects Caused by Incompletely Known Data in Elliptic Problems Generated by Quadratic Energy Functionals Contributed Session CT3.7: Thursday, 17:30 - 18:00, CO122 In mathematical modelling, the data of the problem is often known with limited accuracy. Instead of exact data values, some set of admissible data is known. This set generates a family of problems and the respective set of solutions. We consider linear elliptic problems generated by quadratic energy functionals. The coefficients of the problem and the respective right-hand side are considered to be known by limited accuracy. The knowledge we have is of the form: mean value±variations, which is motivated by the engineering practice. The quantity of interest is the radius of the solution set. It is the distance between the solution related to the “mean” data and the most distant member of the solution set. The relation between the magnitude of variations of the data and the radius of the solution set is of special interest. This question has been studied in [1, 2] for diffusion type problems in terms of the primal variable. Here, we study also the relationship between the set of admissible data and the dual variable as well as the primal–dual pair in a combined norm.

References [1] O. Mali and S. Repin, Estimates of the indeterminacy set for elliptic boundary–value problems with uncertain data, J. Math. Sci. 150, pp. 18691874, 2008. [2] O. Mali and S.Repin, Two-sided estimates of the solution set for the reactiondiffusion problem with uncertain data, Applied and numerical partial differential equations, 183–198, Comput. Methods Appl. Sci., 15, Springer, New York, 2010. Joint work with S. Repin.

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Gunar Matthies Universität Kassel, DE A two-level local projection stabilisation on uniformly refined triangular meshes Contributed Session CT2.3: Tuesday, 15:30 - 16:00, CO3 The local projection stabilisation (LPS) has been successfully applied to scalar convection-diffusion-reaction equations, the Stokes problem, and the Oseen problem. A fundamental tool in its analysis is that the interpolation error of the approximation space is orthogonal to the discontinuous projection space. It has been shown that a local inf-sup condition between approximation space and projection space is sufficient to construct modifications of standard interpolations which satisfy this additional orthogonality. There are different versions of the local projection stabilisation on the market; we will consider the two-level approach based on standard finite element spaces Yh on a mesh Th and on projection spaces Dh living on a macro mesh Mh . Hereby, the finer mesh is generated from the macro mesh by a certain refinement rules. In the usual two-level local projection stabilisation on triangular meshes, each macro triangle M ∈ Mh is divided by connecting its barycentre with its vertices. Three disc triangles T ∈ Th are obtained. Then, the pairs (Pr,h , Pr−1,2h ), r ≥ 1, of spaces of continuous, piecewise polynomials of degree r on Th and discontinuous, piecewise polynomials of degree r − 1 on Mh satisfy the local inf-sup condition and can be used within the LPS framework. One disadvantage of this refinement technique is however that Th contains simplices with large inner angles even in the case of a uniform decomposition Mh into isosceles triangles. Another drawback is that this refinement rule leads to nonnested meshes and spaces whereas the common refinement technique of one triangle into 4 similar triangles (called red refinement in adaptive finite elements) results into nested meshes and spaces. disc We will show that in the two-dimensional case the pairs (Pr,h , Pr−1,2h ), r ≥ 2, satisfy the local inf-sup condition with the refinement of one triangle into 4 triangles. Consequently, the LPS can be also applied on sequences of nested meshes and spaces and keeping the same error estimates. Finally, we compare the properties of the two resulting LPS methods based on the different refinement strategies by means of numerical test examples for convection-diffusion problems with dominating convection. Joint work with Lutz Tobiska.

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Miriam Mehl Technische Universität München, DE Towards massively parallel fluid-structure simulations – two new parallel coupling schemes Minisymposium Session NFSI: Thursday, 10:30 - 11:00, CO122 Multi-physics applications and in particular fluid-structure interactions have dramatically gained importance in various kinds of applications from biomedical computing to engineering design due to both an increasing need for accuracy and, thus, more accurate models, and a huge available computing power on todays supercomputers that allows to tackle the computational challenges of such simulations. Since only a high grid resolution ensures a discretization accuracy that accounts for the increased modelling accuracy as compared to single-physics models, scalability of multi-physics simulation codes on massively parallel machines is mandatory. However, complex multi-physics models not only pose large computational challenges but also implementational, software engineering and maintance challenges. The latter can be eased by using a partitioned approach, i.e., reusing existing and trusted codes for the involved single-physics effects and combining them with suitable coupling methods to a multi-physics simulation environment. This reduces the implementational effort substantially and, if done carefully, allows a flexible exchange of the involved software components. The downside of partitioned approaches are stability issues induced by the high-level coupling of the underlying interaction equations. For fluid-structure interactions, stability issues become more severe with decreasing structural density, decreasing fluid compressibility, and increasing structure size relative to the size of the fluid domain, e.g.. A lot of work has been invested by various groups to overcome these difficulties and, indeed, sophisticated coupling methods have been found that ensure stability even for massless structures and completely incompressible fluids. Hereby, the most common basic scheme is a Gauss-Seidel type coupling executing fluid and structure solver in an alternating manner transfering forces as boundary conditions from fluid to structure and displacements and velocities from the structure to the fluid (Dirichlet-Neumann coupling). For incompressible fluids, a strong coupling has to be ensured in general in order to achieve stability of the transient simulation, that is, several iterations of this staggered fluid-structure solve have to be executed within each time step. Convergence of these methods is ensured either by Aitken underrelaxation [1] or quasi-Newton interface methods [4]. In particular the latter lead to very good convergence rates even in ’hard’ cases. However, there’s one drawback of these schemes for parallel computing: fluid and structure solver have to be executed one after the other which prevents good scalability due to the unbalanced computational needs: the structural solver is in general much cheaper then the fluid part and doesn’t scale on a large number of processors. Numerical methods executing fluid and structure solver in parallel have been applied mostly to problems with compressible fluids. Ross [3] for example solves fluid and structure in parallel followed by a solve step of an interface equation, which, however, needs discretization details of both solvers at the interface and is thus not suited for coupling black-box solver which is our aim. Farhat [2] proposes parallel Jacobian-like time-stepping which works considerably well in a weak coupling setting executing fluid and structure solver only once per time step but turns out to be equivalent to two separate staggered couling schemes if done iteratively in

260

each time step. We propose two new coupling methods combining the coupling ideas of [3] and [2] with the interface quasi-Newton method from [4]. In the presentation, we show a uniform formulation of all three methods – the original staggered scheme and our two parallel schemes – in terms of fixed point problems. The quasi-Newton method then uses a least squares method based on previous iteration data to estimate the effect of an approximate Jacobian. For several test and benchmark cases, we show that we achieve iteration numbers comparable to those achieved in [4] for the staggered approach, which marks and important step towards efficient multi-physics simulations in the ’exascale era’.

References [1] Küttler, U. and Wall, W. Fixed-point fluid-structure interaction solvers with dynamic relaxation. Comput. Mech. (2008) 43:61–72. [2] Farhat, C. and Lesoinne, M. Two efficient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems. Comput. Method. Appl. M. (2000) 182:499–515. [3] Ross, M.R., Felippa, C.A., Park, K.C. and Sprague, M.A. Treatment of acoustic fluid-structure interaction by localized Lagrange multipliers: Formulation. Comput. Methods Appl. Mech. Eng. (2008) 197:305–3079. [4] Degroote, J., Bathe, K.-J. and Vierendeels, J. Performance of a new partitioned procedure versus a monolithic procedure in fluid-structure interaction Comput. Struct. (2009) 87:793–801. Joint work with Hans-Joachim Bungartz, Bernhard Gatzhammer, and Benjamin Uekermann.

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Lina Meinecke Uppsala University, SE Stochastic simulation of diffusion on unstructured meshes via first exit times Contributed Session CT3.2: Thursday, 17:00 - 17:30, CO2 In molecular biology it is of interest to simulate the diffusion and reactions of molecules such as proteins in a cell. When simulating the biochemistry in a biological cell, many molecules are present in only very low copy numbers. As a result a macroscopic or deterministic description with the reaction-diffusion equation is inaccurate and does not reproduce experimental data and a stochastic description is needed [1]. The diffusion of the molecules is then given by Brownian dynamics and the reactions between them occur with certain probability. For stochastic simulation of the diffusion, the cell is partitioned into compartments or voxels in a mesoscopic model. The number of molecules in a voxel is recorded and the molecules can jump between neighbouring voxels to model diffusion. In order to accurately represent the geometry of the cell including outer and inner curved boundaries it is helpful to use unstructured meshes for the voxels. The probabilities to jump between the voxels is given in [2] by a discretization of the Laplacian with the finite element method (FEM) on the mesh. Solutions of the diffusion equation with FEM encounter problems on some unstructured meshes in 3D. If the mesh is of poor quality, the maximum principle may not be satisfied by the FEM solution and the jump coefficients derived from it may be negative. We present a new approach to diffusion simulation using first exit times that for unstructured meshes guarantees positive jump coefficients. These first exit times can be sampled from the survival probability for molecules within a voxel. It will be shown that this approach yields accurate results on multidimensional Cartesian meshes and on meshes with variable mesh size in 1D. This approach is extended to unstructured 2D meshes of varying quality by solving the local equation for the exit time or computing the exit time between the nodes along the edges. The method is compared to the accuracy obtained with FEM coefficients and jump coefficients determined by the finite volume method (FVM). 1. A. Mahmutovic, D. Fange, O. G. Berg, and J. Elf, Lost in presumption: stochastic reactions in spatial models, Nature Methods 9, 1163-1166, 2012. 2. S. Engblom, L. Ferm, A. Hellander, and P. Lötstedt. Simulation of stochastic reaction-diffusion processes on unstructured meshes. SIAM J. Sci. Comput., 31(3):1774-1797, 2009.

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Figure 1: Mesh of poor quality: one angle is bigger than 90 degrees.

Figure 2: Stochastic Simulation of Diffusion on an unstructured mesh of good quality. Joint work with Per Lötstedt.

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Ward Melis KU Leuven, BE A relaxation method with projective integration for solving nonlinear systems of hyperbolic conservation laws Contributed Session CT4.1: Friday, 09:50 - 10:20, CO1

1

Introduction

Hyperbolic conservation laws are ubiquitous in domains such as fluid dynamics, plasma physics, traffic modeling and electromagnetism. We present a general strategy for systems of nonlinear hyperbolic conservation laws: ∂u ∂F (u) + = 0, ∂t ∂x

(1)

in which x ∈ RD contains the independent variables; u ∈ RI holds I conserved quantities ui : RD × [0, T ] → R, i = 1, ..., I; and F ∈ RI corresponds to the vector of flux functions Fi : R → R, i = 1, ..., I which may be nonlinear in each of the functions ui . The method is based on combining a relaxation method with projective integration. In a relaxation method, the nonlinear conservation law is approximated by a system of kinetic equations, in which a small relaxation parameter 0 <   1 is present. The general idea of these methods is to eliminate the nonlinear flux term, at the expense of introducing a stiff nonlinear source term. The kinetic equation describes the evolution of a distribution function f (x, v, t) : RD ×RD ×[0, T ] → RI of particles with positions x ∈ RD and velocities v ∈ RD (see [?]): ∂ ∂ 1 f (x, v, t) + v f (x, v, t) = (M (P (f (x, v, t))) − f (x, v, t)) , ∂t ∂x 

(2)

and is constructed such that, in the hydrodynamic limit ( → 0), the solution converges to that of (1).

2

Method and results

The projective integration algorithm was developed by Gear and Kevrekidis (see [?]). It consists of the following steps: 1. Perform K small steps for equation (2) with a naive explicit integrator with time step δt = O(2 ) (this is called an inner integrator). When ε is small, this will enforce convergence of the fast modes to the slow manifold that is characterized by the density ρ. 2. Subsequently, perform a large time step ∆t by extrapolation in time (this is called an outer integrator). We show that the method allows both ∆t and K to be independent of , while being fully explicit and general. Moreover, the method can be of arbitrary order and its implementation is surprisingly simple, even for complex nonlinear systems. We will present the method and illustrate its performance on the linear advection equation, Burgers’ equation and the Euler equations in fluid dynamics, both in a one and two dimensional domain. 264

Figure 1: Left: stability plot of the projective forward Euler (PFE) method in terms of the amplification factor τ of the inner integrator. Right: Order test for PFE with FE as inner integrator and three different spatial orders. Solid lines represent the calculated error whereas the dotted lines shows the expected error. Joint work with Pauline Lafitte, and Giovanni Samaey.

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Wim Michiels KU Leuven, BE Projection based methods for nonlinear eigenvalue problems and associated distance problems Minisymposium Session NEIG: Thursday, 14:00 - 14:30, CO2 We consider the nonlinear eigenvalue problem ! m X Ai pi (λ) v = 0, λ ∈ C, v ∈ Cn ,

(1)

i=1

where A1 , . . . , Am are complex n × n matrices and the scalar functions pi : C → C, 1 = 1, . . . , m, are entire. Problems of the form (1) do not only include polynomial eigenvalue problems but also eigenvalue problems arising from systems of delay differential equations. In the first part of the talk we present a rational Krylov method for solving Pm the nonlinear eigenvalue problem (1). The method approximates A(λ) = i=1 Ai pi (λ) by polynomial Newton and/or Hermite interpolation, resulting in a generalized eigenvalue problem which is solved by the rational Krylov method. We show that, by matching the interpolation points with the poles of the rational Krylov method, the resulting algorithm can be constructed in a fully dynamic way, in the sense that the degree of the interpolating polynomial does not need to be fixed beforhand. New interpolation points can be added on the run (on the basis of the quality of the obtained eigenvalue approximations), and arbitrary accuracy of eigenvalue approximations can be obtained by a sufficiently large number of iterations. The latter is in contrast with an ‘approximate plus solve’ approach, where the final accuracy is limited by the chosen degree of the polynomial approximation. In case of Hermite interpolation in one point, the Newton rational Krylov method reduces to the infinite Arnoldi method, where the dynamic property is reflected in the interpretation as the standard Arnoldi method applied to an infinite-dimensional linear operator whose spectrum corresponds with the one of A(λ). We illustrate that with an appropriate choice of interpolating points/poles, the method is suitable for a global search for eigenvalues in a region of interest, as well as for local corrections on individual eigenvalues. Finally, for very large problems, we show that the subspace generated by the Newton rational Krylov method can be used to project (1), resulting in a small nonlinear eigenvalue problem, which can be solved using a method of choice. In the second part of the presentation we point out how these nonlinear eigenvalue solvers can be used as building blocks in algorithms for distance problems. More precisely, we consider the situation where (1) is perturbed to ! m X (Ai + δAi )pi (λ) v = 0, λ ∈ C, v ∈ Cn . (2) i=1

Assuming that (1) is stable in the sense that all eigenvalues are confined to the open left half plane or the open unit disk, the distance to instability can be defined as the smallest size of the perturbations in (2) which lead to instability. This definition depends on (i) the class of allowable perturbation and (ii) a global measure of the combined perturbations on the different coefficient matrices. For both real and complex valued allowable perturbations and for various perturbation measures 266

we present numerical algorithms for computing the distance to instability. As a common feature, in all these cases it is sufficient to restrict the perturbations in (2) to rank one or rank two perturbations. This leads to algorithms on manifolds of low rank matrices. Both discrete iteration maps and differential equations on such manifolds will be considered. Joint work with Roel Van Beeumen, Karl Meerbergen, and Dries Verhees.

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Agnieszka Miedlar TU Berlin, DE Multiscale adaptive finite element method for PDE eigenvalue/eigenvector approximations Minisymposium Session NEIG: Thursday, 15:00 - 15:30, CO2 In this talk we present a multiscale adaptive finite element method for PDE eigenvalue problems which will use one scale, e.g., P 1 finite elements, to approximate the solution and finer scale, e.g., P 2 finite elements, to capture the approximate residual. Starting from the results of Grubišić and Ovall [GO09] on the reliable and efficient asymptotically exact a posteriori hierarchical error estimators in the self-adjoint case, we explore the possibility to use the enhanced Ritz values and vectors to restart the iterative algebraic procedures within the adaptive algorithm. Using higher order hierarchical polynomial finite element bases, as indicated by Bank [Ban96] and by Le Borne and Ovall [LO12], our method generates discretization matrices which are almost diagonal. This construction can be repeated for the complements of higher (even) order polynomials and yields a structure which is particularly suitable for designing computational algorithms with low complexity. We present some numerical results for both the symmetric as well as the nonsymmetric eigenvalue problems. 2010 Mathematics Subject Classification. 65F10, 65F15, 65N15, 65N22, 65N25, 65N30, 65M60 Key words. eigenvalue problems, FEM, finite element method, AFEM, adaptive finite element method

References [Ban96] R. E. Bank, Hierarchical bases and the finite element method, Acta numerica, 1996, Acta Numer., vol. 5, Cambridge Univ. Press, Cambridge, 1996, pp. 1–43. [GO09] L. Grubišić and J. S. Ovall, On estimators for eigenvalue/eigenvector approximations, Math. Comp. 78 (2009), no. 266, 739–770. [LO12] S. Le Borne and J. S. Ovall, Rapid error reduction for block GaussSeidel based on p-hierarchical bases, Numer. Linear Algebra Appl. (2012), Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nla.1841. Joint work with Luka Grubisic, and Jeffrey S. Ovall.

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Giovanni Migliorati CSQI, MATHICSE, EPF Lausanne, CH Adaptive polynomial approximation by random projection of multivariate aleatory functions Contributed Session CT1.1: Monday, 17:30 - 18:00, CO1 In this talk we present recent results on polynomial approximation by the Random Discrete L2 Projection (RDP) of functions depending on multivariate random variables distributed with a given probability density. The RDP is computed using point-wise noise-free evaluations of the target function in independent realizations of the random variables. First, we recall the main results achieved in [1, 2, 3, 4] concerning the stability and accuracy of the RDP. In particular, we focus on the relation between the number of sampling points and the dimension of the polynomial space that ensures an accurate RDP, independently of the “shape” of the polynomial space. The effects of the smoothness of the target function and of the number of random variables are addressed as well. Then we focus on the approximation of Quantities of Interest depending on the solution to of a PDE with stochastic coefficients. For a class of isotropic PDE models with “inclusion-type” coefficients parametrized by a moderately large number of random variables we show that, with an a-priori optimal choice of the polynomial space, the RDP approximation error in expectation converges subexponentially w.r.t. the number of sampling points. Moreover, a comparison between the convergence rates of RDP and Stochastic Galerkin is established. Lastly we discuss adaptive polynomial approximation to approximate best N-terms sets of the coefficients in the polynomial expansion of the target function. We employ the results achieved in the theoretical analysis to devise strategies based on RDP that adaptively explore the unknown anisotropy of the target function and adaptively enrich the polynomial space. A critical issue that will be discussed concerns how to increase the number of sampling points during the adaptive algorithm. Numerical results will be presented as well.

References [1] G.Migliorati, F.Nobile, E.von Schwerin, R.Tempone: Analysis of the discrete L2 projection on polynomial spaces with random evaluations, submitted. Also available as MOX-report 46-2011. [2] A.Cohen, M.A.Davenport, D.Leviatan: On the stability and accuracy of Least Squares approximations, to appear on Found. Comput. Math. [3] G.Migliorati, F.Nobile, E.von Schwerin, R.Tempone: Approximation of Quantities of Interest in stochastic PDEs by the random discrete L2 projection on polynomial spaces, to appear on SIAM J. Sci. Comput. [4] A.Chkifa, A.Cohen, G.Migliorati, F.Nobile, R.Tempone: Discrete least squares polynomial approximation with random evaluations; application to parametric and stochastic PDEs, in preparation. Joint work with A.Chkifa, A.Cohen, F.Nobile, and R.Tempone.

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Shinya Miyajima Gifu University, JP Fast verified computation for solutions of generalized least squares problems Contributed Session CT2.1: Tuesday, 14:00 - 14:30, CO1 The generalized least squares problems considered in this talk is to find the nvector x that minimizes (Ax − b)T B −1 (Ax − b),

A ∈ Rm×n ,

b ∈ Rm ,

B ∈ Rm×m ,

(1)

where m ≥ n, A, b and B are given, A has full column rank, and B is symmetric positive definite. This problem arises in finding the least squares estimate of the vector x when we are given the linear model b = Ax + w with w an unknown noise vector of zero mean and covariance B. In several practical problems in econometrics [J. Johnston, Econometric Methods, second ed., McGraw-Hill, New York, (1972)] and engineering [D.B. Duncan, S.D. Horn, Linear dynamic recursive estimation from the viewpoint of regression analysis, J. Amer. Statist. Assoc. 67, 815–821 (1972)], A and B will have special block structure. It is well known that −1 the vector minimizing (1) is (AT B −1 A) AT B −1 b. Since B is symmetric positive definite, there exist matrices L satisfying B = LLT , which can be obtained by Cholesky decomposition or eigen-decomposition. In several applications, L is more basic and important than B, so that it is assumed in several papers (e.g. [C.C. Paige, Computer solution and perturbation analysis of generalized linear least squares problems, Math. Comp. 33(145), 171–183 (1979)]) + + that L is given. Then the solution can be written as (L−1 A) L−1 b, where (L−1 A) denotes the Moore-Penrose inverse of L−1 A. In this talk, we treat both of the cases when B is given and L is given. Stable algorithms for solving (1) have been proposed in [C.C. Paige, Fast numerically stable computations for generalized linear least squares problems, SIAM J. Numer. Anal. 16(1), 165–171 (1979)]. These algorithms are based on the idea that (1) is equivalent to the problem of finding x which minimizes v T v on the equality constraint b = Ax + Lv. In these algorithms, the equivalent problem is solved via orthogonal transformation. −1 In this talk, we consider numerically enclosing (AT B −1 A) AT B −1 b, specifically, computing error bounds of x ˜ using floating point operations, where x ˜ denotes −1 a numerical result for (AT B −1 A) AT B −1 b. As far as the author knows, algo−1 rithms for enclosing (AT B −1 A) AT B −1 b in (1) have not been known. Although −1 (AT B −1 A) AT B −1 b can be enclosed by utilizing the INTLAB [S.M. Rump, INTLAB - INTerval LABoratory, in T. Csendes (ed.), Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, 77–104 (1999)] routine, this approach involves large computational cost, since intervals including B −1 A and B −1 b, or L−1 A and L−1 b are required during the execution. −1 The purpose of this talk is to propose algorithms for enclosing (AT B −1 A) AT B −1 b in both of the cases when B is given and L is given. These algorithms do not require the intervals described above, and allow the presence of underflow in floating point arithmetic. In order to develop these algorithms, we establish theories for computing error bounds of x ˜. The error bounds obtained by the proposed algorithms are “verified” in the sense that all the possible rounding errors have been taken into account. In the case when B is given, the proposed algorithms do not assume but prove A and B to have full rank and to be positive definite, 270

respectively. In the case when L is given, the algorithms do not assume but prove A and L to have full rank and to be nonsingular, respectively. We introduce a technique for obtaining smaller error bounds and report numerical results to show the properties of the proposed algorithms.

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Olga Mula LJLL and CEA, FR The Generalized Empirical Interpolation Method: Analysis of the convergence and application to the Stokes problem Minisymposium Session ROMY: Thursday, 11:30 - 12:00, CO016 The extension of classical lagrangian interpolation is an old problem in approximation theory that remains a field of current active research (see, e.g. [6], or the kriging studies in the stochastic community such as [3]). This development involves two main tasks that must be addressed together: the generalization of the interpolating functions and of the position of the interpolating points so that the interpolation process is at least stable and close to the best approximation in some sense. Indeed, since classical lagrangian interpolation approximates general functions by finite sums of well-chosen, linearly independent interpolating functions (e.g. polynomial functions), the question on how to approximate general functions by general interpolating functions arises. As a consequence, an investigation on how to optimally select the interpolating points needs to be carried out (i.e. the well documented theory about the location of the interpolating points in classical polynomial interpolation needs to be enlarged). One step in this direction is the Empirical Interpolation Method (EIM, [1], [2], [6]) that has been developed in the broad framework where the functions f to approximate belong to a compact set F of a Banach space X . The structure of F is supposed to make any f ∈ F be approximable by finite expansions of small size of given basis functions. This is the case when the Kolmogorov n−width of F in X is small. Indeed, the Kolmogorov n−width of F in X , defined by sup inf kx − ykX (see [4]) measures the extent to which dn (F, X ) := inf Xn ⊂X dim(Xn )=n

x∈F y∈Xn

F can be approximated by some finite dimensional space Xn ⊂ X of dimension n. In general Xn is not known and the Empirical Interpolation Method builds simultaneously and recursively in n the set of interpolating functions and the associated interpolating points by a greedy selection procedure (see [1]), but note however that the approach can be shortcut in case the basis functions are available, then the interpolating points are the only output of EIM. A recent generalization of this interpolation process consists in generalizing the evaluation at interpolating points by application of a class of interpolating continuous linear functionals chosen in a given dictionary Σ ⊂ L(F ) and this gives rise to the so-called Generalized Empirical Interpolation Method (GEIM, [5]). In this newly developed method, the particular case where the space X = L2 (Ω) or X = H 1 (Ω) is considered, with Ω being a bounded spacial domain of Rd and F being a compact set of X . In this context, the aim of the talk is twofold: Since the efficiency of GEIM depends critically on the choice of the interpolating functions, we will first analyze the quality of the finite dimensional subspaces Xn ⊂ F built by the greedy selection procedure of GEIM. For this purpose, the accuracy of the approximation in Xn of the elements of F will be compared to the best possible performance which is the Kolmogorov n− width dn (F, L2 (Ω)). The convergence uses the Lebesgue constant Λn that evaluates the operator norm of the interpolation operator. The second part of the talk will be devoted to a numerical example motivated by an observation made in [5] where it was shown in a simple numerical experiment

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(a parameter dependent elliptic problem) that the GEIM provides cases where the Lebesgue constant Λn is uniformly bounded in n when evaluated in the L(L2 ) norm. We will extend the analysis to the Stokes equations and explain how we can take advantage of this framework in order to use GEIM to approximate a solution in the whole domain from the only knowledge of measurements from sensors.

References [1 ] Barrault, M. and Maday, Y. and Nguyen, N.C. Y. and Patera, A.T., An empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations, C. R. Acad. Sci. Paris, Série I., vol. 339, 667–672, 2004. [2 ] Grepl, M.A. and Maday, Y. and Nguyen, N.C. and Patera, A.T., Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations, M2AN (Math. Model. Numer. Anal.), vol. 41(3), 575-605, 2007. [3 ] Kleijnen, J.P.C. and van Beers, W., Robustness of Kriging when interpolating in random simulation with heterogeneous variances: Some experiments, European Journal of Operational Research, vol. 165, 826 - 834, 2005. [4 ] Kolmogoroff, A., Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse, Annals of Mathematics, vol. 37, 107-110, 1936. [5 ] Maday, Y. and Mula, O., A generalized empirical interpolation method: application of reduced basis techniques to data assimilation, Analysis and Numerics of Partial Differential Equations, vol. XIII, 221-236, 2013. [6 ] Maday, Y. and Nguyen, N.C. and Patera, A.T. and Pau, G.S.H., A general multipurpose interpolation procedure: the magic points, Commun. Pure Appl. Anal., vol. 8(1), 383-404, 2009. Joint work with Y. Maday, O. Mula, and G. Turinici.

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Naofumi Murata Keio University, JP Analysis on distribution of magnetic particles with hysteresis characteristics and field fluctuations Contributed Session CT3.6: Thursday, 17:30 - 18:00, CO017 Numerical approach to analyze magnetic particles’ behavior has been widely developed in the field of magnetic fluid and printer toners. It is well known that chain-like clusters are formed as a result of dipole-dipole interactions between particles. However, in most cases, the applied magnetic fields are constant and non-temporal. In those simulations, hysteresis characteristics of each magnetic particle are often neglected or approximated as constant or linear, not considering the whole hysteresis loop. Famous conventional methods such as free-energy theory or Monte Carlo simulations give practical results under some of those particular conditions. However these approximations smear out the effect of individual mutual interaction between particles, making the results averaged. These methods therefore cannot be applied for problems with field fluctuations and hysteresis characteristics of particles. In this research, analysis method on the behavior of magnetic particles with hysteresis characteristics under spatially and temporally fluctuating fields is proposed. In large system where mutual interactions of particles appear strongly, nonlinearity of hysteresis and driving force from the field fluctuation in addition to the energy dissipation by collisions might bring chaotic behavior and patterns of the particles. This research aims to find the foothold of chaotic behavior which appears under these conditions. The proposed method starts from the discretization and interpolation of fields by means of FEM rectangular elements. To model hysteresis characteristics of each particle, sigmoid functions were used to express the major hysteresis loop while minor loops were expressed with linear recoil lines. The hysteresis characteristic adopted here basically obeys the Madelung’s rules [1]. Collisions and clustering of particles were modeled by treating mechanical contacts, namely by solving Hertz’s contact problem. Time integration was carried out by a fourth order symplectic integrator. In the simulation, behavior of 200 magnetic particles under spatially and temporally fluctuating fields was examined. The results imply the nonlinearity of hysteresis characteristics greatly affect the final clustering patterns. [1] S.E. Zirka, Yu.I. Moroz, Hysteresis modeling based on transplantation, IEEE Transactions on Magnetics 31 (6) (1995) 3509–3511. Joint work with Kenji Oguni.

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Gulcin Mihriye Muslu Istanbul Technical University, TR New Numerical Results on Some Boussinesq-type Wave Equations Contributed Session CT2.2: Tuesday, 14:30 - 15:00, CO2 Boussinesq-type equations were proposed to model bi-directional propagation of nonlinear dispersive waves arising in many areas of science and engineering. Elastic waves and surface water waves are the two most studied phenomena in the literature within the context of a Boussinesq-type equation model. In this talk, we will focus on a Fourier pseudo-spectral method for solving one-dimensional Boussinesqtype equations. Then we will present our preliminary numerical results concerning the two standard test problems: the propagation of a single solitary wave and the collision of two solitary waves. We also compare our numerical results with those given in the literature in terms of both numerical accuracy and computational cost. The numerical comparisons show that the Fourier pseudo-spectral method provides very accurate results, at least for the two test problems stated above, and has a promising potential for handling other problems based on Boussinesq-type equations. Joint work with Handan Borluk.

275

Bayramov Nadir RICAM, AT Finite element methods for transient convection-diffusion equations with small diffusion Contributed Session CT2.3: Tuesday, 15:00 - 15:30, CO3 Transient convection-diffusion or convection-diffusion-reaction equations, with in general small or anisotropic diffusion, are considered. A specific exponential fitting scheme, resulting from finite element approximation, is applied to obtain a stable monotone method for these equations. In the first part of the talk error estimates are dicussed for this method and a comparison with the more commonly known SUPG method is drawn. The second part focuses on the efficient solution of the arising linear systems. A nonlinear algebraic multilevel iteration method is introduced in the framework of flexible GMRES using piecewise constant coarse spaces which are based on matching in graphs. The uniform convergence of this method is demonstrated by various numerical experiments. Joint work with Johannes Kraus.

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Federico Negri EPFL - MATHICSE - CMCS, CH Reduced basis methods for PDE-constrained optimization Contributed Session CT4.4: Friday, 09:20 - 09:50, CO015 We present a reduced framework for the numerical solution of parametrized PDEconstrained optimization problems. In particular, we focus on parametrized quadratic optimization problems constrained by either advection-diffusion, Stokes or NavierStokes equations, where the control (or design, inversion) variables are infinite dimensional functions, distributed in a portion of the domain or along its boundary. Parameters are not the object of the optimization, rather they may represent physical and/or geometrical quantities describing the state system or they can be related to observation measurements in the cost functional. In this context, our goal is to design a strategy for the reduction of the complexity of the optimization problem by treating it as a whole, with respect to all its variables (state and control) simultaneously. This framework is based on a suitable optimize-then-discretize-then-reduce approach which takes advantage of the reduced basis (RB) method for the rapid and reliable solution of parametrized PDEs. Indeed, we build our RB approximation directly upon the “truth” underlying finite element approximation of the optimality system. The saddle-point structure of the reduced optimality system requires a careful design of the reduced spaces for the state, control and adjoint variables, in order to ensure the stability of the RB approximation. We propose an aggregated approach, possibly enriched by supremizer solutions, which enables us to prove the well-posedness of the RB approximation. Then, we derive rigorous a posteriori error estimates on the solution variables as well as on the cost functional: in the linear constraint case we exploit the Babuška stability theory, while in the nonlinear constraint case we rely on the BrezziRappaz-Raviart theory. The link between the sharpness of these error bounds, the conditioning of the optimality system and the use of suitable “robust” norms will be discussed. We assess the properties and numerical performances of the methodology by solving some classical benchmark problems of vorticity minimization through suction/injection of fluid. Then we apply this framework to some problems arising in haemodynamics, dealing with both data assimilation and optimal control of blood flows. Joint work with Andrea Manzoni, Alfio Quarteroni, and Gianluigi Rozza.

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Thi Trang Nguyen FEMTO-ST Institute, University of Franche-Comte, FR Homogenization of the one-dimensional wave equation Contributed Session CT2.2: Tuesday, 15:00 - 15:30, CO2 Homogenization of the wave equation in the bounded domain Ω with the timeindependent coefficients has been carried out in several papers. For example in [2], the solution of the homogenized problem is a weak limit, when period tends to 0, of a subsequence of the solution. The latter has no fast time oscillations. So, it can not model correctly the physical solution. In order to overcome this problem, a method for two-scale model derivation of the periodic homogenization of the wave equation has been developed in [1]. It allows to analyze the oscillations occurring on both time and space for low and high frequency waves. Unfortunately, the boundary conditions of the homogenized model have not been found. Therefore, establishing the boundary conditions of the homogenized model is critical and is the main motivation of our works. In this presentation, we use the same method as in [1] for the homogenization for the wave equation in one dimension. A new result on the asymptotic behavior of waves regarding the boundary conditions has been obtained and will be presented for the first time. Numerical simulations will also be provided. For a bounded open set Ω = (0, 1) and a finite time interval I = [0, T ) ⊂ R+ , we consider the wave equation with Dirichlet boundary conditions, 
2 ε

u (t, x) − ∂x a xε ∂x uε (t, x) = f ε (t, x) in I × Ω,  ρ xε ∂tt uε (t = 0, x) = uε0 (x) and ∂t uε (t = 0, x) = v0ε (x) in Ω, (1)  ε u (t, 0) = uε (t, 1) = 0 in I, where ε >
0 denotes a small
parameter intended to go to zero. The two functions aε = a xε and ρε = ρ xε are Lipschitzian, positive, and periodic with respect to a lattice of reference cell εY ⊂ R. We reformulate (1) under the
√ formε of√a system with unknown the vector of first-order derivatives U ε := aε ∂x u , ρε ∂t uε . Here we study the asymptotic behavior of U ε . For any fixed K ∈ N∗ and any fiber k ∈ L∗K
, with the definition of the set L∗K of ±k the eigenvalues and eigenvectors of fibers introduced in [1], we consider λ±k n , en the Bloch wave spectral problem with ±k-quasi-periodic boundary conditions, and M k a set of indices of all Bloch eigenvalues. We denote by Λ = (0, 1) a time unit k ∗ cell. Starting with the observation that λkn = λ−k n for all n ∈ M and k ∈ LK , we ε ε ε apply to U the sum of the modulated two-scale transforms Wk and W−k , defined 2 ε in [1]. For a given k ≥ 0, Wkε U ε +W−k U ε converges weakly in L2 (I × Λ × Ω × Y ) to U k (t, τ, x, y)which can be decomposed by X U k (t, τ, x, y) = UH (t, x, y) + Unk (t, x) e2iπsn τ ekn (y) (2) n∈M k +Un−k

(t, x) e−2iπsn τ e−k n (y)




The term UH is the low frequency part. The other terms represent the high frequency waves, and Un±k are solution of a system of macroscopic equations which boundary conditions constitute one of the main contributions of this work. We

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deduce an approximation of the physical solution, q    X  x isn λk t/ε k x ε k |n| + Un (t, x) e U (t, x) ≈ UH t, x, en ε ε n∈M k q   t/ε −k x isn λk −k |n| +Un (t, x) e en ε

(3)

which holds in the strong sense. The figures below represent the numerical results.

References [1] M. Brassart, M. Lenczner, A two-scale model for the periodic homogenization of the wave equation, J. Math. Pures Appl. 93 (2010) 474 − 517. [2] S. Brahim-Otsmane, G.A. Francfort, F. Murat, Correctors for the homogenization of the wave and heat equations, J. Math. Pures Appl. 71 (1992) 197 − 231.

Figure 1: At t = 0.665. Figure 1: Comparison of the first component of U ε and the corresponding homogenized solution. Figure 2: The error between the physical solution and the homogenized solution, with the maximal error is 0.011. Joint work with Michel Lenczner, and Matthieu Brassart.

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Kirill Nikitin Institute of Numerical Mathematics of Russian Academy of Sciences, RU A monotone nonlinear finite volume method for diffusion equations and multiphase flows Minisymposium Session SDIFF: Monday, 11:40 - 12:10, CO123 We present a new nonlinear monotone finite volume method for diffusion equation and its application to two-phase black oil model. We consider full anisotropic discontinuous diffusion or permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which provides the conventional 7-point stencil for the discrete diffusion operator on cubic meshes. We show that the quality of the discrete flux in a reservoir simulator has great effect on the front behavior and the water breakthrough time. We compare two two-point flux approximations (TPFA), the proposed nonlinear TPFA and the conventional linear TPFA, and multi-point flux approximation (MPFA). The new nonlinear scheme has a number of important advantages over the traditional linear discretizations. Compared to the linear TPFA, the new nonlinear scheme demonstrates low sensitivity to grid distortions and provides appropriate approximation in case of full anisotropic permeability tensor. For non-orthogonal grids or full anisotropic permeability tensors the conventional linear TPFA provides no approximation, while the nonlinear flux is still first-order accurate. The computational work for the new method is higher than the one for the conventional TPFA, yet it is rather competitive. Compared to MPFA, the new scheme provides sparser algebraic systems and thus is less computational expensive. Moreover, it is monotone which means that the discrete solution preserves the non-negativity of the differential solution. Joint work with K.Terekhov, and Yu.Vassilevski.

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Caroline Nore LIMSI-CNRS and University Paris-Sud, FR Dynamo action in finite cylinders Minisymposium Session MMHD: Thursday, 14:00 - 14:30, CO017 Using numerical simulations, we investigate two magnetohydrodynamics (MHD) problems in a cylindrical cavity, namely a precessing cylinder and a short TaylorCouette set-up, both containers being filled with a conducting fluid. We use a parallel code denoted SFEMaNS (Guermond at al., JCP, 2011) to integrate nonlinear MHD equations for incompressible fluids in heterogenous domains with axisymmetric interfaces embedded in a vacuum. We numerically demonstrate that precession is able to drive a dynamo and that a short Taylor-Couette set-up with a body force can also sustain dynamo action. In the precessing cylinder, the generated magnetic field is unsteady and quadrupolar (Nore et al., PRE, 2011). These numerical evidences may be useful for an experiment now planned at the DRESDYN facility in Germany. In the Taylor-Couette set-up, the nonlinear dynamo state is characterized by fluctuating kinetic and magnetic energies and a tilted dipole whose axial component exhibits aperiodic reversals during the time evolution (Nore et al., PoF, 2012). These numerical evidences may be useful for developing an experimental device. This work was performed using HPC resources from GENCI-IDRIS (Grant 90254).

Joint work with F. Luddens (LIMSI-CNRS and Univ. Paris-Sud, France), L. Cappanera (LIMSI-CNRS and TAMU), J. Leorat (Obs. Meudon, France) and J.-L. Guermond (TAMU, and USA).

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Takeshi Ogita Tokyo Woman’s Christian University, JP Backward error bounds on factorizations of symmetric indefinite matrices Contributed Session CT2.1: Tuesday, 14:30 - 15:00, CO1 In this talk we are concerned with the rounding error analysis on block LDLT factorizations of symmetric matrices. Let A be a real symmetric matrix. Then A can be factorized as P AP T = LDLT , where L is a unit triangular matrix, D is a block diagonal matrix with each block of order 1 or 2, and P is a permutation matrix according to some pivoting strategy. It is called a block LDLT factorization with diagonal pivoting, which is known as a stable numerical algorithm and widely used for solving symmetric and indefinite linear systems. Moreover, it is also useful for checking the inertia of a symmetric matrix. In practice floating-point arithmetic is extensively used for these purposes. Since finite precision numbers are used, rounding errors are involved in computed results. For symmetric and positive definite matrices a backward error bound for a Cholesky factorization has been given, for example, in [Demmel (1989), Higham (2002)]. Moreover, it is modified for sparse cases in [Rump (2006)], which is rigorous and easy to compute efficiently. For symmetric and indefinite matrices, however, Cholesky factorization cannot be applied, but LDLT or block LDLT factorization can. In terms of the stability of the algorithms, block LDLT factorization is preferable. ˜ D ˜ and P˜ be floating-point block LDLT factors of A, which are approximaLet L, tions of L, D and P , respectively. Then the backward error ∆ of the floating-point factorization is defined by ˜D ˜L ˜T . ∆ := P˜ AP˜ T − L

(1)

In some methods of verified numerical computations for the solution of a linear system with A being a coefficient matrix [Rump (1994), Rump (1995), Rump (1999)] and for the inertia of A [Yamamoto (2001)], it is mandatory to compute an upper bound of k∆k, where k · k stands for the spectral norm. A main point of this research is to derive a method of calculating the backward error bound that is easy to compute. There are several methods for block LDLT factorizations of symmetric matrices with different pivoting strategies such as Bunch–Parlett (1971), Bunch–Kaufman (1977) and so forth. There are also useful implementations, e.g. [Duff (2002), Duff–Reid (1982)]. In addition, rounding error analyses are presented in [Fang (2011), Higham (1997), Slapničar (1998)]. See [Fang (2011)] for details. From a qualitative standpoint some rough estimations suffice to show the backward stability of the algorithms. From the viewpoint of verified numerical computations, however, rigorous and computable estimations are necessary, especially precise estimations are preferable. In order to obtain an upper bound of |∆| by backward error analysis it is necessary to derive a backward error bound for 2 × 2 linear systems during a block LDLT factorization. We present two ways for the purpose; One is to use a classical rounding error analysis as in [Higham (2002)]. The other is to apply a direct rounding error analysis. The latter gives much sharper bounds than the former. 282

Numerical results are also presented with some applications. Joint work with Kenta Kobayashi.

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Mario Ohlberger University of Muenster, DE Model reduction for nonlinear parametrized evolution problems Minisymposium Session UQPD: Thursday, 11:00 - 11:30, CO1 In this contribution we present and discuss recent development of the reduced basis method [8] in the context of model reduction for nonlinear parametrized evolution problems. Our approach is based on the POD-Greedy algorithm, first introduced in the linear setting in [6] and then extended to the nonlinear setting in [7, 2]. The model reduction in nonlinear scenarios is based on empirical interpolation of nonlinear differential operators and their Frechet derivatives. As a result, the POD-Greedy algorithm is generalized to the PODEI-Greedy algorithm that simultaneously constructs the reduced and the collateral reduced basis space employing the empirical interpolation of nonlinear differential operators. Efficient online/offline decomposition is obtained for discrete operators that satisfy an Hindependent DOF dependence for a certain set of interpolation functionals, where H denotes the dimension of the underlying high dimensional discretization space. The resulting reduced basis method is applied to nonlinear parabolic and hyperbolic equations based on explicit or implicit finite volume discretizations, as well as to mixed elliptic-parabolic systems modeling two phase flow in porous media [3]. We show that the resulting reduced scheme is able to capture the evolution of both smooth and discontinuous solutions. In case of symmetries of the problem, the approach realizes an automatic and intuitive space-compression or even space-dimensionality reduction. We perform empirical investigations of the error convergence and runtimes. In all cases we obtain a runtime acceleration of at least one order of magnitude. To speed up offline and/or online runtimes, adaptive basis enrichment strategies and multiple bases generation approaches [5] can be combined with the PODEI-Greedy approach. In [4] it has been shown that the POD-Greedy method is optimal in the sense that exponential or algebraic convergence rates of the Kolmogorov n-with are maintained by the algorithm. Although this is a very nice result for situations with fast decay rates of the Kolmogorv n-with, it also shows limitations of the model reduction approach, in particular in nonlinear hyperbolic scenarios with evolving discontinuities. As a first attempt to address such classes of problems we will also present a new model reduction approach that is based on a combination of the PODEI-Greedy approach with the method of freezing that was originally introduced to study relative equilibria of evolution problems [9, 1]. Given the action of a Lie group on the solution space, in this approach the original problem is reformulated as a partial differential algebraic equation system by decomposing the solution into a group component and a spatial shape component and imposing appropriate algebraic constraints on the decomposition. The system is then projected onto a reduced basis space. We show that efficient online evaluation of the scheme is possible and study a numerical example showing its strongly improved performance in comparison to a scheme without freezing.

References [1] W.-J. Beyn, V. Thümmler. Freezing Solutions of Equivariant Evolution Equations. SIAM J. Appl. Dyn. Syst. 3:85–116, 2004. 284

[2] M. Drohmann, B. Haasdonk, and M. Ohlberger. Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput., 34:A937-A969, 2012. [3] M. Drohmann, B. Haasdonk, and M. Ohlberger. Reduced basis model reduction of parametrized two-phase fow in porous media. In: Proccedings of the 7th Vienna International Conference on Mathematical Modelling (MathMod), Vienna, 2012. [4] B. Haasdonk. Convergence rates of the pod-greedy method. M2AN Math. Model. Numer. Anal., doi:10.1051/m2an/2012045, 2013. [5] B. Haasdonk, M. Dihlmann, and M. Ohlberger. A training set and multiple bases generation approach for parametrized model reduction based on adaptive grids in parameter space. Math. Comput. Model. Dyn. Syst., 17(4):423–442, 2011. [6] B. Haasdonk, and M. Ohlberger. Reduced basis method for finite volume approximations of parametrized evolution equations. M2AN Math. Model. Numer. Anal., 42(2):277-302, 2008. [7] B. Haasdonk, M. Ohlberger, and G. Rozza. A reduced basis method for evolution schemes with parameter-dependent explicit operators. Electronic Transactions on Numerical Analysis, 32: 145–161, 2008. [8] A.T. Patera, and G. Rozza. Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equations. MIT, 2007. Version 1.0, Copyright MIT 2006-2007, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering. [9] C. W. Rowley, I. G. Kevrekidis, J. E. Marsden, K. Lust. Reduction and reconstruction for self-similar dynamical systems. Nonlinearity 16:1257–1275, 2003. Joint work with Martin Drohmann, Bernard Haasdonk, and Stephan Rave.

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Rikard Ojala Dept. of Numerical Analysis, KTH, Stockholm, SE Accurate bubble and drop simulations in 2D Stokes flow Contributed Session CT2.3: Tuesday, 14:30 - 15:00, CO3 This talk will be on moving interfaces and free boundaries in two dimensional Stokes flow, where the flow is due to surface tension. For such flows in the Stokesian regime, with small Reynolds numbers, the resulting linear governing equations can be recast as an integral equation. This is a well-known and widely used fact. What is frequently overlooked, however, is how to deal with interfaces that are close to each other. In this case, the integral kernels are near-singular, and standard quadrature approaches do not give accurate results. Phenomena such as lubrication are then not captured correctly. We will discuss how to apply a general special quadrature approach to resolve this problem. The result is a robust and accurate solver capable of handling a wide range of bubble and drop configurations. An example of a fairly complex drop setup that can be treated is displayed below. Joint work with Anna-Karin Tornberg.

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Maxim Olshanskii University of Houston, US An adaptive finite element method for PDEs based on surfaces Minisymposium Session ADFE: Wednesday, 11:30 - 12:00, CO016 An adaptive finite element method for numerical treatment of elliptic partial differential equations defined on surfaces is discussed. The method makes use of a standard outer volume mesh to discretize an equation on a two-dimensional surface embedded in R3 . The reliability of a residual type a posteriori error estimator is proved and both reliability and efficiency of the estimator are studied numerically in a series of experiments. A simple adaptive refinement strategy based on the error estimator is demonstrated to provide optimal convergence rate in the H 1 and L2 norms.

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Maxim Olshanskii University of Houston, US Preconditioners for the linearized Navier-Stokes equations based on the augmented Lagrangian Minisymposium Session PSPP: Thursday, 14:00 - 14:30, CO3 We discuss block preconditioners based on the augmented Lagrangian formulation of the algebraic equations of the linearized Navier-Stokes equations. We consider incompressible fluids, and the resulting algebraic problems are of generalized saddle point type. The talk reviews variants of augmented Lagrangian preconditioner based on different forms of augmentation and certain simplifications to make the approach computationally efficient. The preconditioned systems admit eigenvalue and field-of-value analysis. We include numerical results for several fluid problems. The talk reports on a joint work with Michele Benzi (Emory).

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Christoph Ortner University of Warwick, GB Optimising Multiscale Defect Simulations Minisymposium Session MSMA: Monday, 11:40 - 12:10, CO3 A universal quality measure for any numerical approximation scheme is its accuracy relative to its computational cost. This point of view seems to have gone largely unnoticed in the analysis of atomistic-continuum multiscale simulations, but it guarantees an unbiased approach to the construction and evaluation of computational schemes. In this talk, I will focus on atomistic-to-continuum (quasicontinuum) methods for lattice defects, and some related schemes. I will first review how the framework of numerical analysis leads to error estimates (accuracy) in terms of the various approximation parameters such as domain size, atomistic region size, finite element mesh, or interface correction. I will then discuss how these estimates can be recast as error estimates in terms of computational cost. Finally, this can be used to optimise the various approximation parameters. Interesting comparisons are, e.g., between the choices of coupling mechanisms or the usage of nonlinear versus linear elasticity. Joint work with Virginie Ehrlacher, Helen Li, Mitch Luskin, Alex Shapeev, and Brian Van Koten.

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Abderrahim Ouazzi wiss.Ang., DE Newton-Multigrid Least-Squares FEM for V-V-P and S-V-P Formulations of the Navier-Stokes Equations Contributed Session CT2.9: Tuesday, 14:30 - 15:00, CO124 Least squares finite element methods are motivated, beside others, by the fact that in contrast to standard mixed finite element methods, the choice of the finite element spaces is not subject to the LBB stability condition and the corresponding discrete linear system is symmetric and positive definite. We intend to benefit from these two positive attractive features, in one hand, to use different types of elements representing the physics as for instance the capillary forces and mass conservation and, on the other hand, to show the flexibility of the geometric multigrid methods to handle efficiently the resulting linear systems. We numerically solve the V-V-P, Vorticity-Velocity-Pressure, and S-V-P, Stress-Velocity-Pressure, formulations of the incompressible Navier-Stokes equations based on the least squares principles using different types of finite elements, conforming, nonconforming and discontinuous of low as well as high order. For the discrete systems, we use a conjugate gradient (CG) solver accelerated with a geometric multigrid preconditioner. In addition, we employ a Krylov space smoother which allows a parameter-free smoothing. Combining this linear solver with the Newton linearization results in a robust and efficient solver. We analyze the application of this general approach, of using different types of finite elements, and the efficient solver, geometric multigrid, for several prototypical benchmark configurations (driven cavity, flow around obstacles), and we investigate the effects of pressure jumps for the capillary force in multiphase flow simulations (static bubble configuration). Key words: Least Squares EFM, Geometric multigrid, First-order system least squares, Capillary force, Mass conservation, Navier-Stokes equations. Joint work with M. Sc. Masoud Nickaeen, and Prof. Dr. Stefan Turek.

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Katsuhisa Ozaki Shibaura Institute of Technology, JP Fast Interval Matrix Multiplication by Blockwise Computations Contributed Session CT2.1: Tuesday, 15:00 - 15:30, CO1 Keywords: interval arithmetic, enclosure methods, a priori error analysis Interval arithmetic [1] is widely applied into so-called verified numerical computations, which discuss reliability of approximate results by numerical computations. This talk is concerned with interval matrix multiplication. Let F be a set of floating-point numbers. Let IF be a set of midpoint-radius interval: hc, ri = {x ∈ R | c − r ≤ x ≤ c + r, r ≥ 0, c, r ∈ F}, where c and r are center and radius of the interval, respectively. For interval matrices A ∈ IFm×n and B ∈ IFn×p , enclosure of interval matrix multiplication can be obtained by hAm , Ar i ∗ hBm , Br i ⊆ hAm ∗ Bm , |Am |Br + Ar (|Bm | + Br )i,

(1)

where | · | returns a matrix by taking an absolute value elementwise. For study of interval matrix multiplication according to (1), algorithms are characterized by the number of matrix products: • 4 matrix products: Rump [2] • 3 matrix products: Rump [4] and Ozaki-Ogita-Oishi-Rump [5] • 2 matrix products: Ogita-Oishi [3], Rump [4] and Ozaki-Ogita-Oishi-Rump [5] • a matrix product: Ozaki-Ogita-Oishi-Rump [5] Basically, there is a tradeoff between the number of matrix products and tightness of computed intervals. In this talk, we introduce how to improve tightness of the intervals without a slowdown of computational performance. In the fastest method in [5], a priori error analysis for a floating-point result AB ≈ C ∈ Fm×p is used: |C − AB| ≤ γn |A||B|, γn =

nu , 1 − nu

(2)

where u is the unit roundoff, especially, u = 2−53 for binary64 in the IEEE 754 standard. If the matrix multiplication is computed by blockwise computation, it is known in [6] that the bound (2) can be significantly improved. Therefore, first we implement blockwise matrix multiplication suited for the a priori error analysis. Our simple implementation of block matrix multiplication does not slow the performance down, compared to optimized BLAS (Basic Linear Algebra Subprograms). In addition, numerical results illustrate that tightness of the interval can be significantly improved. References: [1] A. N EUMAIER, Interval Methods for Systems of Equations, Cambridge University Press, 1990. 291

[2] S. M. R UMP, Fast and parallel interval arithmetic, BIT Numerical Mathematics, 39(3):539–560, 1999. [3] T. O GITA , S. O ISHI, Fast Inclusion of Interval Matrix Multiplication, Reliable Computing, 11(3):191–205, 2005. [4] S. M. R UMP, Fast Interval Matrix Multiplication, Numerical Algorithms, 61(1):1-34, 2012. [5] K. O ZAKI , T. O GITA , S.M. R UMP, AND S. O ISHI, Fast algorithms for floatingpoint interval matrix multiplication, Journal of Computational and Applied Mathematics, 236(7):1795-1814, 2012. [6] N.J. H IGHAM, Accuracy and Stability of Numerical Algorithms, second edition, SIAM Publications, Philadelphia, 2002. Joint work with Takeshi Ogita.

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Jan Papez Faculty of Mathematics and Physics, Charles University in Prague, CZ Distribution of the algebraic, discretization and total errors in numerical PDE model problems Contributed Session CT1.5: Monday, 17:30 - 18:00, CO016 The finite element method (FEM) is widely used in numerical solution of partial differential equations. This method generates an approximate solution in form of a linear combination of basis functions with local supports. Each basis function (multiplied by the proper coefficient) thus approximates the desired solution only locally. The global approximation property of the FEM discrete solution is then ensured by solving a linear algebraic system for the unknown coefficients. If this system is solved exactly, then the FEM discrete solution is obtained and its difference from the true solution is given by the discretization error. But in practice we do not solve exactly. In hard problems we even do not want to aim at a small algebraic error as it might be too costly or even impossible to get. One should therefore take into consideration also the error caused by the inexact algebraic computation. In particular, such consideration should include the spatial distribution of the algebraic error in the domain. There is no a priori evidence that this distribution is analogous to the distribution of the discretization error. On the contrary, as demonstrated in [1, Section 5.1] and [2], the spatial distribution of the algebraic error can significantly differ from the distribution of the discretization error. The results presented there for the FEM discretization of the simplest Poisson boundary value problem demonstrate that the algebraic error can have large local components which can dominate the total error in parts of the domain. In this contribution we further elaborate on results from [1, Section 5.1] and [2]. Using various iterative and direct algebraic solvers, we compare spatial distribution of algebraic and discretization errors in numerical solution of several boundary value problems present in literature. Acknowledgment: This work was supported by the ERC-CZ project LL1202 and by the GAUK grant 695612.

References [1] J. Liesen and Z. Strakoš. Krylov subspace methods: principles and analysis. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, 2012. [2] J. Papež, J. Liesen, and Z. Strakoš. Distribution of the discretization and algebraic error in numerical solution of partial differential equations. Preprint MORE/2012/03, submitted for publication, 2013. Joint work with Zdenek Strakos.

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Luca Pavarino University of Milan, IT Isogeometric Schwarz preconditioners for mixed elasticity and Stokes systems Minisymposium Session PSPP: Thursday, 12:00 - 12:30, CO3 Overlapping Schwarz preconditioners for the isogeometric mixed formulation of almost incompressible linear elasticity and Stokes systems are here presented and studied. The preconditioner is based on partitioning the domain of the problem into overlapping subdomains, solving local isogeometric mixed problems on these subdomains and solving an additional coarse isogeometric mixed problem associated with the subdomain mesh. Numerical results in 2D and 3D tests show that this preconditioner is scalable in the number of subdomains and optimal in the ratio between subdomain and overlap sizes. The numerical tests also show a good convergence rate with respect to the polynomial degree p and regularity k of the isogeometric basis functions, as well as with respect to the presence of discontinuous elastic coefficients in composite materials and to domain deformation. References: L. Beirao da Veiga, D. Cho, L. F. Pavarino, S. Scacchi. Overlapping Schwarz methods for Isogeometric Analysis. SIAM J. Numer. Anal., 50 (3): 13941416, 2012. L. Beirao da Veiga, D. Cho, L. F. Pavarino, S. Scacchi. Isogeometric Schwarz preconditioners for linear elasticity systems. Comput. Meth. Appl. Mech. Engrg., 253: 439-454, 2013. Joint work with L. Beirao da Veiga, D. Cho, L. F. Pavarino, and S. Scacchi.

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Bengisen Pekmen Atilim University, TR Steady Mixed Convection in a Heated Lid-Driven Square Cavity Filled with a FluidSaturated Porous Medium Contributed Session CT1.3: Monday, 17:00 - 17:30, CO3 Steady mixed convection flow in a lid-driven porous square cavity is studied numerically using the dual reciprocity boundary element method (DRBEM). Two-dimensional, steady, laminar flow of an incompressible fluid is considered in a homogeneous, isotropic porous medium. Viscosity, thermal conductivity, specific heat, thermal expansion coefficient, and permeability (except the density variation in the buoyancy term) are assumed to be constant with a body force term in the momentum equations according to Boussinessq approximation. The governing non-dimensional equations in terms of stream function ψ-temperature T -vorticity w are ∇2 ψ = −w ∂w ∂w Gr ∂T 1 1 2 ∇ w=u +v − + w 2 Re ∂x ∂y Re ∂x Da Re 1 ∂T ∂T ∇2 T = u +v P r Re ∂x ∂y

(1) (2) (3)

where u = ∂ψ/∂y, v = −∂ψ/∂x, w = ∂v/∂x − ∂u/∂y, and Re, Gr, Da, P r are Reynolds, Grashof, Darcy and Prandtl numbers, respectively. Left and right lids of the cavity move with a velocity v = 1 while u = ψ = 0, and u = v = ψ = 0 on the other walls. The left wall is the cold wall Tc = 0 and the right wall is the hot wall Th = 1. Adiabatic condition (∂T /∂n = 0) is imposed on the top and bottom walls. Applying the DRBEM with linear boundary elements to the non-dimensional governing equations (1)-(3), the following matrix-vector equations are obtained Hψ m+1 − Gψqm+1 = −Swm ∂F −1 m+1 ∂F −1 m+1 um+1 = F ψ , v m+1 = − F ψ ∂y ∂x   Gr ∂F −1 m+1 1 S wm+1 − Gwqm+1 = − S F T H − ReSM − Da Re ∂x (H − P rReSM ) T m+1 − GTqm+1 = 0 ˆ − GQ)F ˆ −1 , M = where S = (H U



∂F [u]m+1 d

−1

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∂F [v]m+1 d

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F + F . H and G ∂x ∂y ∗ are BEM matrices containing integrals of fundamental solution u = ln r/2π and its normal derivative, respectively. F is the coordinate matrix formed from the radial basis functions approximating the inhomogeneities of the equations (1)-(3). ˆ and Q ˆ matrices are built from particular solutions and their derivatives. U In the computations, radial basis function f = 1+r is used (where r is the distance between source and field points), N = 96, L = 625 are taken, P r = 0.71 is fixed, and the porosity of the medium is assumed to be unity. In Figure 1, columnwise contours are streamlines, isotherms and vorticity, respectively. A decrease in Da is shown as Da = 0.01 and Da = 0.001 from top to

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bottom. Fluid flows slowly as Da decreases since the porosity of the medium increases, (ψmax = 0.0405 for Da = 0.01, and ψmax = 0.0099 for Da = 0.001), and the heat transfer passes to the conductive mode as can be seen from isotherms. The effect of moving lids diminishes. Vorticity becomes stagnant at the center forming strong boundary layers through the vertical walls. Apart from this, the increase in buoyancy effect with the increase in Gr causes the heat transfer to become conductive dominant. DRBEM as a boundary only nature gives very good accuracy at a small computational expense for solving mixed convection flow.

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Juan Manuel Pena Universidad de Zaragoza, ES Accurate computations for some classes of matrices Contributed Session CT2.1: Tuesday, 15:30 - 16:00, CO1 A square matrix is called a P-matrix if all its principal minors are positive. Subclasses of P-matrices very important in applications are the nonsingular totally nonnegative matrices and the nonsingular M-matrices. For diagonally dominant M-matrices and some subclasses of nonsingular totally nonnegative matrices, accurate methods for computing their singular values, eigenvalues or inverses have been obtained, assuming that adequate natural parameters are provided. We present some recent extensions of these methods to other related classes of matrices.

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Simona Perotto MOX, Dipartimento di Matematica, Politecnico di Milano, IT Recent developments of Hierarchical Model (HiMod) reduction for boundary value problems Minisymposium Session SMAP: Monday, 12:40 - 13:10, CO015 The construction of surrogate models is a crucial step for bringing computational tools to practical applications within the appropriate timeline. This can be accomplished by taking advantage of specific features of the problem at hand. For instance, when solving flow problems in networks (in the modeling of blood, oil, water, or air dynamics), the local dynamics is expected to develop mainly along the edges of the network. The interaction between the local and network dynamics calls often for appropriate model reduction techniques. A possible approach consists of introducing a modal discretization for computing the edge-transversal dynamics and classical discretization methods (such as finite elements) for the prevalent (mainstream) dynamics. The former is anticipated to be computed with an acceptable accuracy by just a few modes. This approach is called HierarchicalModel (Hi-Mod) reduction since it can be regarded as a way for generating a hierarchy of one-dimensional models, locally improved, for the leading dynamics. More specifically, the number of employed modes decides the improvement of the model. In an adaptive framework, the number of modes for the transverse solution is automatically detected by the solver, on the basis of a suitable a posteriori estimator. In this talk, we present recent developments of this approach towards the effective solution of real problems. Joint work with Alessandro Veneziani, Department of Mathematics and Computer Science, Emory University, Atlanta, GA, and USA.

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Ilaria Perugia University of Pavia, IT Trefftz-discontinuous Galerkin methods for time-harmonic wave problems Plenary Session: Wednesday, 09:10 - 10:00, Rolex Learning Center Auditorium Several finite element methods used in the numerical discretization of wave problems in frequency domain are based on incorporating a priori knowledge about the differential equation into the local approximation spaces. This can be done by using Trefftz-type basis functions, namely functions which belong to the kernel of the considered differential operator (e.g., plane, circular/spherical and angular waves). The resulting methods feature enhanced convergence properties with respect to standard polynomial finite elements. Prominent examples of such methods are the ultra weak variational formulation (UWVF) by Cessenat and Després, the partition of unit finite element method (PUFEM) by Babuška and Melenk, the discontinuous enrichment method (DEM/DGM) by Farhat and co-workers, the variational theory of complex rays (VTCR) by Ladevèze, and the wave based method (WBM) by Desmet. In this talk, we focus on a family of Trefftz-discontinuous Galerkin (TDG) methods, which includes the UWVF as a special case. For the Helmholtz problem −∆u − k 2 u = 0 in a bounded domain with connected boundary and impedance boundary condition, TDG methods are proved to be unconditionally well-posed and quasi-optimal a in a mesh-dependent energy-type norm, i.e., well-posedness and quasi-optimality hold for any value of the wave number and of the mesh size. High-order convergence is obtained by using new approximation estimates for plane and spherical waves. These methods and their analysis framework can be generalized to the time-harmonic Maxwell equation and to the Navier equation. By duality arguments, L2 -norm error estimates can be obtained for both the hand the p-version of the TDG methods, with a (more or less) standard choice of DG numerical flux parameters, in the former case, and with constant flux parameters (like in the UWVF) in the latter case. On the other hand, for scattering problems with complicated geometries, an hp-approach is advisable. In this case, a special choice of the numerical flux parameters has been devised, which allows to prove a priori error estimates on locally refined meshes, with explicit dependence on the local mesh size, local number of degrees of freedom and local regularity of the analytical solution. Establishing the exponential convergence in the number of degrees of freedom of a full hp-version of the TDG method would complete the picture. Preliminary results in this directions will be presented.

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Steffen Peter Technische Universität München, DE Damping Noise-Folding and Enhanced Support Recovery in Compressed Sensing Minisymposium Session ACDA: Monday, 15:30 - 16:00, CO122 The practice of compressive sensing suffers very importantly in terms of the efficiency/accuracy trade-off when acquiring noisy signals prior to measurement. It is rather common to find results treating the noise affecting the measurements, avoiding in this way to face the so-called noise-folding phenomenon, related to the noise in the signal, eventually amplified by the measurement procedure. In this talk we present a new decoding procedure, combining `1 -minimization followed by a selective least p-powers, which not only is able to reduce this component of the original noise, but also has enhanced properties in terms of support identification with respect to the sole `1 -minimization. We prove such features, providing relatively simple and precise theoretical guarantees. We additionally confirm and support the theoretical estimates by extensive numerical simulations, which give a statistics of the robustness of the new decoding procedure with respect to more classical `1 minimization. Joint work with Marco Artina, and Massimo Fornasier.

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Johannes Pfefferer Universität der Bundeswehr München, DE On properties of discretized optimal control problems with semilinear elliptic equations and pointwise state constraints Minisymposium Session FEPD: Monday, 14:30 - 15:00, CO017 This talk is concerned with the analysis of finite element discretized optimal control problems governed by a semilinear elliptic state equation and subject to pointwise state constraints. In this context two issues mainly arise: the convergence of the discrete locally optimal controls to the related continuous ones and the convergence of the solution algorithm such as the SQP method. Imposing second-order sufficient conditions (SSC) for the continuous problem allows us to prove a rate of convergence of the discrete local solutions to the related continuous ones. Moreover, we elucidate that the SSC postulated for continuous locally optimal solutions transfer to the discrete level. This contributes to the second issue since for instance the proof of convergence of the SQP method relies on SSC. Joint work with Ira Neitzel, and Arnd Rösch.

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Marco Picasso EPFL-MATHICSE, CH Numerical simulation of extrusion with viscoelastic flows Minisymposium Session MANT: Wednesday, 12:00 - 12:30, CO017 Numerical simulation of extrusion is important for the food processing industry, pasta, chocolate, cereals, for instance. Extrusion is difficult to simulate since free surfaces with complex shapes are involved. Using the numerical method proposed in Bonito Picasso Laso, J. Comp. Phys. 2006, numerical experiments are reported for several extrusion geometries and several viscoelastic fluids. Joint work with Alexandre Caboussat, Alexandre Masserey, and Gilles Steiner.

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Konstantin Pieper Technische Universität München, DE Finite element error analysis for optimal control problems with sparsity functional Minisymposium Session FEPD: Monday, 11:40 - 12:10, CO017 We consider an elliptic optimal control problem with a sparsity functional, where the control variable is searched for in the space of regular Borel measures. Minimize

1 2 ku

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− ud kL2 (Ωo ) + αkqkM(Ωc ) ,

q ∈ M(Ωc ),

subject to A(u) = q in Ω. Under suitable conditions on Ωo and Ωc the optimal solutions have highly sparse structure, which suggests applications for the optimal placement of actuators. For practical computations we discretize the elliptic equation with finite elements, where the control is approximated by a sum of nodal Dirac delta functions. Using this discretization concept introduced by Casas, Clason and Kunisch, we are able to obtain improved rates of convergence in the case of two and three spacial dimensions. The new results agree with the generic regularity of the solutions as well as with the numerical observations. In the case Ωc ⊂ Ωo additional regularity for the optimal controls can be obtained by careful inspection of the optimality system, which results in improved convergence estimates. We also develop an a posteriori error estimator for this problem: Therefore an additional regularized problem is introduced, where a Tichonov regularization term is added to the objective functional. For the regularized problem the discretization error can then be estimated with a “dual weighted residual” type estimator to provide indicators for local mesh refinement. The error introduced by the regularization is estimated with an asymptotic model. This error can be controlled by the regularization parameter, which is chosen within an adaptive algorithm to balance both contributions of the error. Joint work with Boris Vexler.

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Petra Pořízková Czech Technical University in Prague, CZ Compressible and incompressible unsteady flows in convergent channel Contributed Session CT4.3: Friday, 08:50 - 09:20, CO3 This study deals with the numerical solution of a 2D unsteady flow of a viscous fluid in a channel for low inlet airflow velocity. The unsteadiness of the flow is caused by a prescribed periodic motion of a part of the channel wall with large amplitudes, nearly closing the channel during oscillations. The channel is a simplified model of the glottal space in the human vocal tract and the flow can represent a model of airflow coming from the trachea, through the glottal region with periodically vibrating vocal folds to the human vocal tract. Goal of this work is numerical simulation of flow in the channel which involves attributes of real flow causing acoustic perturbations as is “Coandă phenomenon” (the tendency of a fluid jet to be attracted to a nearby surface), vortex convection and diffusion, jet flapping etc. along with lower call on computer time, due to extension in 3D channel flow. Particular attention is paid to the acoustic analysis of pressure signal from the channel. Four governing systems are considered to describe the unsteady laminar flow of a viscous fluid in a channel: 1. Full system - 2D system of Navier-Stokes equations closed with static pressure expression for ideal gas p = f (ρ, u, v, e) describes flow of compressible viscous fluid, 5 equations. 2. Iso-energetic system - 2D system of Navier-Stokes equations closed with pressure expression which is independent on total energy variable e (p = f (ρ, u, v)) describes flow of compressible viscous fluid, 4 equations. 3. Adiabatic system - 2D system of Navier-Stokes equations closed with pressure expression which is independent on variables e, u, v describes flow of compressible viscous fluid, 4 equations. 4. Incompressible system - 2D system of Navier-Stokes equations where density ρ = const describes steady state flow of incompressible viscous fluid, 3 equations. Solution is computed using Artificial Compressibility Method. The numerical solution is implemented using the finite volume method (FVM) and the predictor-corrector MacCormack scheme with Jameson artificial viscosity using a grid of quadrilateral cells. The unsteady grid of quadrilateral cells is considered in the form of conservation laws using Arbitrary Lagrangian-Eulerian method. The numerical simulations of flow fields in the channel, acquired from a developed program, are presented for inlet velocity u ˆ∞ = 4.12ms−1 and Reynolds number Re∞ = 4481 and the wall motion frequency 100 Hz. Joint work with Karel Kozel, and Jaromir Horacek.

304

Stefan Possanner Université Paul Sabatier, FR Numerical integration of the MHD equations on the resistive timescale Minisymposium Session ASHO: Tuesday, 11:30 - 12:00, CO123 The two-dimensional magneto-hydrodynamic (MHD) equations constitute a relatively simple, low-cost model for describing the interplay between plasma motion and magnetic field dynamics in laboratory (Tokamaks) and in astrophysical plasmas. In these situations, the resistivity is usually small. Hence, resistive effects such as the tearing mode instability occur on large timescales of order −1 , where the asymptotic parameter is the inverse Lundquist number. In this talk, we elaborate on the MHD equations rescaled to the resistive time, which leads to a singularly perturbed problem as  goes to zero. We present two reformulations giving a well-posed problem in the limit, the first being based on a micro-macro decomposition and the second stemming from a reordering of the equations. Finite difference method is applied for numerical studies. We shall discuss to what degree the obtained schemes can be categorized as ’asymptotic-preserving’, which is not trivial because we observe boundary layers in time as well as in space. Finally, simulation results for the magnetic reconnection process in the non-linear tearing mode are presented. The significant reduction in computational cost due to the new schemes with respect to conventional explicit schemes is highlighted.

305

Jerome Pousin Université de Lyon ICJ INSA UMR CNRS 5208, FR A posteriori estimate and adaptive partial domain decomposition Contributed Session CT1.5: Monday, 18:00 - 18:30, CO016 The method of asymptotic partial decomposition of a domain (MAPDD) originates with the works of G.Panasenko [1]. The idea is to replace an original 3D or 2D problem by a hybrid one 3D −1D; or 2D −1D where the dimension of the problem decreases in part of the domain. In the problem considered here, due to geometrical considerations concerning the domain Ω it is assumed that the solution does not differ very much from a function which depends only on one variable in a part of the domain (subdomain Ω2 ). The a posteriori error estimate proved in this paper, is able to measure the discrepancy between the exact solution and the hybrid solution. Moreover, the method proposed is able to determine the location of the junction (i.e the location of the boundary Γ in the example treated) by using optimization techniques combined with an a posteriori error estimate and an error indicator. Let us also mention the interest of locating with accuracy the position of the junction in blood flows simulations [2]. The domain Ω = (0, 1)×(0, 1) is decomposed in two sub domains Ω1 = (0, a)×(0, 1) and Ω2 = (a, 1) × (0, 1), the boundary Γ = Ω1 ∩ Ω2 , and the boundary ∂Ω is divided into four subparts γ1 = {0} × (0, 1) γ2 = (0, 1) × {0} γ3 = {1} × (0, 1) γ4 = (0, 1) × {1} 2. Define the following functional spaces: (Ω1 ) = {ϕ ∈ H 1 (Ω1 ); ϕ|γ1 = 0}; 0 H 1 (Ω2 ) = {ϕ ∈ H 1 (Ω2 ); ϕ|γ3 = 0}; V =0 H 1 (Ω1 ) ×0 H 1 (Ω2 ); Λ = span{1}. 0H

1

Let us define (u1 , u2 , λ) ∈ V × Λ solution to  Z 2 Z 2 Z X   X   f vi dx1 dx2 ; ∇ui · ∇vi dx1 dx2 + λ(v1 − v2 ) dx2 = Γ i=1 Ωi i=1 Ωi Z    ξ(u1 − u2 ) dx2 = 0 ∀ξ ∈ Λ. 

∀v ∈ V

Γ

(1) Lemma Assume f ∈ L2 (Ω) and f|Ω2 = f (x1 ), then, there exists a unique (u1 , u2 , λ) ∈ V × Λ solution to Problem (1). Moreover, u2 depends only on x1 and we have Z 1 ∂n1 u1 = −∂n2 u2 in L2 (Γ); u2 |Γ = u1 dx2 . |Γ| Γ

Let (w1 , w2 , λ0 ) ∈ V × 2 (Γ) be solution to Problem (1) where the mortar space is 2 (Γ), then the error e = (w − u, λ0 − λ) satisfies: Z 1 1 1 kek ≤ ku1 − u2 k0,Γ = ku1 − u1 dx2 k0,Γ . (2) β β |Γ| Γ Let a denote the position of the boundary Γ. Due to relation (2), the proposed strategy is to minimize with respect to a the functional J(a) defined by: Z 1 J(a) = k˜ u1 (a, x2 ) − u ˜1 (a, x2 ) dx2 k20,Γ |Γa | Γa

in order to locate precisely the position of the interface. In this presentation I will discuss some numerical results, and I will show how to combine mesh refinement and localisation of the interface in order to reduce the error. 306

References [1]

G.P.Panasenko, title of a book, Multi-scale modelling for structures and composites, Springer, the Netherlands, 2005.

[2]

A. Quarteroni and A. Veneziani, Analysis of a geometrical multiscale model based on the coupling of PDE’s and ODE’s for Blood Flow Simulations, SIAM J. on MMS. No. 2 Vol 1 (2003), pp. 173-195.

307

Catherine Powell University of Manchester, GB Fast solvers for stochastic FEM discretizations of PDEs with uncertainty Minisymposium Session CTNL: Tuesday, 10:30 - 11:00, CO015 In modelling most physical processes we encounter uncertainties, both in the mathematical models we use as well as in the input data required to solve them. A common approach is to view unknown inputs as stochastic processes (in one dimension) or random fields (in higher dimensions), giving rise to stochastic differential equations. Starting from a statistical description of the data, our task is to obtain statistical information about output quantities of interest. This is known as Uncertainty Quantification (UQ) and contrasts with traditional deterministic modelling, where we simulate specific events corresponding to hypothesized models with certain data and seek to assess only discretisation errors. Extensive work has been carried out in the last decade to develop accurate numerical methods (stochastic Galerkin, stochastic collocation, Quasi-Monte Carlo) for PDEs with uncertainty. Galerkin-based schemes lead to linear systems with much higher complexity than their deterministic counterparts. The matrices are sums of Kronecker products of smaller matrices associated with distinct discretizations and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. For stochastically nonlinear problems, this is compounded by the fact that the matrices are block-dense and the cost of a matrix vector product is non-trivial. On the other hand, sampling methods lead to extremely long sequences of small similar linear systems. Challenges for the linear algebra community include: coping with high dimensionality, non-assembled coefficient matrices, intriguing block structures, exploiting similarity and the influence of statistical as well as discretization parameters on robustness of solvers. In this talk, we discuss some of the linear algebra issues involved in applying stochastic Galerkin and collocation schemes to saddle point problems with uncertain data. Model problems include: a mixed formulation of a second-order elliptic problem (with uncertain diffusion coefficient), the steady-state Navier-Stokes equations (with uncertain Reynolds number) and a second-order elliptic PDE, formulated on an uncertain domain.

308

Vladimír Prokop CTU in Prague, CZ Numerical Simulation of Generalized Oldroyd-B Fluid Flows in Bypass Contributed Session CT1.7: Monday, 17:00 - 17:30, CO122 In this paper the numerical solution of viscous incompressible generalized OldroydB fluids is described. The motivation for this work is to model blood flow in vessels of small diameter and to evaluate the importance of taking into account shear thinnning behaviour of blood in this case. The flows of Oldroyd-B fluids are described by the system of conservation laws of mass and momentum. The extra stress tensor is decomposed into Newtonian and elastic part. The later part is described by the Oldroyd-B model. In the generalized case of flows of Oldroyd-B fluids, the viscosity function is specified to describe shear-thinning behaviour of blood. In this case, the modified Cross model is used, where constants of the model, such as asymptotic viscosity values at zero and infinite shear rates, are taken from literature. Energy conservation is not taken into account because the temperature variations are in our case negligible.Steady numerical solution of incompressible generalized Oldroyd-B flows is sought in the geometry of stenotic channel with bypass in 2D. An artificial compressibility method is used in numerical solution. In this case one can use marching in time to find steady solution with steady boundary conditions in the same manner as in the case of compressible flow. The system of governing equations is discretized by the finite volume method in space. The viscous fluxes are computed using dual finite volumes cells of the diamond type. The convective fluxes are discretized in a central manner. The resulting system of ordinary differential equations is then solved by the threestage Runge-Kutta method. In the case of higher Reynolds numbers an artificial viscosity of Jameson’s type is added to maintain stability of the numerical computation of the system of Navier-Stokes equations.The comparison of Newtonian, generalized Newtonian, Oldroyd-B and generalized Oldroyd-B flows is presented in the geometry of stenotic channel with bypass. Joint work with Karel Kozel.

309

Maria Adela Puscas Université Paris Est, FR 3d conservative coupling method between a compressible fluid flow and a deformable structure Minisymposium Session NFSI: Thursday, 11:00 - 11:30, CO122 ABSTRACT In this work, we present a conservative method for three-dimensional inviscid fluidstructure interaction problems. On the fluid side, we consider an inviscid Euler fluid in conservative form. The Finite Volume method uses the OSMP high-order flux with a Strang operator directional splitting [1]. On the solid side, we consider an elastic deformable solid. In order to examine the issue of energy conservation, the behavior law is here assumed to be linear elasticity. In order to ultimately deal with rupture, we use a Discrete Element method for the discretization of the solid [2]. Body-fitted methods are not well-suited for this type of problem or even for large displacements of the structure, since they involve possibly costly remeshing of the fluid domain. We use an immersed boundary technique through the modification of the finite volume fluxes in the vicinity of the solid. The method is tailored to yield the exact conservation of mass, momentum and energy of the system and exhibits consistency properties. Since both fluid and solid methods are explicit, the coupling scheme is designed to be globally explicit too. The computational cost of the fluid and solid methods lies mainly in the evaluation of fluxes on the fluid side and of forces and torques on the solid side. It should be noted that the coupling algorithm evaluates these only once every time step, ensuring the computational efficiency of the coupling. Our approach is an extension to the three-dimensional deformable case of the conservative method developed in [3]. We will present numerical results assessing the robustness of the method in the case of a deformable solid with large displacements coupled with a compressible fluid flow. REFERENCES [1] V. Daru and C. Tenaud. High-order one-step monotonicity-preserving schemes for unsteady com- pressible flow calculations. Journal of Computational Physics, 193:563-594, 2004. [2] L. Monasse and C. Mariotti. An energy-preserving Discrete Element Method for elastodynamics. ESAIM: Mathematical Modelling and Numerical Analysis, 46:1527-1553, 2012. [3] L. Monasse, V. Daru, C. Mariotti, S. Piperno and C. Tenaud. An embedded Boundary method for the conservative coupling of a compressible flow and a rigid body. Journal of Computational Physics, 231:2977-2994, 2012. Joint work with Alexandre ERN, Laurent MONASSE, Virginie DARU, Christian TENAUD, and Christian MARIOTTI.

310

Qingguo Qingguo Hong RICAM, Austrian Academy of Science, AT A multigrid method for discontinuous Galerkin discretizations of Stokes equations Minisymposium Session PSPP: Thursday, 14:30 - 15:00, CO3 In this talk, a multigrid algorithm for discontinuous Galerkin(DG) H(div)-conforming discretizations of the Stokes equations is presented. Using the Augmented Uzawa method to solve this saddle point problem, a linear elasticity problem needs to be solved efficiently. A variable V- cycle and a W-cycle are designed for this purpose since the bilinear forms arising from DG disretizations are nonnested. The proposed method is proved to converge uniformly independent of the Poisson ratio and mesh size which shows its robustness and optimality. Joint work with Johannes Kraus, Jinchao Xu, and Ludmil Zikatanov.

311

Andreas Rademacher Mathematical Institute, University of Cologne, DE Model and mesh adaptivity for frictional contact problems Contributed Session CT4.7: Friday, 08:20 - 08:50, CO122 Frictional contact problems play an important role in many production processes. Here, the use of complex frictional laws to ensure an accurate modelling leads to an high computational effort. One approach to reduce the effort is given by mesh adaptivity based on goal oriented a posteriori error estimation, which is discussed, for instance, in [1]. An advanced idea is now not only to adaptively modify the mesh but also the underlying models based on a posteriori error estimators. In this note, we shortly describe the approach in the case of Signorini’s problem with friction in mixed form using the notation of [1]: (σ(u), ε(v))+ < λn , vn > +(λt , svt )0,ΓC

=

< l, v >,

< µn − λn , un − g > +(µt − λt , sut )0,ΓC

≤

0,

for all v ∈ V , all µn ∈ Λn and all µt ∈ Λt . Here, s specifies the reference friction model. This problem is discretized with a mixed finite element approach leading to a discrete solution (uh , λn,H , λt,H ), where the usual nodal low order finite element approach is used to discretize the displacement. For the discretization of the Lagrange multipliers piecewise constant basis functions on coarser meshes as in [1] or biorthogonal basis functions leading to Mortar methods, see, e.g., [2], are applied. The first and essential step for model adaptivity is to specify an admissible and consistent model hierarchy. One example of such a model hierarchy for friction laws is given by the following models: frictionless contact, Tresca friction, Coulomb friction, friction model by Betten. We refer to [3, Section 4.2] to a detailed description of the single models. Now, a simplified friction model sm is locally composed by choosing one of the models out of the hierarchy. The corresponding discrete m m solution is given by (um h , λn,H , λt,H ). The aim is now to derive a posteriori error estimates for the error J(u, λn , λt ) − m m J(um h , λn,H , λt,H ), where J is an user specified possibly nonlinear output functional. To this end, the contact conditions are formulated with the help of a nonlinear complementarity (NCP) function such that we arrive at a semilinear problem. Here, the NCP function is given by Z D(uh , λt,H )(µt,H ) := µt,H (max{s, kλt,H + ut,h k} − s · (λt,H + ut,h )) do ΓC

for the reference friction model s. For the model adaptive friction law sm , it is given by Dm . The approach presented in [4] to derive a posteriori error estimates concerning the model and the discretization error is applied on the given problem formulation. However, we have to pay special attention to the remainder terms due to the nondifferentiability of the NCP function. At last, we obtain the model error estimator η m = Dm (uh , λt,H )(ξt,H ) − D(uh , λt,H )(ξt,H ) and the usual discretization error estimator η h . The function ξt,H is the Lagrange multiplier concerning the frictional variable of the dual problem, which corresponds in this case to the last step of a primal dual active set method for solving the reduced problem. The error estimators η m and η h are localized and normalized to obtain model and refinement indicators, respectively. In the adaptive strategy, the values 312

η h and η m are compared and using a balancing strategy, it is decided, whether the mesh, the modeling or both should be improved. Then, standard techniques are applied for model improvement respectively mesh refinement. In Figure 1, we present some numerical results. Here, the adaptively chosen friction model for a three dimensional contact problem is depicted.

References [1] Blum, H., Rademacher, A. and Schröder, A., Goal oriented error control for frictional contact problems in metal forming, Key Engineering Materials, 504506, 987-992 (2012). [2] Wohlmuth, B., Variationally consistent discretization schemes and numerical algorithms for contact problems, Acta Numerica, 20, 569-734 (2011). [3] Wriggers, P., Computational Contact Mechanics, John Wiley & Sons, Chichester (2002). [4] Braack, M. and Ern, A., A posteriori control of modelling errors and discretisation errors, Multiscale Model. Simul., 1, 221-238 (2003).

Figure 1: Exemplarily results of the adaptive algorithm

313

Istvan Reguly Oxford e-Research Centre, University of Oxford, GB OP2: A library for unstructured grid applications on heterogeneous architectures Minisymposium Session PARA: Monday, 15:30 - 16:00, CO016 Due to the physical limitations in building faster single core microprocessors, the development and use of multi- and many-core architectures for general purpose scientific and engineering applications has received increasing attention for the past few years. The greatest obstacle to the widespread usage of parallel computing is the difficulty to program such devices in an efficient and scalable manner. It is unreasonable to expect domain experts who want to write efficient applications to learn different complex parallel programming languages and create hardware specific code. In the past, traditional programming languages such as C and Fortran scaled well over time with increasingly higher processor frequencies, however this is no longer the case, because of the inevitable heterogeneity of high-performance architectures. A more efficient solution to this issue is to provide high-level programming abstractions to the application developers, which permit them to focus on the mathematical aspects of the problem, leaving the optimisation issue to a corresponding framework that, thanks to the insight into the high-level program abstractions, is capable of solving the performance portability issue across heterogeneous architectures, and other aspects such as code longevity. OPlus (Oxford Parallel Library for Unstructured Solvers) [1], a research project that had its origins in 1993 at the University of Oxford, provided such an abstraction framework for performing unstructured mesh based computations. OP2 [2] is the successor of OPlus, bringing support for state-of-the-art hardware such as many-core processors and heterogeneous systems. OPlus and OP2 can be viewed as an instantiation of the AEcute (access-execute descriptor) programming model that separates the specification of a computational kernel with its parallel iteration space, from a declarative specification of how each iteration accesses its data. We present the design of the current OP2 library, starting with the API that uses the notion of sets and mappings between sets to define the mesh and its components. Data of arbitrary dimension can be assigned to the elements of any set. Loops over these sets are defined through OP2’s API by specifying the set itself and the datasets accessed either directly or indirectly via a mapping. The framework takes care of data dependencies and indirect access, thus the operation performed on each set element can be oblivious to the underlying mesh. OP2 utilizes sourceto-source translation and compilation so that a single application code written using the OP2 API can be transformed into different parallel implementations for execution on different back-end hardware platforms. We briefly describe our code generation technique that only involves static parsing of OP2 API calls, which define the access to all data in a given parallel loop. We discuss how execution maps to different hardware and multiple levels of parallelism: on multicore CPUs, different generations of GPUs and across MPI, and what parameters are involved in this mapping that can have an impact on performance. One of the core issues is the handling of data dependencies in accessing indirectly referenced data. The OP2 run-time support solves this parallelism control problem in different ways, depending on the target back-end architecture: race conditions are handled using an owner-compute approach over MPI, a block coloring scheme on the coarse shared-memory level, and for vector machines, suchs as the GPU, an additional set element based coloring is also required. Since the difference

314

between the computing capacity and the bandwidth to off-chip memory has been increasing on modern hardware, we discuss the impact of unstructured mesh data layouts (array of structs vs. struct of arrays) on different architectures. Similarly, we show how multi-level memory hierarchies can be exploited, such as the on-chip cache and the explicitly managed scratch-pad memory. Data locality is one of the most important factors affecting performance, by dividing the execution set into mini-partitions and staging data we show how to improve data locality. One of the key challenges is the ever-changing hardware landscape. We aim to achieve near-optimal performance on different CPUs, GPUs and future many-core architectures, but their parameters change even from generation to generation: a good example is the shift in required levels of parallelism and amount of resources used between the Fermi and the Kepler generation of GPUs. For this reason, we have to re-evaluate and re-tune the code we generate for different back-ends for new hardware. Additionally, there are parameters that depend on the application; the tuning of these have to be carried out in the context of the application. We demonstrate the tools available, and show the advantages of tuning. Finally, through Volna [3], a tsunami simulation code that was ported to OP2, we provide a contrasting benchmarking and performance analysis study on a range of multi-core/many-core systems. Acknowledgments This research is funded by the UK Technology Strategy Board and Rolls-Royce plc. through the Siloet project, the UK Engineering and Physical Sciences Research Council pro jects EP/I006079/1, EP/I00677X/1 on Multi-layered Abstractions for PDEs and the Algorithms and Software for Emerging Architectures (ASEArch) EP/J010553/1 project

References [1] Crumpton, P. I. and Giles, M. B. Multigrid aircraft computations using the OPlus parallel library. Parallel Computational Fluid Dynamics: Implementations and Results Using Parallel Computers (1998), pp. 339 - 346. [2] M.B. Giles, G.R. Mudalige, Z. Sharif, G. Markall, P.H.J. Kelly. Performance Analysis and Optimization of the OP2 Framework on Many-Core Architectures. The Computer Journal (2011). ISSN 0010-4620 [3] D. Dutykh, R. Poncet and F. Dias, The VOLNA code for the numerical modeling of tsunami waves: Generation, propagation and inundation, European Journal of Mechanics - B/Fluids (2011), vol. 30, issue 6, pp. 598-615 Joint work with M. B. Giles, G. R. Mudalige, C. Bertolli, and P.H.J. Kelly.

315

Gunhild Allard Reigstad NTNU, NO Numerical investigation of network models for Isothermal junction flow Contributed Session CT4.3: Friday, 09:20 - 09:50, CO3 This paper deals with the issue of how to properly model fluid flow in pipe junctions. In particular we investigate the numerical results from three alternative network models, all three based on the Isothermal Euler equations. Using two different test cases we will focus on the physical validity of simulation results from each of the models. We will as well show how the different models may produce results that are fundamentally different for a given set of initial data. Finally we will give some attention to the selection of suitable test cases for network models. Network models are used to find global weak solutions for hyperbolic conservation laws defined on N segments of the real line, where all segments are connected by junctions. In addition to flow in pipelines, such models are used to describe for example traffic flow, data networks, and supply chains. Each pipe in a network model is modelled along a local axis (x ∈ R+ ) and the pipe-junction interface is at x = 0. Presupposing constant initial conditions, the flow condition in each pipe may be found from the half-Riemann problem: ∂ ∂Uk + F(Uk ) = 0 ∂t ( ∂x ¯ k if x > 0 U Uk (x, 0) = U∗k if x < 0,

(1)


¯ 1, . . . , U ¯ N = lim Uk (x, t), U∗k U +

(2)

where x→0

¯ k by a Lax wave of the 2nd family and obey a set is by definition connected to U of coupling conditions. For the Isothermal Euler equations, the coupling conditions are related to mass and momentum: CC1: Mass is conserved at the junction: N X

ρ∗k vk∗ = 0.

(3)

k=1

CC2: There is a unique, scalar momentum related coupling constant at the junction: ˜ Hk∗ (ρ∗k , vk∗ ) = H ∀k ∈ {1, . . . , N }. (4) Three different expressions for the momentum related coupling constant are considered in this paper. Equal pressure (CCp) and equal momentum flux (CCMF) have been frequently used in the litterature 1 2 3 4 . Equal Bernoulli (CCBI) 1 M. K. Banda, M. Herty and A. Klar, Gas flow in pipeline networks, Netw. Heterog. Media 1, 41–56, (2006). 2 M. K. Banda, M. Herty and A. Klar, Coupling conditions for gas networks governed by the isothermal Euler equations, Netw. Heterog. Media 1, 295–314, (2006). 3 R. M. Colombo and M. Garavello, A well posed Riemann problem for the p-system at a junction, Netw. Heterog. Media 1, 495–511, (2006). 4 M. Herty and M. Seaïd, Simulation of transient gas flow at pipe-to-pipe intersections, Netw. Heterog. Media 56, 485–506, (2008).

316

was recently proposed in Reigstad et al. 5 The physical validity of the network model results is evaluated by an entropy condition 3 . Analytical investigations in Reigstad et al. showed that for certain ranges of flow rates in a junction with three connected pipes, the coupling constants CCp and CCMF would produce unphysical results. Using equal Bernoulli (CCBI) would yield physical results for all subsonic initial data. In the present paper, the numerical test cases will be used to verify this analysis and to explore the behaviour of the different models. The first case consists of a junction connecting 5 pipes. The case illustrates how the network model easily may be applied on a large number of pipes connected at a junction. We will as well show how the results of the three models relate to the entropy condition. The second case consists of three pipes connected by two junctions such that a closed system is constructed. We will show how the different models produce fundamentally different results in terms of rarefaction and shock waves. The total energy of the system as a function of time will as well be presented in order to display the effect of having non-entropic solutions.

Acknowledgements This work was financed through the research project Enabling low emission LNG systems. The authors acknowledge the project partners; Statoil and GDF SUEZ, and the Research Council of Norway (193062/S60) for support through the Petromaks programme. Joint work with Tore Flåtten.

5 G. A. Reigstad, T. Flåtten, N. E. Haugen and T. Ytrehus, Coupling constants and the generalized riemann problem for isothermal junction flow, Submitted (2012).

317

Knut Reimer Christian Albrechts Universität zu Kiel, DE H2 -matrix arithmetic and preconditioning

Contributed Session CT4.2: Friday, 09:50 - 10:20, CO2 The discretisation of integral and partial differential equations leads to highdimensional systems of linear equations. Usually these systems are ill-conditioned. Thus preconditioners are needed to ensure fast convergence for iterative methods. The approximated inverse and LU-decomposition are established approaches. For this purpose H-matrix arithmetic is commonly known as an efficient technique with log-linear complexity. A further development of the H-matrices are the H2 matrices, which enable storage and evaluation in linear complexity, instead of log-linear. To adapt the H-matrix technique to H2 -matrices, it is essentially to design an efficient low-rank-update for every block of an H2 -matrix. The talk presents the idea for the low-rank-update and a sketch of the inversion and the LU-decomposition. It concludes with some numerical results for both, the inversion and the LU-decomposition. Joint work with Steffen Börm.

318

Sergey Repin University of Jyvaskyla, FI On Poincaré Type Inequalities for Functions With Zero Mean Boundary Traces and Applications to A Posteriori Analysis of Boundary Value Problems Contributed Session CT4.7: Friday, 08:50 - 09:20, CO122 We discuss Poincaré type inequalities for the functions having zero mean value on the whole boundary of a Lipschitz domain or on a measurable part of the boundary. For some basic domains (rectangles, cubes, and right triangles) exact and easily computable constants in these inequalities can be derived [1]. With the help of the inequalities a new type of a posteriori estimates are derived. The major difference with respect to well known error a posteriori estimates of the functional type (see an overview in [2]) consists of that new estimates are applicable to a much wider set of approximations. They can be useful for approximations violating boundary conditions and for nonconforming approximations. One other application is related to modeling errors arising as a result of coarsening of a boundary value problem. In this case, the estimates yield directly computable bound of the modeling error encompased in the coarsened solution. Constants in Poincaré type inequalities enter all these estimates, which provide guaranteed error control of the corresponding approximation and modeling errors. Finally, we discuss possible applications to a posteriori estimates of nonlinear elliptic problems.

References [1] A. Nazarov, S. Repin Exact constants in Poincare type inequalities for functions with zero mean boundary traces. Preprint V.A. Steklov Inst. Math. St. Petersburg, 2012 (arXiv [math. AP], 1211.2224) [2] S. Repin. A posteriori error estimates for partial differential equations. Walter de Gruyter, Berlin, 2008.

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Thomas Richter Universität Heidelberg, DE A Fully Eulerian Formulation for Fluid-Structure Interactions Minisymposium Session NFSI: Thursday, 11:30 - 12:00, CO122 In this contribution, we present a monolithic formulation for fluid-structure interactions, where both subproblems - fluid and solid - are given in Eulerian coordinates on a fixed background mesh. This formulation comes without the necessity of artificial transformations of domains and meshes. It can be used to describe problems with very large deformation, free motion of the solid in the fluid and it can model contact problems. As a front-capturing method on a fixed background mesh, the interface moves freely throughout the mesh. We present the Initial Point Set as an alternative to Levelset formulations capturing this interface. Joint work with Thomas Wick.

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Marco Rozgic Helmut Schmidt University of the Federal Armed Forces Hamburg, DE Mathematical optimization methods for process and material parameter identification in forming technology Contributed Session CT3.1: Thursday, 16:30 - 17:00, CO1 Parameter identification plays a crucial role in various mathematical applications and technological fields. Both to determine good parameter sets and to judge the quality of a computed set of parameters a rigorous mathematical theory is needed. A common method to determine optimal parameters is to solve an inverse problem. Typical inverse problems that arise in forming processes are material and process parameter identification [2]. The framework presented by Taebi et al. [3] shows the impact of the ability to choose good process parameters when exploring a new technology. The presented methodology finds optimal parameters in a quasi static forming process combined with an electromagnetic high speed forming method in order to extend classical forming limits. The parameter space comprises contributions of the triggering current (e.g., frequency, amplitude, damping, etc.), geometric descriptions of the tool coil as well as deep drawing parameters (e.g., drawing radii or tribological parameters). The quality of a given parameter set is determined by computing the distance of the simulated forming result to the prescribed ideal shape. Every evaluation of the objective function requires a full coupled (mechanical and electromagnetic) finite element simulation. Reliable and fast computable material models are needed to perform efficient numerical finite element simulations. Recently introduced anisotropic models [4] take into account nonlinear kinematic and isotropic hardening. To identify material parameters we introduce an optimization method based on a simulation of a uniaxial tensile test. The objective is to fit the simulated data to the experimental results. Again the objective function is only accessible by a finite element simulation, which makes its evaluation expensive. Opposed to optimal control theory approaches often used in the context of finite element based problems [1] we state the inverse problem as a classical discretized nonlinear optimization problem where objective and constrain evaluation require full simulations of the underlying differential equations. The arising nonlinear optimization problems can be solved by various methods. Derivative free methods like genetic algorithms or simulated annealing usually require more function evaluations and parameter tuning than gradient descent based methods, on the other hand gradient information is not always accessible. In our approach we focus on the use of inner point methods as proposed in [5]. Inner point methods are known to be fast and efficient. Furthermore the fact that the parameters computed by the algorithm lie in the interior region defined by constrains is often beneficial in technological applications. The required derivative information is computed by the finite difference method. To tackle the resulting increase in the number of objective and constrain function evaluations adaptive control heuristics are needed. Based on the duality gap, such a heuristic could for example decide if either the full model or a reduced function, as in active set methods, should be used to perform a Newton step. Further the size of the duality gap can be used to control if a rough discretized finite element simulation or a more precise, fine discretized simulation should be performed. In case of a large duality gap a less accurate simulation can yield sufficiently good descent directions in less time. Finally the underlying elastoviscoplastic models have to be investigated in order to assess the quality of an identified parameter set. Questions about con-

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strain qualifications and necessary optimality conditions can only be answered by a close observation of the whole discretization scheme. We will discuss the use of inner point methods in the scope of material and process parameter identification for technological forming. A systematic approach to study the properties of the resulting optimization problems is introduced. Further we will point towards a duality gap based heuristic that can eventually help to reduce the number of expensive function calls.

References [1] R. Becker and R. Rannacher. An optimal control approach to a posteriori error estimation in finite element methods. Acta numerica, 10(1):1–102, 2001. [2] J.-L. Chenot, E. Massoni, and J. Fourment. Inverse problems in finite element simulation of metal forming processes. Engineering computations, 13(2/3/4):190–225, 1996. [3] F. Taebi, O. Demir, M. Stiemer, V. Psyk, L. Kwiatkowski, A. Brosius, H. Blum, and A. Tekkaya. Dynamic forming limits and numerical optimization of combined quasi-static and impulse metal forming. Computational Materials Science, 54(0):293 – 302, 2012. [4] I. Vladimirov and S. Reese. Anisotropic finite plasticity with combined hardening and application to sheet metal forming. International Journal of Material Forming, 1:293–296, 2008. [5] A. Wächter and L. T. Biegler. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming, 106(1):25–57, 2006. Joint work with Robert Appel, and Marcus Stiemer.

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Gianluigi Rozza SISSA, International School for Advanced Studies, mathLab, IT A reduced computational and geometrical framework for viscous optimal flow control in parametrized systems Minisymposium Session ROMY: Thursday, 15:30 - 16:00, CO016 Any computational problem is filled with uncertain elements, such as (i) material parameters and coefficients, boundary conditions, and (ii) geometrical configurations. Usually such factors cannot be completely identified to the point of absolute certainty; the former may be recovered from measurements, while the latter can be obtained as a result of a shape identification process. In general, inverse identification problems entail very large computational efforts, since they involve iterative optimization procedures that require several input/output evaluations. Incorporating geometrical configurations into the framework, e.g. when dealing with problems or optimal shape design, makes the inverse problem even less affordable. Given a parameterized PDE model of our system, the forward problem consists in evaluating outputs of interest (depending on the state solution of the PDE) for specified parameter inputs. On the other hand, whenever some parameters are uncertain, we aim at inferring their values (and/or distributions) from indirect observations (and/or measures) by solving an inverse problem: given an observed output, can we deduce the value of the parameters that resulted in this output? Such problems are often encountered as problems of parameter identification, variational data assimilation, flow control or shape optimization. Computational inverse problems are characterized by two main difficulties: 1. The forward problem is typically a nonlinear PDE, possibly incorporating coupled multiphysics phenomena. State-of-the-art discretization methods and parallelized codes are therefore required to solve them up to a tolerable accuracy. This is exacerbated by the fact that solving the inverse problem requires multiple solutions of the forward problem. Hence, if the forward problem can be replaced with an inexpensive (but reliable) surrogate, solving the inverse problem is much more feasible. 2. Uncertainty in the model parameters can be large when the parameters describe geometric quantities such as shape. This is especially true in biomedical applications, where observed geometries are often patient-specific and only observable through medical imaging procedures which are highly susceptible to measurement noise. In this talk we propose a general framework for computationally solving inverse and optimal flow control problems using reduced basis methods, and apply it to some inverse identification problems in haemodynamics. An implementation of the reduced basis method is presented by considering different shape or domain parametrizations by non-affine maps with flexible techniques, such as free-form deformations or radial basis functions. In order to develop efficient numerical schemes for inverse problems related with shape variation such as shape optimization, geometry registration and shape analysis through parameter identification, we combine a suitable low-dimensional parametrization of the geometry (yielding a geometrical complexity reduction) with reduced basis methods (yielding a reduction of computational complexity). The analysis will focus on the general properties (stability, reliability, accuracy) and performance of the reduced basis 323

method for Stokes and Navier-Stokes equations and we will highlight its special suitability for the numerical study of viscous flows in parametrized geometries with emphasis in cardiovascular problems. Joint work with Andrea Manzoni, Federico Negri, and Alfio Quarteroni.

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Karl Rupp Argonne National Laboratory, US ViennaCL - Portable High Performance at High Convenience Minisymposium Session PARA: Monday, 12:10 - 12:40, CO016 High-level application programming interfaces (API) are said to be the natural enemy of performance. Even though suitable programming techniques as well as just-in-time compilation approaches have been developed in order to overcome most of these limitations, the advent of general purpose computations on graphics processing units (GPUs) has lead to a renaissance of a wide-spread use of low-level programming languages such as CUDA and OpenCL. Porting existing code to GPUs is, however, in many cases a very time consuming process if low-level programming languages are used. They require the programmer to understand many details of the underlying hardware and often consume a larger amount of development time than what is saved by a reduced total execution time. On the other hand, high-level libraries for GPU computing can significantly reduce the porting effort without changing too much of existing code. ViennaCL is one of the most widely used library offering a high-level C++ API for linear algebra operations on multi-core CPUs and many-core architectures such as GPUs. In particular, we demonstrate that a high-level API for linear algebra operations can still be provided without sacrificing performance on GPUs. Furthermore, the generic implementations of algorithms such as iterative solvers allow for code reuse beyond device and library boundaries, making the transition from purely CPU-based code to GPU-accelerated code as seamlessly as possible. Also, we explain why and how ViennaCL manages different parallel computing backends and assess the role of autotuning for achieving portable performance. Benchmark results for GPUs from NVIDIA and AMD as well as for Intel’s MIC platform are presented along with a discussion of techniques for achieving portable high performance.

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Figure 1: STREAM-like benchmark for the performance of vector additions (top) and performance comparison for 50 iterations of the conjugate gradient method (bottom) on different computing hardware. Joint work with Philippe Tillet, Florian Rudolf, and Josef Weinbub.

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Daniel Ruprecht Institute of Computational Science, University of Lugano, CH Convergence of Parareal for the Navier-Stokes equations depending on the Reynolds number Contributed Session CT1.6: Monday, 18:00 - 18:30, CO017

1

Introduction

The number of cores in state-of-the-art high-performance computing systems is rapidly increasing and has reached the order of millions already. This requires new inherently parallel algorithms that feature a maximum degree of concurrency. A promising approach for time-dependent partial differential equations are method that parallelize in time. The Parareal parallel-in-time method has been introduced in (Lions et al., 2001) and the first study considering its application to the NavierStokes equations is (Fischer et al., 2003). Another study reporting speedups for experiments on up to 24 processors with a Reynolds number of Re = 1000 can be found in (Trindade and Pereira, 2004). A larger scale study using up to 2,048 cores is conducted in (Croce et al., 2012), but also only for Re = 1000. It is has been demonstrated numerically and theoretically that Parareal can exhibit instabilities when applied to advection-dominated problems, see e.g. (Ruprecht and Krause, 2012) and references given there. Thus it can be expected that Parareal will cease to converge for increasing Reynolds numbers.

2

Intended Experiments and Preliminary Results

The present study aims at providing a detailed numerical investigation of the convergence behavior of the Parareal method over a wide range of Reynolds numbers. The setup will be a two-dimensional driven cavity flow. The employed code solves the incompressible Navier-Stokes equations in dimensionless form, that is 1 ∆u Re ∇·u=0

ut + u · ∇u + ∇p =

(1) (2)

so that the Reynolds number can be directly controlled as a parameter. Based on the documented instability of Parareal for advection-dominated flows, we expect an instability to arise for increasing Reynolds number. A first hint is given in Figure 1: It shows the "residuals" of the Parareal, that is the maximum change from the previous to the current iteration, for different Reynolds numbers. While for small Reynolds numbers the iteration converges very quickly, for larger Reynolds numbers the iteration stalls for several iterations before starting to converge. The simulations used a small time-step of ∆t = 1/250 in the coarse method and half this step size in the fine propagator. The spatial resolution of ∆x = 1/32. The coarse propagator run alone is stable and provides a reasonable solution for all depicted Reynolds numbers.

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3

Outlook

The talk will present a comprehensive study of the behavior of Parareal for large Reynolds numbers. It will illustrate how the convergence behavior depends on the Reynolds number and also explore the effect of spatial and temporal resolution. Also, studies with Parareal using implicit as well as explicit integrators will be performed. For the linearized Navier-Stokes equations, the effect of adding a stabilization as in (Ruprecht and Krause, 2012) will be analyzed.

Figure 1: Maximum defect between current and previous iteration in Parareal for Reynolds numbers ranging from Re = 102 to Re = 105 . Joint work with Johannes Steiner, Robert Speck, and Rolf Krause.

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Oxana Sadovskaya Institute of Computational Modeling SB RAS, RU Parallel Software for the Analysis of Dynamic Processes in Elastic-Plastic and Granular Materials Contributed Session CT3.6: Thursday, 17:00 - 17:30, CO017 The universal mathematical model for numerical solution of 2D and 3D problems of the dynamics of deformable media with constitutive relationships of rather general form is worked out [1]. The model for description of the process of deformation of elastic bodies can be represented as the system of equations: n

A

X ∂U ∂U = + QU + G , Bi ∂t ∂xi i=1

(1)

where U is unknown vector–function, A is a symmetric positive definite matrix of coefficients under time derivatives, B i are symmetric matrices of coefficients under derivatives with respect to the spatial variables, Q is an antisymmetric matrix, G is a given vector, n is the spatial dimension of a problem (2 or 3). The dimension of the system (1) and concrete form of matrices–coefficients are determined by the used mathematical model. When taking into account the plastic deformation of a material, the system of equations (1) is replaced by the variational inequality:   n ∂U X i ∂U e − B (U − U ) A − QU − G ≥ 0 , ∂t ∂xi i=1

e, U ∈ F , U

(2)

where F is a given convex set, by means of which some constraints are imposed e is an arbitrary admissible element of F . In the on possible states of a medium, U problems of mechanics of granular media with plastic properties a more general variational inequality   n ∂U X i ∂V B − (Ve − V ) A − QV − G ≥ 0 , Ve , V ∈ F , (3) ∂t ∂xi i=1 takes place, where the vector–functions V and U are related by the equations V = λ U + (1 − λ) U π ,

U=

1−λ π 1 V − V . λ λ

Here λ ∈ (0, 1] is the parameter of regularization of the model characterizing the ratio of elastic moduli in tension and compression, U π is the projection of the vector of solution onto the given convex cone K, by means of which the different resistance of a material to tension and compression is described. The set F of admissible variations, included in (2) n o and (3), can be defined by the von Mises yield condition: F = U : τ (σ) ≤ τs , where σ is the stress tensor, τ (σ) is the intensity of tangential stresses, τs is the yield point of particles. As a convex cone K of stresses, criterion, the von Mises–Schleicher circular cone n allowed by the strength o K = U : τ (σ) ≤ æ p(σ) can be used, where p(σ) is the hydrostatic pressure, æ is the parameter of internal friction. In the framework of considered mathematical model the parallel computational algorithm is proposed for numerical analysis of dynamic processes in elastic-plastic and granular materials. The system of equations (1) is solved by means of the 329

splitting method with respect to spatial variables. An explicit monotone finitedifference ENO–scheme is applied for solving 1D systems of equations at the stages of splitting method. Variational inequalities (2) and (3) are solved by splitting with physical processes, which leads to the system (1) and the procedure of solution correction, taking into account plastic properties of a material. This procedure consists of determining a fixed point of the contractive mapping and is implemented by the method of successive approximations. Granularity of materials is accounted by means of the procedure for finding the projection onto the convex cone K of admissible stresses. The parallelizing of computations is carried out using the MPI library. The data exchange between processors occurs at step “predictor” of the finite-difference scheme. At first each processor exchanges with neighboring processors the boundary values of their data, and then calculates the required quantities in accordance with the difference scheme. Mathematical models are embedded in programs by means of software modules that implement the constitutive relationships, the initial data and boundary conditions of problems. The universality of programs is achieved by a special packing of the variables, used at each node of the cluster, into large 1D arrays. Detailed description of the parallel algorithm one can found in [1]. Program system allows to simulate the propagation of elastic-plastic waves produced by external mechanical effects in a medium body, aggregated of arbitrary number of heterogeneous blocks. Some computations of dynamic problems were performed on the cluster MVS–100k of Joint Supercomputer Center of RAS (Moscow). In Fig. 1 one can see the examples of computations for 2D Lamb’s problem about the action of concentrated impulsive load on the boundary of an elastic medium. Level curves of the normal stress for a homogeneous material (left) and for a block medium consisting of 6 blocks with interlayers from a more pliant material (right) are shown. This work was supported by RFBR (grant no. 11–01–00053) and Complex Fundamental Research Program no. 18 of the Presidium of RAS.

References [1] O. Sadovskaya, V. Sadovskii. Mathematical Modeling in Mechanics of Granular Materials. Ser.: Advanced Structured Materials, Vol. 21. Springer, Heidelberg – New York – Dordrecht – London (2012).

Figure 1: 2D Lamb’s problem for a homogeneous elastic medium (left) and for a block medium consisting of 6 blocks with pliant interlayers (right)

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Vladimir Sadovskii Institute of Computational Modeling SB RAS, RU Hyperbolic Variational Inequalities in Elasto-Plasticity and Their Numerical Implementation Contributed Session CT3.6: Thursday, 18:00 - 18:30, CO017 Thermodynamically consistent systems of conservation laws were firstly obtained by Godunov for the models of reversible thermodynamics – elasticity theory, gas dynamics and electrodynamics [1]. Such form of equations assumes the setting socalled generating potentials Φ(U ) and Ψj (U ), depending on the unknown vector– function U of the state variables. The first of them must be strongly convex function of U . By means of generating potentials the governing system is written in the next form: n ∂ ∂Φ X ∂ ∂Ψj = , (1) ∂t ∂U ∂xj ∂U j=1 or in a more general form, including terms that are independent of derivatives. The additional conservation law   X   n ∂Φ ∂ ∂Ψj ∂ U −Φ = − Ψj U (2) ∂t ∂U ∂xj ∂U j=1 is valid for this system, which may be a conservation law of energy or of entropy. Thermodynamically consistent system of conservation laws in the form (1), (2) turn out to be very useful in justification of the mathematical correctness of models. It is intended for the integral generalization, which allows to construct discontinuous solutions. For numerical analysis of the system (1), (2) the effective shock-capturing methods, such as Godunov’s method, adapted to the computation of solutions with discontinuities, may be applied. This talk addresses to generalization and application of this approach for the analysis of thermodynamically irreversible models of mechanics of deformable media taking into account plastic deformation of materials. For such models the governing systems are formulated as variational inequalities for hyperbolic operators with one-sided constraints, describing the transition in plastic state:   n ∂ ∂Φ X ∂ ∂Ψj ˜ (U − U ) − ≥ 0, ∂t ∂U j=1 ∂xj ∂U

˜ ∈ F. U, U

(3)

Here F is a convex set, whose boundary describes the yield surface of a material in the space of stresses. On this basis a priori integral estimates are constructed in characteristic cones of operators, from which follows the uniqueness and continuous dependence on initial data of solutions of the Cauchy problem and of the boundaryvalue problems with dissipative boundary conditions. With the help of an integral generalization of variational inequalities the relationships of strong discontinuity in dynamic models of elastic-plastic and granular media are obtained, whose analysis allows us to calculate velocities of shock waves and to construct discontinuous solutions. Original shock-capturing algorithms are developed, which can be considered as a realization of the splitting method with respect to physical processes [2]. Such algorithms automatically satisfy the properties of monotonicity and dissipativity

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on discreet level. They are applicable for computation of the solutions with singularities of the type of strong discontinuities (elastic-plastic shock waves) and of discontinuities of displacements. The approximation of differential operator and constraint on an example of the inequality (3) leads to the following discrete problem: ˜ −U ˆ k+1 ) (U



 n X ∂Ψkj ∂Φk+1 ∂Φk − − ∆t ≥ 0, Λj ∂U ∂U ∂U j=1

ˆ k+1 , U ˜ ∈ F, U

ˆ k+1 is a special combination of U k+1 and U k , ∆t is the time step of a grid, where U and Λj is the differential operator approximating partial derivative with respect ˆ k+1 = U k+1 there is the most simple to the spatial variable xj . In the case of U problem. Its solution can be found in two steps: at first the vector n X ¯ k+1 ∂Ψkj ∂Φk ∂Φ = − ∆t , Λj ∂U ∂U ∂U j=1

implementing the explicit finite-difference scheme at each time step for the system of equations (1), is calculated and then the solution correction is made in accordance with the variational inequality  k+1 ¯ k+1  ∂Φ ˜ − U k+1 ) ∂Φ ˜ ∈ F, − (U ≥ 0, U k+1 , U ∂U ∂U which is equivalent by convexity of Φ(U ) to the problem of conditional minimiza¯ k+1 /∂U under the constraint U k+1 ∈ F . tion of the function Φ(U k+1 ) − U k+1 ∂ Φ If the generating potential Φ(U ) is a quadratic function, then the solution correction is reduced to determining the projection onto the convex set F with respect to the corresponding norm. This method of correction was first used by Wilkins under numerical solution of elastic-plastic problems, it is widespread now. However there exist more accurate algorithms, which are realized rather simple in [3]. With the help of these algorithms the results of simulation of the wave motion in elastic-plastic and granular media are obtained. This work was supported by the Russian Foundation for Basic Research (grant no. 11–01–00053) and the Complex Fundamental Research Program no. 18 of the Presidium of RAS.

References [1] S.K. Godunov, E.I. Romenskii. Elements of Continuum Mechanics and Conservation Laws. Kluwer Academic / Plenum Publishers, New York – Boston – Dordrecht – London – Moscow (2003). [2] O. Sadovskaya, V. Sadovskii. Mathematical Modeling in Mechanics of Granular Materials. Ser.: Advanced Structured Materials, Vol. 21. Springer, Heidelberg – New York – Dordrecht – London (2012). [3] V.M. Sadovskii. Discontinuous Solutions in Dynamic Elastic-Plastic Problems. Fizmatlit, Moscow (1997) [in Russian].

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Davood Saffar Shamshirgar PhD student, Applied and Computational Mathematics, KTH, SE The Spectrally Fast Ewald method and a comparison with SPME and P3M methods in Electrostatics Contributed Session CT1.8: Monday, 17:30 - 18:00, CO123 The Ewald summation formula is the basis for different methods used in molecular dynamic simulation in which the computation of the long range interactions in a periodic setting is important. The P3M method by Hockney and Eastwood (1981) and SPME method by Essmann et al. (1995) are FFT-based Ewald methods that scale as O(N log(N )), where N is the number of particles. These methods are used in several major molecular dynamics packages e.g. GROMACS, NAMD and AMBER. The new spectrally accurate FFT based algorithm developed by Lindbo and Tornberg (2011), is similar in structure to the above mentioned methods. In the new method, approximation errors can however be controlled without increasing the resolution of the FFT grid, in contrast to the other methods. Hence, the Spectral Ewald (SE) method significantly reduces both the computational cost for the FFTs and the associated memory use. The price you pay is an increased “spreading" or “interpolation" cost when evaluating the grid functions, arising from the larger support of the truncated Gaussians used for the spreading. The use of Gaussians instead of polynomial interpolation is what allows for the spectral accuracy. Another key aspect of the new method is the error estimate yielding a straight forward parameter selection. The SE method has earlier been shown by Lindbo and Tornberg (2011) to be more efficient than the SPME and P3M methods for all but very low accuracies. To have a fair comparison, the new algorithm is implemented as a plug-in into GROMACS, a molecular dynamics package which primarily developed for simulation of lipids and proteins, where the other methods are already available and highly optimized. Joint work with Anna-Karin Tornberg.

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Mehmet Sahin Istanbul Technical University, TR Parallel Large-Scale Numerical Simulations of Purely-Elastic Instabilities with a Template-Based Mesh Refinement Algorithm Minisymposium Session MANT: Tuesday, 10:30 - 11:00, CO017 The parallel large-scale unstructured finite volume method proposed in [Sahin, A stable unstructured finite volume method for parallel large-scale viscoelastic fluid flow calculations. J. Non-Newtonian Fluid Mech., 166 (2011) 779–791] has been incorporated with a template-based mesh refinement algorithm in order to investigate the viscoelastic fluid flow instabilities. The numerical method based on side-centered finite volume method where the velocity vector components are defined at the mid-point of each cell face, while the pressure term and the extra stress tensor are defined at element centroids. The present arrangement of the primitive variables leads to a stable numerical scheme and it does not require any ad-hoc modifications in order to enhance the pressure-velocity-stress coupling. The combination of the present numerical method with the log-conformation representation proposed in [R. Fattal, R. Kupferman, Constitutive laws for the matrix-logarithm of the conformation tensor. J. Non-Newtonian Fluid Mech. 123 (2004) 281–285] and the geometric non-nested multilevel preconditioner for the Stokes system have enabled us to simulate large-scale viscoelastic fluid flow problems on highly parallel machines. The calculations are presented for an Oldroyd-B fluid past a confined circular cylinder in a rectangular channel and the sphere falling in a circular tube at relatively high Weissenberg numbers. The present numerical calculations reveal three-dimensional purely-elastic instabilities in the wake of a confined single cylinder which is in accord with the earlier experimental results in the literature. In addition, the flow field is found out to be no longer symmetric at high Weissenberg numbers.

Figure 1: Computed surface streamtraces at W e = 2.0 on the cylinder surface (r = 1.01R) for an Oldroyd-B fluid past a confined circular cylinder in a rectagular channel with a periodic boundary condition in the spanwise direction (β = 0.59). The streamtrace color indicates v−velocity components. Joint work with Evren ONER.

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Giovanni Samaey Numerical Analysis and Applied Mathematics, Dept. Computer Science, KU Leuven, BE A micro-macro parareal algorithm: application to singularly perturbed ordinary differential equations Contributed Session CT4.1: Friday, 09:20 - 09:50, CO1 We introduce a micro-macro parareal algorithm for the time-parallel integration of multiscale-in-time systems. The algorithm first computes a cheap, but inaccurate, solution using a coarse propagator (simulating an approximate slow macroscopic model), which is iteratively corrected using a fine-scale propagator (accurately simulating the full microscopic dynamics). This correction is done in parallel over many subintervals, thereby reducing the wall-clock time needed to obtain the solution, compared to the integration of the full microscopic model. We provide a numerical analysis of the algorithm for a prototypical example of a micro-macro model, namely singularly perturbed ordinary differential equations. We show that the computed solution converges to the full microscopic solution (when the parareal iterations proceed) only if special care is taken during the coupling of the microscopic and macroscopic levels of description. The convergence rate depends on the modeling error of the approximate macroscopic model. We illustrate these results with numerical experiments. Joint work with Frederic Legoll, and Tony Lelievre.

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Mattias Sandberg KTH Mathematics, SE An Adaptive Algorithm for Optimal Control Problems Contributed Session CT4.7: Friday, 09:50 - 10:20, CO122 The analysis and performance of numerical computations for optimal control problems is complicated by the fact that they are ill-posed. It is for example often the case that optimal solutions depend discontinuously on data. Moreover, the optimal control, if it exists, may be a highly non-regular function, with many points of disontinuity etc. On the other hand optimal control problems are well-posed in the sense that the associated value function is well-behaved, with such properties as continuous dependence on data. I will present an error representation for approximation of the value function when the Symplectic Euler scheme is used to discretize the Hamiltonian system associated with the optimal control problem. It is given by X u ¯(x0 , t0 ) − u(x0 , t0 ) = ∆t2n ρn + R, (1) n

where u ¯ is the approximation of the value function u, and the term ρn is an error density which is computable from the Symplectic Euler solution. I will show a theorem which says that the remainder term R is small compared with the error density sum in (1). The proof uses two fundamental facts: 1. The value function solves a non-linear PDE, the Hamilton-Jacobi-Bellman equation. When this property is used, we take advantage of the well-posed character of the optimal control problem. 2. The Symplectic Euler scheme corresponds to the minimization of a discrete optimal control problem. Using this error representation I will show an example of an adaptive algorithm, and illustrate its performance with numerical tests. I will also discuss the applicability of the adaptive algorithm in cases where the Hamiltonian is non-smooth. Joint work with Jesper Karlsson, Stig Larsson, Anders Szepessy, and Raul Tempone.

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Giancarlo Sangalli Universita‘ di Pavia, IT Isogeometric elements for the Stokes problem Minisymposium Session ANMF: Monday, 16:00 - 16:30, CO1 In this work I discuss the application of IsoGeometric Analysis to incompressible viscous flow problems. We consider, as a prototype problem, the Stokes system and we propose various choices of compatible Spline spaces for the approximations to the velocity and pressure fields. The proposed choices can be viewed as extensions of the Taylor-Hood, Nédélec and Raviart-Thomas pairs of finite element spaces, respectively. We study the stability and convergence properties of each method and discuss the conservation properties of the discrete velocity field in each case. Joint work with Andrea Bressan, Annalisa Buffa, and Carlo De Falco.

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Dmitry Savostyanov University of Southampton, GB Alternating minimal energy methods for linear systems in higher dimensions. Part I: the framework and theory for SPD systems Contributed Session CT2.8: Tuesday, 14:30 - 15:00, CO123 We propose a new algorithm for the approximate solution of large-scale highdimensional tensor-structured linear systems. It can be applied to high-dimensional differential equations, which allow a low-parametric approximation of the multilevel matrix, right-hand side and solution in the tensor train format. We combine the Alternating Linear Scheme approach with the basis enrichment idea using Krylov–type vectors. We obtain the rank–adaptive algorithm with the theoretical convergence estimate not worse than the one of the steepest descent. The practically observed convergence is significantly faster, comparable or even better than the convergence of the DMRG–type algorithm. The complexity of the method is still at a level of ALS. The method is successfully applied for a high–dimensional problem of quantum chemistry, namely the NMR simulation of a large peptide. Keywords: high–dimensional problems, tensor train format, ALS, DMRG, steepest descent, convergence rate, superfast algorithms, NMR. Joint work with Sergey Dolgov.

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Robert Scheichl University of Bath, GB Hierarchical Multilevel Markov Chain Monte Carlo Methods and Applications to Uncertainty Quantification in Subsurface Flow Minisymposium Session UQPD: Wednesday, 11:30 - 12:00, CO1 In this talk we address the problem of the prohibitively large computational cost of Markov chain Monte Carlo (MCMC) methods for large–scale PDE applications with high dimensional parameter spaces. We propose a new multilevel version of a standard Metropolis-Hastings algorithm, and give an abstract, problem dependent theorem on the cost of the new multilevel estimator. The parameters appearing in PDE models of physical processes are often impossible to determine fully and are hence subject to uncertainty. It is of great importance to quantify the resulting uncertainty in the outcome of the simulation. A popular way to incorporate uncertainty is to model the input parameters in terms of random processes. Based on the information available, a probability distribution (the prior) is assigned to the input parameters. If in addition to this assumed distribution, we have some dynamic data (or observations) related to the model outputs, it is possible to condition on this data to reduce the overall uncertainty (the posterior). However, in most situations, this posterior distribution is intractable and exact sampling from it is unavailable. One way to circumvent this problem, is to generate samples using a Metropolis-Hastings type MCMC approach, which consists of two main steps: proposing a new sample, e.g. using a random walk from a previous sample, and then comparing the likelihood (i.e. the data fit) with that of the previous sample. The proposed sample gets accepted and used for inference, or rejected and a new sample is proposed. A major problem with this approach, e.g. in subsurface applications, is that each evaluation of the likelihood involves the numerical solution of a PDE with highly varying coefficients on a fine spatial grid. The likelihood has to be calculated also for samples that end up being rejected, and so the overall cost of the algorithm becomes extremely expensive. This is particularly true for high-dimensional parameter spaces, typical in realistic subsurface flow problems, where the acceptance rate of the algorithm can be very low. The key ingredient in our new multilevel MCMC algorithm is a two-level proposal distribution that ensures (as in the case of multilevel Monte Carlo based on i.i.d. samples) that we have a dramatic variance reduction on the finer levels of the multilevel estimator, leading to an overall variance reduction for the same computational cost, or conversely to a significantly lower computational cost for the same variance. For a typical model problem in subsurface flow with lognormal prior permeability, we then provide a detailed analysis of the assumptions in our abstract complexity theorem and show gains in the ε-cost of at least one whole order over the standard Metropolis-Hastings estimator. This requires a judicious "partitioning" of the prior space across the levels. One of the crucial theoretical observations is that on the finer levels the acceptance probability tends to 1 as the mesh is refined and as the dimension of the prior is increased. Numerical experiments confirming the analysis and demonstrating the effectiveness of the method are presented with consistent gains of up to a factor 50 in our tests. Joint work with C. Ketelsen, and A.L. Teckentrup.

339

Friedhelm Schieweck Department of Mathematics, University of Magdeburg, DE An efficient dG-method for transport dominated problems based on composite finite elements Minisymposium Session ANMF: Monday, 15:30 - 16:00, CO1 The discontinuous Galerkin (dG) method applied to transport dominated problems has the big advantage that it delivers with its upwind version for convective terms a parameter-free stabilization of higher order. On the other hand, it has compared to continuous finite element methods the disadvantages that it needs much more unknowns as well as much more couplings between the unknowns and that a static condensation for higher order elements is not possible. In this talk, we propose a modification of the underlying finite element space that keeps the advantage and removes the disadvantages of the usual dG-method. The idea is to use composite finite elements, i.e., for instance, quadrilateral or hexahedral elements each of which is composed of a fixed number of triangular or tetrahedral sub-elements. Then, the finite element space is constructed from functions whose restrictions to the sub-elements are polynomials of some maximal order k and which are continuous along the common faces of neighboured quadrilateral or hexahedral elements. Jumps are allowed only along common faces of sub-elements which are inside of the same composite element. Thus, the total number of unknowns is reduced essentially compared to the classical discontinuous finite element space. Moreover, all the degrees of freedom which are related to the interior of the composite elements can be removed from the global system of equations by means of static condensation. Finally, we can prove for the convection-diffusion-reaction equation that the usual (upwind) dG-method applied to such composite finite element space has the same good stability and convergence properties as for the classical discontinuous finite element space. For this model problem, we show some numerical examples in the convection dominated case and compare the classical with our new approach.

340

Claudia Schillings SAM, ETH Zuerich, CH Sparsity in Bayesian Inverse Problems Minisymposium Session UQPD: Thursday, 12:00 - 12:30, CO1 We present a novel, deterministic approach to inverse problems for identification of unknown, parametric coefficients in differential equations from noisy measurements. Based on new sparsity results on the density of the Bayesian posterior, we design, analyze and implement a class of adaptive, deterministic sparse tensor Smolyak quadrature schemes for the efficient numerical evaluation of expectations under the posterior. Convergence rates for the quadrature approximation are shown, both theoretically and computationally, to depend only on the sparsity class of the unknown and, in particular, are provably higher than those of Monte-Carlo (MC) and Markov-Chain Monte-Carlo methods. This work is supported by the European Research Council under FP7 Grant AdG247277. Joint work with Christoph Schwab.

341

Karin Schnass University of Sassari, IT Non-Asymptotic Dictionary Identification Results for the K-SVD Minimisation Principle Minisymposium Session ACDA: Monday, 12:10 - 12:40, CO122 In this presentation we give theoretical insights into the performance of K-SVD, a dictionary learning algorithm that has gained significant popularity in practical applications. The particular question studied is when a dictionary Φ ∈ Rd×K can be recovered as local minimum of the minimisation criterion underlying K-SVD from a set of N training signals yn = Φxn . A theoretical analysis of the problem leads to two types of identifiability results assuming the training signals are generated from a tight frame with coefficients drawn from a random symmetric distribution. First asymptotic results showing, that in expectation the generating dictionary can be recovered exactly as a local minimum of the K-SVD criterion if the coefficient distribution exhibits sufficient decay. This decay can be characterised by the coherence of the dictionary and the `1 -norm of the coefficients. Based on the asymptotic results it is further demonstrated that given a finite number of training samples N , such that N/ log N = O(K 3 d), except with probability O(N −Kd ) there is a local minimum of the K-SVD criterion within distance O(KN −1/4 ) to the generating dictionary.

342

Reinhold Schneider TU Berlin , DE Convergence of dynamical low rank approximation in hierarchical tensor formats Minisymposium Session LRTT: Tuesday, 11:00 - 11:30, CO3 In tensor product approximation, Hierarchical Tucker tensor format (Hackbusch) and Tensor Trains (TT) (Tyrtyshnikov) have been introduced recently offering stable and robust approximation by a low order cost . For many problems, which could not be handled so far, this approach has the potential to circumvent from the curse of dimensionality. For numerical computations, we cast the computation of an approximate solution into an optimization problems constraint by the restriction to tensors of prescribed ranks r. For approximation by elements from this highly nonlinear manifold , we apply the Dirac Frenkel variational principle by observing the differential geometric structure of the novel tensor formats. We analyse the (open) manifold of such tensors and its projection onto the tangent space, and investigate the convergence and possibly convergence rates in this framework. Literature: 1. C. Lubich, T. Rohwedder, R. Schneider and B. Vandereycken Dynamical approximation of hierarchical Tucker and tensor train tensors SPP1324 Preprint (126/2012) 2. B. Khoromskij, I. Oseledets and R. Schneider Efficient time-stepping scheme for dynamics on TT-manifolds, MIS Preprint 80/2011

343

Katharina Schratz INRIA and ENS Cachan Bretagne, FR Efficient numerical time integration of the Klein-Gordon equation in the nonrelativistic limit regime Minisymposium Session TIME: Thursday, 14:00 - 14:30, CO015 We consider the Klein-Gordon equation in the non-relativistic limit regime, i.e. the speed of light c formally tending to infinity. Due to the highly-oscillatory nature of the solution in this regime, its numerical simulation is very delicate. Here we will construct an asymptotic expansion for the exact solution in terms of the small parameter c−2 which allows us to filter out the highly-oscillatory phases. We will see that in the first approximation the numerical task reduces to the time integration of a system of non-linear c-independent Schrödinger equations. Thus, this approach allows us to construct numerical schemes that are robust with respect to the large parameter c producing high oscillations in the exact solution.

Joint work with Erwan Faou.

344

Mauricio Sepúlveda Universidad de Concepción, CL Convergent Finite Volume Schemes for Nonlocal and Cross Diffusion Reaction Equations. Applications to biology Contributed Session CT2.6: Tuesday, 14:00 - 14:30, CO017 In this work, we consider reaction-diffusion systems with nonlocal and cross diffusion. We construct a finite volume scheme for this system, we establish existence and uniqueness of the discrete solution, and it is also showed that the scheme converges to the corresponding weak solution for the model studied. The convergence proof is based on the use of the discrete Sobolev embedding inequalities with general boundary conditions and a space-time L1 compactness argument that mimics the compactness lemma due to S. N. Kruzhkov. The first example of application is the description of three interacting species in a HP food chain structure. The second example of application corresponds to a mathematical model with cross-diffusion for the indirect transmission between two spatially distributed host populations having non-coincident spatial domains, transmission occurring through a contaminated environment. We give also, several numerical examples.

References [1] M. Bendahmane and M. Sepúlveda, Convergence of a finite volume scheme for nonlocal reaction-diffusion systems modelling an epidemic disease. Discrete and Continuous Dynamical Systems - Series B. Vol. 11, 4 (2009) 823-853. [2] V. Anaya, M. Bendahmane and M. Sepúlveda, Mathematical and numerical analysis for reaction-diffusion systems modeling the spread of early tumors. Boletin de la Sociedad Espanola de Matematica Aplicada. Vol 47, (2009), 55-62. [3] V. Anaya, M. Bendahmane and M. Sepúlveda, A numerical analysis of a reaction-diffusion system modelling the dynamics of growth tumors. Mathematical Models and Methods in Applied Sciences. Vol. 20, 5 (2010) 731-756. [4] V. Anaya, M. Bendahmane and M. Sepúlveda, Mathematical and numerical analysis for predator-prey system in a polluted environment. Networks and Heterogeneous Media. Vol. 5, 4 (2010) 813-847. [5] V. Anaya, M. Bendahmane and M. Sepúlveda, Numerical analysis for HP food chain system with nonlocal and cross diffusion. Submitted. Prepublicación 2011-11, DIM, Universidad de Concepción. Joint work with Verónica Anaya, and Mostafa Bendahmane.

345

Alexander Shapeev University of Minnesota, US Atomistic-to-Continuum coupling for crystals: analysis and construction Minisymposium Session MSMA: Monday, 11:10 - 11:40, CO3 Atomistic-to-continuum (AtC) coupling is a popular approach of utilizing an atomistic resolution near the defect core while using the continuum model to resolve the elastic far-field. In my talk I will (1) give a brief introduction to AtC coupling, (2) present one of the recent developments in construction of a consistent energybased AtC coupling method, (3) and discuss the complexity of computations.

346

Natasha Sharma University of Heidelberg, DE Convergence Analysis of an Adaptive Interior Penalty Discontinuous Galerkin Method for the Helmholtz Problem Minisymposium Session MMHD: Thursday, 15:00 - 15:30, CO017 We consider the numerical solution of the 2D Helmholtz equation by an adaptive Interior Penalty Discontinuous Galerkin method based on adaptively refined simplicial triangulations of the computational domain. The a posteriori error analysis involves a residual type error estimator consisting of element and edge residuals and a consistency error which, however, can be controlled by the estimator. The refinement is taken care of by the standard bulk criterion (Dörfler marking) known from the convergence analysis of adaptive finite element methods for linear second order elliptic PDEs. The main result is a contraction property for a weighted sum of the energy norm of the error and the estimator which yields convergence of the adaptive IPDG approach. Numerical results are given that illustrate the performance of the method. Joint work with Ronald H.W. Hoppe.

347

Zhiqiang Sheng Institute of Applied Physics and Computational Mathematics, CN The nonlinear finite volume scheme preserving maximum principle for diffusion equations on polygonal meshes Minisymposium Session SDIFF: Monday, 12:10 - 12:40, CO123 We further develop the nonlinear finite volume scheme of diffusion equation on polygonal meshes, and construct a nonlinear finite volume scheme which satisfies the discrete maximum principle. Our scheme is locally conservative and has only cell-centered unknowns. Numerical results are presented to show how our scheme works for preserving discrete maximum principle and positivity on various distorted meshes. Joint work with Guangwei yuan.

348

Corina Simian University of Zurich, CH Conforming and Nonconforming Intrinsic Discretization for Elliptic Partial Differential Equations Contributed Session CT4.6: Friday, 09:20 - 09:50, CO017 The aim of this presentation is to introduce a general method for the construction of intrinsic conforming and non-conforming finite element spaces. As a model problem we consider the Poisson equation, however this approach can be applied for the discretization of more general elliptic equations. We will derive piecewise polynomial intrinsic conforming and non-conforming finite element spaces and local basis functions for these spaces. In the conforming case our method leads to a finite element space spanned by the gradients of the well known hp-finite elements. In the non-conforming case we employ the stability and convergence theory for non-conforming finite elements based on the second Strang Lemma and derive, from these principles, weak compatibility conditions for non-conforming finite elements across the boundary, for domains Ω ⊂ Rd , d ∈ {2, 3}. For d = 2 our space contains all gradients of hp-finite element basis functions enriched by some edge-type non-conforming basis functions for even polynomial degree and by some triangle-type non-conforming basis functions for odd polynomial degree. Joint work with Stefan Sauter.

349

Valeria Simoncini Universita’ di Bologna, IT Solving Ill-posed Linear Systems with GMRES Minisymposium Session CTNL: Wednesday, 11:30 - 12:00, CO015 Almost singular linear systems arise in discrete ill-posed problems. Either because of the intrinsic structure of the problem or because of preconditioning, the spectrum of the coefficient matrix is often characterized by a sizable gap between a large group of numerically zero eigenvalues and the rest of the spectrum. Correspondingly, the right-hand side has leading eigencomponents associated with the eigenvalues away from zero. In this talk the effect of this setting in the convergence of the Generalized Minimal RESidual (GMRES) method is considered. It is shown that in the initial phase of the iterative algorithm, the residual components corresponding to the large eigenvalues are reduced in norm, and these can be monitored without extra computation. The analysis is supported by numerical experiments on singularly preconditioned ill-posed Cauchy problems for partial differential equations with variable coefficients. Joint work with Lars Eldén, Linköping University, Sweden.

350

Jonathan Skowera ETH Zurich, CH Entanglement via algebriac geometry Minisymposium Session LRTT: Monday, 11:40 - 12:10, CO1 Tensor rank generalizes matrix rank and admits a formulation in algebraic geometry in terms of secant varieties of Segre embeddings. Entangled quantum states appear as states of tensor rank greater than one. States related by stochastic local operations and classical communication fall into the same entanglement class. A unitary group action preserves tensor rank while orbits of a semisimple group action correspond to entanglement classes. Entanglement classes are furthermore characterized by polytopes arising as images under symplectic moment maps. We review these notions in elementary terms and discuss connections.

351

Iain Smears University of Oxford, GB Discontinuous Galerkin finite element approximation of HJB equations with Cordès coefficients Minisymposium Session NMFN: Monday, 12:40 - 13:10, CO2 Hamilton–Jacobi–Bellman (HJB) equations are fully nonlinear second order PDE that arise in the study of optimal control of stochastic processes. For problems with Cordès coefficients, we present an hp-version discontinuous Galerkin FEM that is consistent, stable and high-order, with convergence rates that are optimal with respect to mesh size, and suboptimal in the polynomial degree by only half an order. The scheme is obtained by coupling the residual of the numerical solution to discrete analogues of identities that are central to the analysis of the continuous problem. Numerical experiments illustrate the accuracy and computational efficiency of the scheme, with particular emphasis on problems with strongly anisotropic diffusions. I. S MEARS AND E. S ÜLI, Discontinuous Galerkin finite element approximation of Hamilton–Jacobi–Bellman equations with Cordès coefficients, Tech. Report NA 13/03, Univ. of Oxford, 2013. In Review. http://eprints.maths.ox.ac.uk/1671/ I. S MEARS AND E. S ÜLI, Discontinuous Galerkin finite element approximation of non-divergence form elliptic equations with Cordès coefficients, Tech. Report NA 12/17, Univ. of Oxford, 2012. In Review. http://eprints.maths.ox.ac.uk/1623/ Joint work with Endre Suli.

352

Kathrin Smetana Massachusetts Institute of Technology, US The Hierarchical Model Reduction-Reduced Basis approach for nonlinear PDEs Minisymposium Session SMAP: Monday, 16:00 - 16:30, CO015 Many phenomena in fluid dynamics have dominant spatial directions along which the essential dynamics occur. Nevertheless, the processes in the transverse directions are often too relevant for the whole problem to be neglected. For such situations we present a new problem adapted version of the hierarchical model reduction approach. The hierarchical model reduction approach (see [3] and references therein) uses a truncated tensor product decomposition of the solution and hierarchically reduces the full problem to a small lower dimensional system in the dominant directions, coupled by the transverse dynamics. In previous approaches [3] these transverse dynamics are approximated by a reduction space constructed from a priori chosen basis functions such as trigonometric or Legendre polynomials. We present the hierarchical model reduction-reduced basis approach [2] where the reduction space is constructed a posteriori from solutions (snapshots) of appropriate reduced parametrized problems in the transverse directions. To get an efficient lower-dimensional approximation also for nonlinear PDEs we introduce for the approximation of the nonlinear operator the adaptive Empirical Projection Method which employs the Empirical Interpolation Method [1]. An a posteriori error estimator which includes both the errors caused by the model reduction and the approximation of the nonlinear operator is presented. Numerical experiments demonstrate that the hierarchical model reduction-reduced basis approach converges exponentially fast with respect to the model order for problems with smooth solutions but also for some test cases where the source term belongs to C 0 (Ω) only. Run-time experiments verify a linear scaling of the proposed method in the number of degrees of freedom used for the computations in the dominant direction.

References [1] M. Barrault, Y. Maday, N. C. Nguyen, and A. T. Patera, An ’empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations, C. R. Math. Acad. Sci. Paris Series I, 339 (2004), pp. 667–672. [2] M. Ohlberger and K. Smetana, A new problem adapted hierarchical model reduction technique based on reduced basis methods and dimensional splitting, Preprint FB 10, University Munster num. 03/10 (2010). [3] S. Perotto, A. Ern, and A. Veneziani, Hierarchical local model reduction for elliptic problems: a domain decomposition approach, Multiscale Model. Simul. 8 (2010), pp. 1102–1127. Joint work with Mario Ohlberger.

353

Alexandra Smirnova Georgia State Unniversity, US A Novel Stopping Criterion for Iterative Regularization with Undetermined Reverse Connection Minisymposium Session STOP: Thursday, 14:30 - 15:00, CO1 In our talk, we address a problem of minimizing a (non)linear functional, Φ(u) := ||F (u) − y||2 ,

F : DF ⊂ X → Y,

on a pair of Hilbert spaces. We investigate the generalized Gauss-Newton scheme [B09, BS10a, BS10b] un+1 = ξn − θ(F 0∗ (un )F 0 (un ), τn )F 0∗ (un ){F (un ) − yδ − F 0 (un )(un − ξn )}, u0 , ξn ∈ DF ⊂ X ,

(1)

and consider three basic groups of generating functions. In this framework, we present a nonstandard approximation of pseudoinverse through a "gentle" iterative truncation, and prove its optimality on the class of generating functions with the same correctness coefficient. The convergence analysis of (1) in the noise-free case has been carried out in [B09], an a priori and a posteriori stopping rules have been justified in [BS10a] and [BS10b], respectively. In [B09, BS10a, BS10b], the modified source condition 1 , (2) 2 which depends on the current iteration point un , is proposed. We call condition (2) the undetermined reverse connection. It has been shown in [B09, BS10a, BS10b] that even though (2) still contains the unknown solution u ˆ, the norm of ωn in (2) is greater than the norm of ω in the case of the source condition with ξ fixed. Moreover, the norm of ωn can even tend to infinity as n → ∞. Specifically, at every step of iterative process (1), the element ξn may be such that u ˆ − ξn = (F 0∗ (un )F 0 (un ))p ωn ,

||wn || ≤

ε , τnk

1 ≤ p − k, 2

p≥

ε ≥ 0.

(3)

The main disadvantage of undetermined reverse connection (2) is the need to find ξn satisfying (2) in each step of the iteration. How can such a ξn be found in practice? Well, the problem is similar to the one with single ξ: no general recipe is known and we just hope to get lucky after trying different ξ’s. In case of (2), one can argue that the set of potential candidates for the test function is larger due to (3). Still, with n source conditions in place of one, it is unlikely that (2) will hold at every step "by chance". Therefore in this talk, we look into the convergence analysis of algorithm (1) under a more realistic "noisy" source condition: u ˆ − ξn = (F 0∗ (un )F 0 (un ))p ωn + ζn , ε ≥ 0,

p≥

1 , 2

||ωn || ≤

||ζn || ≤ ∆.

ε , τnk

1 ≤ p − k, 2 (4)

To that end, we introduce a novel a posteriori stopping rule. Let N = N (δ, ∆, yδ ) be the number of the first transition of ||F (un ) − yδ || through the level σnµ , 21 ≤ µ < 1, i.e., µ ||F (xN (δ,∆,yδ ) ) − yδ || ≤ σN

and σnµ < ||F (un ) − yδ ||, 354

0 ≤ n < N (δ, ∆, yδ ), (5)

where

√ C3 ∆ τn , σn := δ + C1

and ||y − yδ || ≤ δ.

(6)

The constant ∆ in (6) is usually harder to estimate than the noise level δ. However for the asymptotic behavior of the approximate solution u = uN (δ,∆,yδ ) as δ and √ ∆ tend to zero, this is not relevant. It follows from (6) that due to the factor τn , the contribution of ∆ to the total error of the model approaches zero as n → ∞. In other words, error in the source condition disappears in the overall noise as we iterate. Notice that if F 0∗ (·)F 0 (·) is compact and the null space of F 0∗ (·)F 0 (·) is {0}, then the range of F 0∗ (·)F 0 (·) is dense in X , so in any neighborhood of un there are points ξn for which (5) holds with ∆ = 0. On the other hand, since the range of F 0∗ (·)F 0 (·) is not closed, in the same neighborhood there are also points ξn for which (5) holds with ∆ 6= 0. In practice, one can try different ξn ’s and choose those for which the iterative scheme works better, that is convergence is more rapid and the algorithm is more stable. To illustrate the practical aspects related to the converge result, numerical simulations for a large-scale image de-blurring system are presented.

References [B09] Bakushinsky, A. B. [2009] Iterative methods with fuzzy feedback for solving irregular operator equations, Dokl. Russian Acad. Sci. 428 N5, 1-3. [BS10a] Bakushinsky, A. B. and Smirnova, A. [2010] Irregular operator equations by iterative methods with undetermined reverse connection, Journal of Inverse and Ill-Posed Problems, 18 N2, 147–165. [BS10b] Bakushinsky, A. B. and Smirnova, A. [2010] Discrepancy Principal for Generalized GN Iterations Combined with the Reverse Connection Control. Inverse and Ill-Posed Problems, 18, N4, 421-432. Joint work with A. Bakushinsky and Hui Liu.

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Benjamin Stamm Laboratoire J.-L. Lions, Paris 6 and CNRS, FR Domain decomposition for implicit solvation models Minisymposium Session MSMA: Monday, 12:40 - 13:10, CO3 In this talk, we present a domain decomposition algorithms for implicit solvent models that are widely used in computational chemistry. We show that, in the framework of the COSMO model, with van der Waals molecular cavities and classical charge distributions, the electrostatic energy contribution to the solvation energy, usually computed by solving an integral equation on the whole surface of the molecular cavity, can be computed more efficiently by using an integral equation formulation of Schwarz’s domain decomposition method for boundary value problems. In addition, the so-obtained potential energy surface is smooth, which is a critical property to perform geometry optimization and molecular dynamics simulations. We present the methodology, set up the mathematical foundations of the approach, and present a numerical study of the accuracies and convergence rates of the resulting algorithm. If time permitting, we present the applicability of the method to large molecular systems of biological interest and illustrate that computing times and memory requirements scale linearly with respect to the number of atoms. Joint work with E. Cancès, F. Lipparini, Y. Maday, and B. Mennucci.

356

Simeon Steinig University of Stuttgart, DE Convergence Analysis and A Posteriori Error Estimation for State-Constrained Optimal Control Problems Minisymposium Session FEPD: Monday, 16:00 - 16:30, CO017 In our talk we present a convergence result for finite element discretisations on non-quasi-uniform meshes of optimal control problems with constraints involving the state or the gradient of the state. In a second step we present an a posteriori error estimator involving the actually computed discrete solutions to a regularised problem that provides an upper bound for the error up to constants. Joint work with Prof. A. Roesch, and Prof. K.G. Siebert.

357

Rolf Stenberg Department of Mathematics and Systems Analysis, FI Mixed Finite Element Methods for Elasticity Plenary Session: Wednesday, 08:20 - 09:10, Rolex Learning Center Auditorium During the last decade the theory of mixed finite element methods have been recast with the aid of differential geometry. This was first done for methods for scalar second order elliptic equation, e.g. the Raviart-Thomas-Nedelec and Brezzi-Douglas-Marini-Duran-Fortin families. Lately, the theory has been extended to methods for linear elasticity. Both methods with a symmetric approximation for the stress tensor (Fraijs de Veubeke, Watwood-Hartz, Johnson-Mercier, Arnold-Douglas-Gupta,. . . ), and methods where the symmetry is imposed weakly (Fraijs de Veubeke, Arnold-Brezzi-Douglas, Stenberg, Arnold-Falk-Winther,. . . ), have been analyzed. The purpose of this talk is to highlight an alternative and more elementary way of analysis, which, nevertheless, gives optimal error estimates. The approach is that of using mesh dependent norms, first used by Babuska, Osborn and Pitkäranta in 1980. In this, the norm used for the “stress” variable is the L2 norm, which has the physical meaning of energy. For the “displacement” variable the broken H 1 norm (now well-known from Discontinuous Galerkin Methods) is used. The stability of the methods follows directly from local scaling arguments. The second ingredient is the so-called “equilibrium condition”, which the methods fulfil. Using these, the quasi-optimal error estimate for the stress follows by the classical saddle point theory. For the displacement the analysis yields a superconvergence result for the distance between the L2 projection onto the discrete space and the finite element solution. This is utilized to postprocess the displacement yielding an approximation of two polynomial degrees higher, with an optimal convergence rate. This postprocessing turns out to be crucial in a posteriori estimates. Based on the hypercircle idea of Prager and Synge one obtains a posteriori estimates with explicitly computable constants.

358

Christian Stohrer Department of Mathematics and Computer Science, University of Basel, CH Micro-Scales and Long-Time Effects: FE Heterogeneous Multiscale Method for the Wave Equation Contributed Session CT2.2: Tuesday, 15:30 - 16:00, CO2 For limited time the propagation of waves in a highly oscillatory medium is welldescribed by the non-dispersive homogenized wave equation. With increasing time, however, the true solution deviates from the classical homogenization limit, as a large secondary wave train develops unexpectedly. In [1] a finite element heterogeneous multiscale method (FE-HMM) was proposed and convergence to the homogenized solution was shown. However, neither the homogenized solution, nor the FE-HMM of [1] capture these dispersive effects. We propose a new FE-HMM which is able to recover not only the short-scale macroscopic behavior of the wave field, but also those secondary long-time dispersive effects. Effective dispersive equation Let Ω ⊂ Rd be a domain and T > 0. We consider the wave equation ( ∂tt uε − ∇ · (aε ∇uε ) = F in Ω × (0, T ), uε (x, 0) = f (x), ∂t uε (x, 0) = g(x) in Ω

where aε (x) ∈ (L∞ (Ω))d×d is highly oscillatory and where we suppose that aε is symmetric, uniformly elliptic and bounded. Various formal asymptotic arguments suggest, that the linearized improved Boussinesq equation may serve as an effective dispersive equation which describes well the long-time macroscopic behavior of wave propagation, e.g. [3]. Moreover, for d = 1 and ε-periodic aε it was shown in [4] that uε converges to the solution of ∂tt (ueff − ε2 b∂xx ueff ) − a0 ∂xx ueff = F

(1)

for T ∈ O(ε−2 ). Here a0 denotes the homogenized coefficient from classical homogenization theory and b > 0. The weak formulation of this dispersive effective equation motivates the design of an FE-HMM, where not only an effective bilinear but in addition an effective inner product is used. Multiscale Algorithm We now give a brief description of our new FE-HMM scheme, more details are given in [2]. First, we generate a macro triangulation TH and choose an appropriate macro FE space S(Ω, TH ). By macro we mean that H   is allowed. The FE-HMM solution uH is given by the following problem: Find uH : [0, T ] → S(Ω, TH ) such that ( (∂tt uH , vH )Q + BH (uH , vH ) = (F, vH ) for all vH ∈ S(Ω, TH ), (2) uH (0) = fH , ∂t uH (0) = gH in Ω, where the initial data fH and gh are approximations of f and g in S(Ω, TH ). The bilinear form BH is the standard FE-HMM bilinear as in [1], but the effective inner product (·, ·)Q is novel. It consists of two parts: The first part corresponds to a standard approximation of the L2 -inner product by numerical quadrature, whereas the second part is a correction, needed to capture the long-time dispersive effects. It can be shown, that this correction is of order O(ε2 ), which is in 359

good correspondence with (1). The computation of BH and (·, ·)Q relies on the numerical solution of micro problems in sampling domains. Since BH is elliptic and bounded and (·, ·)Q is a true inner product, the FE-HMM is well-defined for all H > 0. √ The usefulness of the method can be seen in Figure 1. We set aε (x) = 2 + sin(2πx/ε) with ε = 1/50 and computed a reference solution by fully resolving the micro structure. The new FE-HMM succeeds in capturing, the long-time effects. In contrast, the solution of the FE-HMM of [1] is unable to capture those dispersive effects.

References [1] A. Abdulle and M. J. Grote, Finite Element Heterogeneous Multiscale Method for the Wave Equation, Multiscale Model. Simul., 9 (2011), pp. 766–7921. [2] A. Abdulle, M. J. Grote and C. Stohrer, Finite Element Heterogeneous Multiscale Method for the Wave Equation: Long-Time Effects, in prep. [3] J. Fish, W. Chen and G. Nagai, Non-local dispersive model for wave propagation in heterogeneous media. Part 1: one-dimensional case, Int. J. Numer. Meth. Eng., 54 (2002), pp. 331–346. [4] A. Lamacz, Dispersive Effective Models for Waves in Heterogeneous Media, Math. Models Methods Appl. Sci., 21 (2011), pp. 1871–1899.

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Figure 1: Reference solution (ref.), FE-HMM from [1] and new FE-HMM. Joint work with Assyr Abdulle, and Marcus J. Grote.

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Martin Stoll MPI Magdeburg, DE Fast solvers for Allen-Cahn and Cahn-Hilliard problems Minisymposium Session CTNL: Tuesday, 12:00 - 12:30, CO015 We consider the efficient solution of an Allen-Cahn variational inequality subject to volume constraints as well as a Cahn-Hilliard variational inequality both obtained from the gradient flow of a Ginzburg-Landau energy. Using an implicit time discretization this is formulated as an optimal control problem with pointwise constraints. For both problems we employ a non-smooth potential, which in turn requires the solution of the optimal control problem via a semi-smooth Newton method. This method then requires the efficient solution of large structured linear systems. For realistic problems the matrix size easily becomes intractable for direct methods and we propose the use of preconditioned Krylov subspace methods. Our goal is to present preconditioners that are tailored towards both the Allen-Cahn and the Cahn-Hilliard equations and show robust performance with respect to the crucial parameters such as mesh-size or value of the regularization parameter. Numerical results illustrate the competitiveness of this approach. Joint work with Luise Blank, Lavinia Sarbu, Jessica Bosch, and Peter Benner.

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Zdenek Strakos Faculty of Mathematics and Physics, Charles University in Prague, CZ Remarks on algebraic computations within numerical solution of partial differential equations Minisymposium Session CTNL: Wednesday, 12:00 - 12:30, CO015 Numerical solution of partial differential equations (PDE) starts with a finite dimensional approximation of the mathematical model. This is typically done (e.g. in the finite element method) using some spatial meshes over the given domain and by some form of time discretization. The unknown functions are then approximated as linear combinations of a finite number of basis functions, which leads to a finite dimensional representation of the original model. As the mesh refines, the state-of-the-art paradigm investigates convergence of the finite dimensional solution to the solution of the original model. Proving such convergence often requires fine mathematical techniques. Here a priori error analysis indicates how the error (asymptotically) decreases as the mesh is refined. Resulting bounds are not computable because they typically involve the unknown solution of the problem. A posteriori error analysis estimates the size of the actual error of the computed solution, and it can provide a tool for stopping the computations when the sufficient accuracy is reached. The decisive criterion should be accuracy to which the results of computation reflect the properties of the genuine (analytical) solution of the given PDE. The numerical solution process represents in case of difficult problems a challenge. Despite the fact that PDE discretisation, a priori and a posteriori error analysis and algebraic (matrix) computations represent well-established fields, there are important issues which are under investigation. They should not be studied separately within the particular fields. Modeling with its mathematical analysis together with discretisation, error estimation and solving the resulting finite dimensional discrete problems should be considered closely related tasks of a single solution process. A failure in a subtask may not be identifiable within the same subtask. It may show up later in the form of difficulties in numerical computations and/or in interpretation of the obtained numerical approximations. The fact that the state-of-the-art results may offer only partial answers can be documented on the approach to proving convergence of the discrete approximate solution when the mesh refines using some form of adaptation. The proofs are based on seeing individual mesh refinement steps (adaptation cycles) as contractions for some error estimators, where the contraction parameter is independent of the refinement step. This seemingly allows reaching an arbitrary prescribed accuracy in a finite number of contraction steps. In practical computations, however, an arbitrary accuracy can not be reached simply due to the fact that the discretized algebraic problem cannot be solved exactly, and the restriction on a maximal attainable accuracy can be for difficult problems significant. In practical computations we may not aim at highly accurate numerical solutions of the discretized problems since that could make the whole solution process unfeasible. The principal questions which may occur is therefore what is the maximal attainable accuracy of numerical computations, whether the prescribed user-specified accuracy can be reached and at which price. Construction of efficient numerical algorithms requires for challenging problems a global communication between the information obtained at different (possibly distant) parts of the solution domain. This can be achieved via incorporating

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coarse space components (e.g. using domain decomposition or multigrid methods). Efficient preconditioning can be seen as another tool for achieving the same goal. Preconditioning should reflect the physical nature of the problem expressed in the mathematical model. It can be motivated using a functional analytic operator description (operator preconditioning). Finally, construction of fully computable a posteriori error estimators which allows for the local error control and comparison of the size of the error from different sources (discretisation, linearization, inexact algebraic computation) is a prerequisite for reliable, robust and efficient adaptive approaches. This requires combination of rather diverse techniques from functional analysis through numerical analysis to analysis of iterative matrix computations including effects of rounding errors. This contribution will review some approaches to the questions mentioned above. It will use a combination of the function spaces and algebraic settings, and it will illustrate recent progress as well as difficulties which need to be resolved.

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Hiroaki Sumitomo Keio University, JP GPU accelerated Symplectic Integrator in FEA for solid continuum Contributed Session CT1.6: Monday, 18:30 - 19:00, CO017 GPU accelerated Symplectic Integrator in FEA for solid continuum is proposed. Time integration methods such as Newmark-beta method are widely used to evaluate the dynamic response of solid continuum in FEA. A set of simultaneous equations should be solved for every time step in these time integration methods. This results in high computational cost. PDS (Particle Discretized Scheme)-FEM[1] could be a solution to avoid this intensive computation. It applies particle discretization to a displacement field; the domain is decomposed into a set of Voronoi blocks and the non-overlapping characteristic functions for the Voronoi blocks are used to discretize the displacement function. Each block is connected to adjacent blocks by springs in this discretized field. Spring constants are equivalent to the corresponding components of the stiffness matrix of general FEM. Thus, PDS-FEM enables us to compute a deformation of solid continuum using particle simulation, instead of solving the simultaneous equations. Therefore, time integration required for analysis of deformable solid continuum can be handled in the framework of analytical dynamics. Symplectic Integrator is a numerical integration scheme for particle simulation and is widely used in discrete element method and molecular dynamics. The Hamiltonian is conserved in Symplectic Integrator. The interaction between 2 particles is computed by matrix operation. A particle is affected only by the adjacent particles in FEA for solid continuum. Thus, matrix operation is represented as SpMV (Sparse Matrix Vector product). SpMV spends most of the computational time in Krylov subspace solvers and numerous researches for GPU algorithm have been proposed[2]. In this study, GPU accelerated Symplectic Integrator was implemented focusing on the acceleration of SpMV to reduce the computational time. The main points are shown below: 1. The performance of SpMV was improved compared with the conventional method. We focused on the form of matrix in the 3D problem to decrease the amount of transferring data between device memory and cores. 2. All calculations including SpMV, vector update, and applying boundary conditions were implemented only on the GPU. No communication between host and device memory via a PCI-Express is necessary. 3. Approximately 90 % efficiency was achieved when 3 GPUs were used. Communication was overlapped with computation in SpMV and the total amount of communication data was reduced by using domain decomposition method. The performance of the Symplectic Integrator using 3 GPUs improved up to 121× acceleration compared with Intel Xeon CPU.

References [1] M. Hori, K. Oguni and H. Sakaguchi: Proposal of FEM implemented with particle discretization for analysis of failure phenomena, Journal of the Mechanics and Physics of Solids, 53, 3, 681-703, (2006). 364

[2] F.Vazquez, J.J.Fernandez, E.M. Garzon: A new approach for sparse matrix vector product on NVIDIA GPUs, Concurrency Computat.: Pract. Exper. Vol.23, 815-826, (2010). Joint work with Kenji Oguni.

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Anders Szepessy Royal Institute of Technology, SE How accurate is molecular dynamics for crossings of potential surfaces? Part I: Error estimates Contributed Session CT4.9: Friday, 08:20 - 08:50, CO124 The difference of the value of observables for the time-independent Schrödinger equation, with matrix valued potentials, and the values of observables for ab initio Born-Oppenheimer molecular dynamics, of the ground state, depends on the probability to be in excited states. In this talk I will present a method to determine the probability to be in excited states from Landau-Zener like dynamic transition probabilities, based on Ehrenfest molecular dynamics and stability analysis of a perturbed eigenvalue problem. A perturbation pE , in the dynamic transition probability for a time-dependent Schrödinger WKB-transport equation, yields through 1/2 resonances a larger probability of the order O(pE ) to be in an excited state for the time-independent Schrödinger equation, in the presence of crossing or nearly crossing electron potential surfaces. The stability analysis uses Egorov’s theorem 1/2 and shows that the approximation error for observables is O(M −γ/2 + pE ) for large nuclei-electron mass ratio M , provided the molecular dynamics has an ergodic limit which can be approximated with time averages over the period T and convergence rate O(T −γ ), for some γ > 0. Numerical simulations verify that the transition probability pE can be determined from Ehrenfest molecular dynamics simulations. Joint work with Håkon Hoel (KAUST), Ashraful Kadir (KTH), Petr Plechac (Univ. Delaware), and Mattias Sandberg (KTH).

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Lorenzo Tamellini Ecole Polytechnique Fédérale de Lausanne / Politecnico di Milano (Italy), CH Quasi-optimal polynomial approximations for elliptic PDEs with stochastic coefficients Contributed Session CT3.7: Thursday, 18:00 - 18:30, CO122 Partial differential equations with stochastic coefficients conveniently model problems in which the data of a given PDE (coefficients, forcing terms, boundary conditions) are affected by uncertainty, due e.g. to measurement errors, limited data availability or intrinsic variability of the described system. In this talk we consider a particular case that arises in a number of different engineering fields, i.e. the case of an elliptic PDE with diffusion coefficient depending on N random variables y1 , . . . , yN . In this context, the solution u of the PDE at hand can be seen as a random function, u = u(y1 , . . . , yN ), and common goals include computing its mean and variance, or the probability that it exceeds a given threshold; such analysis is usually referred to as “Uncertainty Quantification”. This could be achieved with a straightforward Monte Carlo method, that may however be very demanding in terms of computational costs. Methods based on polynomial approximations of u(y1 , . . . , yN ) have thus been introduced, aiming at exploiting the possible degree of regularity of u with respect to y1 , . . . , yN to alleviate the computational burden. Such polynomial approximations can be obtained e.g. with Galerkin projections or collocation methods over the parameters space. Although effective for problems with a moderately low number of random parameters, these methods suffer from a degradation of their performance as the number of random parameters increase (“curse of dimensionality”). Minimizing the impact of the “curse of dimensionality” is therefore a key point for the application of polynomial methods to high-dimensional problems. In this talk we will explore possible strategies to determine efficient polynomial approximations of u with given computational cost (the so-called “best M terms” approximation of u). In particular, we will consider a “knapsack approach”, in which we estimate the cost and the “error reduction” contribution of each possible component of the polynomial approximation, and then we choose the components with the highest “error reduction”/cost ratio. The estimates of the “error reduction” are obtained by a mixed “a-priori”/“a-posteriori” approach, in which we first derive a theoretical bound and then tune it with some inexpensive auxiliary computations. We will present theoretical convergence results obtained for some specific problems as well as numerical results showing the efficiency of the proposed approach. Extension to the case where N → ∞ will also be discussed.

References [1] J. Beck, F. Nobile, L. Tamellini, R. Tempone, “On the optimal polynomial approximation of stochastic PDEs by Galerkin and collocation methods”, Math. Mod. Methods Appl. Sci. (M3AS), 22, 2012. [2] J. Beck, F. Nobile, L. Tamellini, R. Tempone, “Convergence of quasi-optimal Stochastic Galerkin Methods for a class of PDEs with random coefficients”, to appear in Comput. Math. Appl. Also available as MATHICSE Technical report 24/2012, Ecole Politechnique Fédérale Lausanne - Switzerland.

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[3] J. Beck, F. Nobile, L. Tamellini, R. Tempone, “A quasi-optimal sparse grids procedure for groundwater flows”, to appear in Proceedings of the International Conference on Spectral and High-Order Methods 2012 (ICOSAHOM’12), Lecture Notes in Computational Science and Engineering, Springer, 2012. Also available as MATHICSE Technical report 46/2012, Ecole Politechnique Fédérale Lausanne - Switzerland. [4] L. Tamellini, “Polynomial approximation of PDEs with stochastic coefficients”, Ph.D. thesis, Politecnico di Milano - Italy. Joint work with J. Beck, R. Tempone, and F. Nobile.

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Mattia Tani Università di Bologna, IT CG methods in non-standard inner product for saddle-point algebraic linear systems with indefinite preconditioning Contributed Session CT4.2: Friday, 09:20 - 09:50, CO2 Developing a good solver for saddle-point algebraic linear systems is often a challenging task, due to indefiniteness and poor spectral properties of the coefficient matrix. In the past few years, the employment of indefinite preconditioners leading to systems which are symmetric (and sometimes even positive definite) in a non-standard inner product has drawn significant attention. In its basics, the method works as follows: given the linear system Ax = b, let P be a preconditioner and D be a symmetric and positive definite matrix such that the preconditioned system is symmetric in the inner product defined by D, that is, DP −1 A = (P −1 A)T D. If, in addition, DP −1 A is positive definite, then the Conjugate Gradients method in the D−inner product can be employed on the preconditioned system, and the rate of convergence of the method, measured in the error DP −1 A−norm, only depends on the (all) real eigenvalues of P −1 A. The aim of this presentation is twofold. Firstly, we report on some advances in the spectral estimates of one of the preconditioned matrix known in literature [1]. Secondly, we explore the sometimes overlooked relation between the non-standard minimized norm of the error and the Euclidean one. Particular emphasis is given to the case when D is close to singular.

References [1] M. Tani and V. Simoncini, Refined spectral estimates for preconditioned saddle point linear systems in a non-standard inner product pp. 1-12, November 2012 Joint work with Valeria Simoncini.

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Raul Tempone MATHEMATICS, KAUST, SA Numerical Approximation of the Acoustic and Elastic Wave Equations with Stochastic Coefficients Minisymposium Session UQPD: Thursday, 10:30 - 11:00, CO1 Partial Differential Equations with stochastic coefficients are a suitable tool to describe systems whose parameters are not completely determined, either because of measurement errors or intrinsic lack of knowledge on the system. In the case of linear elliptic PDEs with random inputs, an effective strategy to approximate the state variables and their statistical moments is to use polynomial based approximations like Stochastic Galerkin or Stochastic Collocation method. These approximations exploit the high regularity of the state variables with respect to the input random parameters and for a moderate number of input parameters, are remarkably more effective than classical sampling methods. However, the performance of polynomial approximations deteriorates as the number of input random variables increases, an effect known as the curse of dimensionality. To address this issue, we proposed strategies to construct optimal polynomial spaces and related generalized sparse grids. In this talk we focus instead on the second order wave equation with a random wave speed and a related generalization to elastodynamics, presenting our recent results from [1] and [2]. Here, the propagation speed is piecewise smooth in the physical space and depends on a finite number of random variables. In particular, we show that, unlike for elliptic and parabolic problems, the solution to hyperbolic problems is not in general analytic with respect to the input random variables. Therefore, the rate of convergence for a Stochastic Collocation method may in principle only be algebraic. We show that faster convergence rates are still possible for some quantities of interest and for the wave solution with particular types of data. These theoretical results agree with our numerical examples. References: [1] A Stochastic Collocation Method fo the Second Order Wave Equation with a Discontinuous Random Speed, by M. Motamed, F. Nobile and R. Tempone. Numerische Mathematik, Volume 123, Issue 3, pp. 493-536, 2013. [2] Analysis and computation of the elastic wave equation with random coefficients, by M. Motamed, F. Nobile, R. Tempone, 2012. Joint work with Mohammad Motamed, and Fabio Nobile.

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Jan ten Thije Boonkkamp Eindhoven University of Technology, NL Harmonic complete flux schemes for conservation laws with discontinuous coefficients Contributed Session CT2.6: Tuesday, 14:30 - 15:00, CO017 The complete flux scheme is a discretization method for conservation laws of advection-diffusion-reaction type. Basically, the numerical flux is determined from the solution of a local boundary value problem for the entire equation, including the source term. Consequently, the integral representation of the flux contains a homogeneous and inhomogeneous part, corresponding to the advection-diffusion operator and the source term, respectively. Suitable quadrature rules give the numerical flux. We distinguish complete flux schemes for scalar equations and systems of equations. In the latter case, the coupling between the constituen equations is taken into account in the discretization. In this talk we consider conservation laws where the diffusion coefficient/matrix is a discontinuous function of the space coordinate. From its integral representation, we show that the scalar numerical flux at an interface can be considered as the constant-coefficient complete flux with the diffusion coefficient replaced by the harmonic average of the diffusion coefficients in the adjacent grid points. Likewise, for systems of equations, we obtain a similar expression for the numerical flux vector at an interface, containing the (matrix) harmonic average of the diffusion matrices in the adjacent grid points. We collectively refer to these schemes as harmonic complete flux schemes. The harmonic complete flux schemes turn out to be more accurate than the standard complete flux schemes. We will demonstrate the performance of the schemes for several test problems. Joint work with L. Liu, and J. van Dijk.

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Francesco Tesei École Polytechnique Fédérale de Lausanne, CH Multi Level Monte Carlo methods with Control Variate for elliptic SPDEs Contributed Session CT4.8: Friday, 08:20 - 08:50, CO123 We consider the numerical approximation of a partial differential equation (PDE) with random coefficients. Nowadays such problems can be found in many applications in which the lack of available measurements make an accurate reconstruction of the coefficients involved in the mathematical model unfeasible. In particular we focus on a model problem given by an elliptic partial differential equation in which the randomness is given by the diffusion coefficient, modeled as a random field with limited spatial regularity. This approach is inspired by the groundwater flow problem which has a great importance in hydrology: in this context the diffusion coefficient is given by the permeability of the subsoil and it is often modeled as a lognormal random field. Several models have been proposed in the literature leading to realizations having varying spatial smoothness for the covariance functions. In particular, a widely used covariance model is the exponential one that has realizations with Hölder continuity C 0,α with α < 12 . Models with low spatial smoothness pose great numerical challenges. The first step of their numerical approximation consists in building a series expansion of the input coefficient; we use here a Fourier expansion; whenever the random field has low regularity, such expansions converge very slowly and this makes the use of deterministic methods such as Stochastic Collocation on sparse grids highly problematic since it is not possible to parametrize the problem with a relatively small number of random variables without a significant loss of accuracy. A natural choice is to try to solve such problems with a Monte Carlo type method. On the other hand it is well known that the convergence rate of the standard Monte Carlo method is quite slow, making it impractical to obtain an accurate solution since the associated computational cost is given by the number of samples of the random field multiplied by the cost needed to solve a single deterministic PDE which require a very fine mesh due to the roughness of the coefficient. Multilevel Monte Carlo methods have already been proposed in the literature in order to reduce the variance of the Monte Carlo estimator, and consequently reduce the number of solves on the fine grid. In this work we propose to use a multilevel Monte Carlo approach combined with an additional control variate variance reduction technique on each level. The control variate is obtained as the solution of the PDE with a regularized version of the lognormal random field as input random data and its mean can be successfully computed with a Stochastic Collocation method on each level. The solutions of this regularized problem turn out to be highly positively correlated with the solutions of the original problem. Within this Monte Carlo framework the choice of a suitable regularized version of the input random field is the key element of this method; we propose to regularize the random field by convolving the log-permeability with a Gaussian kernel. We analyze the mean square error of the estimator and the overall complexity of the algorithm. We also propose possible choices of the regularization parameter and of the number of samples per grid so as to equilibrate the space discretization error, the statistical error and the error in the computation of the expected value of the control variate by Stochastic Collocation. Numerical examples demonstrate the effectiveness of the method. A comparison with the standard Multi Level Monte

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Carlo method is also presented. Joint work with Fabio Nobile, Raul Tempone, and Erik von Schwerin.

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Benjamin Tews University of Kiel, DE Optimal control of incompressible two-phase flows Minisymposium Session ANMF: Monday, 15:00 - 15:30, CO1 We consider an optimal control problem of two incompressible and immiscible Newtonain fluids. The motion of the interface between these two fluids can be captured by a phase field models or level set method. Both methods are subject to this talk. The state equation includes surface tension and is discretized by a discontinuous Galerkin scheme in time and a continuous Galerkin scheme in space. In order to resolve the interface propagation we also apply adaptive finite elements in space and time. We derive first order optimality conditions including the adjoint equation, which is also formulated in a strong sense. The optimality system on the discrete level is solved by Newton’s method. In the numerical examples we compare level sets with a phase field model. Joint work with Malte Braack.

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Münevver Tezer-Sezgin Middle East Technical University, TR DRBEM Solution of Full MHD and Temperature Equations in a Lid-driven Cavity Contributed Session CT1.3: Monday, 17:30 - 18:00, CO3 This study proposes the dual reciprocity boundary element method (DRBEM) solution for full magnetohydrodynamics (MHD) equations coupled with the heat transfer in a lid-driven square cavity by means of the Boussinessq approximation. The two-dimensional, unsteady, laminar, incompressible MHD flow and energy equations are given in terms of non-dimensional stream function ψ, temperature T , induced magnetic field components Bx , By , and vorticity w as ∇2 ψ = −w ∂T ∂T 1 ∂T ∇2 T = +u +v P rRe ∂t ∂x ∂y ∂B ∂B ∂Bx ∂u ∂u 1 x x ∇2 B x = +u +v − Bx − By Rem ∂t ∂x ∂y ∂x ∂y 1 ∂By ∂By ∂By ∂v ∂v ∇ 2 By = +u +v − Bx − By Rem ∂t ∂x ∂y ∂x ∂y 1 2 ∂w ∂w ∂w Ra ∂T ∇ w= +u +v − Re ∂t ∂x ∂y P rRe2 ∂x      2 ∂Bx ∂Bx Ha ∂ ∂By ∂ ∂By Bx − + By − − ReRem ∂x ∂x ∂y ∂y ∂x ∂y

(1) (2) (3) (4) (5)

where u = ∂ψ/∂y, v = −∂ψ/∂x and w = ∂v/∂x − ∂u/∂y. The bottom wall of the unit square cavity is the cold wall Tc = −0.5, and the top wall is hot, Th = 0.5. The top lid moves with a velocity u = 1 and the no-slip condition is imposed on the other walls. Externally applied magnetic field with an intensity B0 = (0, 1) is in +y-direction. In the DRBEM procedure, the right hand sides of equations (1)-(5) are approximated by using radial basis functions f = 1 + r + . . . + rn which are related to Laplacian with particular solution u ˆ as ∇2 u ˆ = f . Thus, fundamental solution of ∗ Laplace equation (u = ln r/2π) is made use of for both sides of equations (1)-(5). Discretization of the boundary of the cavity by using N linear boundary elements and taking arbitrarily required L interior points, systems of equations   ˆ − GQ ˆ F −1 b, Hϕ − Gϕq = H U (6) are obtained, where H and G are BEM matrices containing the boundary integrals of u∗ and q ∗ = ∂u∗ /∂n evaluated at the boundary nodes, respectively. The vectors ϕ and ϕq = ∂ϕ/∂n represent the known and unknown information of ψ, T, Bx , By ˆ and Q ˆ are constructed from u or w at the nodes. U ˆj and then qˆj = ∂ u ˆj /∂n columnwise, and are matrices of size (N + L) × (N + L). The vector b represents collocated values of right hand sides of equations (1)-(5), and F is the (N + L) × (N + L) coordinate matrix containing radial basis functions fj ’s as columns evaluated at N + L points. All the space derivatives are calculated by using coordinate matrix F , and the time derivatives are discretized using BackwardEuler formula. f = 1 + r, N = 160, and L = 1521 are used with 16−point Gaussian integration in the construction of H and G matrices. Computations are carried for Prandtl 375

number P r = 0.1. Unknown boundary conditions for vorticity are extracted from its definition by using coordinate matrix F . The results are depicted with respect to varying physical parameters such as Reynolds (Re), magnetic Reynolds (Rem), Hartmann (Ha) and Rayleigh (Ra) numbers. The increase in Re causes to emerge the new cells at the bottom corners of the cavity, and the convective heat transfer is developed. Heat transfer passes to the conductive mode due to the decrease in buoyancy, and streamlines are divided into counter rotating cells inside the cavity as Ra increases. The circulation in induced magnetic field lines with the increase in Rem shows the dominance of convection terms in the induction equations. The well known MHD characteristics with the increase in Ha which are the flattening tendency in the velocity, and the suppression on the convective heat transfer are well observed (Figure 1). In the figure, the visualized contours are streamlines, isotherms, vorticity and induced magnetic field lines from left to right, and the increase in Ha, as Ha = 10 and Ha = 100 from top to bottom. As Ha increases, boundary layer formation starts in the flow and vorticity concentrates completely on the moving lid. Induced magnetic field weakens due to the dominance of external magnetic field applied in +y direction. DRBEM is an efficient and computationally boundary only numerical scheme in solving the MHD heat transfer problem in a lid-driven cavity.

Figure 1: Re = 400, Rem = 40, Ra = 1000 Joint work with Bengisen Pekmen.

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Mechthild Thalhammer University of Innsbruck , AT Multi-revolution composition methods for time-dependent Schrödinger equations Minisymposium Session ASHO: Tuesday, 11:00 - 11:30, CO123 The error behaviour of the recently introduced multi-revolution composition methods is analysed for a class of highly oscillatory evolution equations posed in Banach spaces. The scope of applications in particular includes time-dependent linear Schrödinger equations, where the realisation of the composition approach is based on time-splitting pseudo-spectral methods. The theoretical error bounds for the resulting space and time discretisations are confirmed by numerical examples. Joint work with Philippe Chartier and Florian Mehats.

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Lutz Tobiska Otto von Guericke University, DE On stability properties of different variants of local projection type stabilizations Minisymposium Session ANMF: Monday, 14:30 - 15:00, CO1 The local projection stabilization (LPS) is one way to stabilize standard Galerkin finite element methods for solving convection-dominated convection-diffusion equations. In recent years, different variants have been developed and analyzed, e.g., the one-level LPS, the two-level LPS, the LPS with exponential enrichments, the LPS with overlapping projection spaces. In the talk we will discuss the different stabilization properties and compare them with the popular streamline diffusion method (SDFEM).

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Lutz Tobiska Otto von Guericke University, DE Influence of surfactants on the dynamics of droplets Minisymposium Session GEOP: Wednesday, 10:30 - 11:00, CO122 We propose a finite element method for studying the influence of surfactants on the dynamic of droplets. The mathematical model for a free surface flow with surfactants consists of the Navier-Stokes equation and the surfactant concentration equation in the bulk coupled with the transport equation on the evolving free surface [1,2]. In the proposed finite element scheme, the free surface is tracked by an arbitrary Lagrangian-Eulerian (ALE) approach, and the coupled partial differential equations are spatially discretized by finite elements. This approach can be extended to consider two-phase flows. We prefer discontinuous pressure approximations to suppress spurious velocities and to get a better local mass conservation. However, the use of fully discontinuous pressure approximations leads to too many additional degrees of freedom to satisfy the Babuška-Brezzi-condition between the spaces approximating velocity and pressure. Therefore, a relaxed discontinuous pressure approximation is used for which the pressure in each phase is continuous. Numerical experiments for a growing droplet below a capillary, for a rising bubble, and for Taylor flows [3] will be presented. References [1] S. Ganesan, L. Tobiska, Arbitrary Lagrangian-Eulerian finite element method for computation of two-phase flows with soluble surfactants. J. Comp. Physics 231(2012), 3685–3702 [2] S. Ganesan, A. Hahn, K. Held, L. Tobiska, An accurate numerical method for computation of two-phase flows with surfactants. European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012), J. Eberhardsteiner et. al. (eds.), Vienna, Austria, September 10-14, CD-ROM, ISBN:9783-9502481-9-7 [3] S.Aland, S. Boden, A. Hahn, F. Klingbeil, M. Weismann, S. Weller, Quantitative comparision of Taylor Flow simulations based on sharp- and diffuse-interface models. Int. J. Numer. Methods in Fluids (submitted) Joint work with S. Ganesan, A. Hahn, and K. Held.

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Rony Touma Lebanese American University, LB Central finite volume schemes on nonuniform grids and applications Contributed Session CT2.6: Tuesday, 15:30 - 16:00, CO017 In this work we develop a new unstaggered central scheme on nonuniform grids for the numerical solution of general hyperbolic systems of conservation laws in one space dimension. Many problems arising in physics and engineering sciences can be formulated mathematically using hyperbolic systems of conservation laws or, in the case of systems with a source term, hyperbolic systems of balance laws. Such problems occur for example in aerodynamics, magnetohydrodynamics (MHD), hydrodynamics and many more. Central schemes are particularly attractive for solving hyperbolic systems as they avoid the resolution of the Riemann problems arising at the cell interfaces, thanks to a layer of staggered cells. Central schemes first appeared with the staggered version of Lax-Friedrichs’ scheme, where a piecewise constant numerical solution was alternatingly evolved on two staggered grids. The resulting scheme is first-order accurate with a stability number of 0.5. In 1990 Nessyahu and Tadmor (NT) [2] presented a predictor-corrector type, second-order accurate scheme that is an extension of the Lax-Friedrichs scheme [1] which evolves a piecewise linear numerical solution on two staggered grids. The NT scheme uses a first-degree Taylor expansion in time to determine the numerical solution at the intermediate time; furthermore slopes limiting reduces spurious oscillations in the vicinity of discontinuities. In this work we propose a new one-dimensional unstaggered central scheme on nonuniform grids for the numerical solution of homogeneous hyperbolic systems of conservation laws of the form ( ut + f (u)x = 0, (1) u(x, t = 0) = u0 (x). where u(x, t) = (u1 , u2 , · · · , up ) is the unknown p−components vector and f (u) is the flux vector. System (1) is assumed to be hyperbolic, i.e., the Jacobian matrix ∂f /∂u has p real eigenvalues and p linearly independent eigenvectors. We discretize the computational domain [a, b] using n subintervals centered at the nodes xk with different lengths ∆xk for k = 1, · · · , n. The proposed scheme evolves a piecewise linear numerical solution Li (x, t) defined at the cell centers xi of the control cells Ci = [xi−1/2 , xi+1/2 ] of a nonuniform grid, and avoids the resolution of the Riemann problems arising at the cell interfaces, thanks to a layer of staggered cells used implicitly. The evolved piecewise linear interpolant is defined by: u(x, tn ) ≈ Li (x, tn ) = uni + (x − xi )(uni )0 , ∀x ∈ Ci ,

(2)

∂ u(x, tn )|x=xi approximates the slope to first-order accuracy. Spuwhere (uni )0 ≈ ∂x rious oscillations are avoided using a slopes limiting procedure. The developed scheme is then validated and used to solve classical problems arising in gas dynamics and in hydrodynamics. The obtained numerical results are in perfect agreement with corresponding ones appearing in the recent literature, thus confirming the efficiency and potential of the proposed method to handle two-phase gas solid flows problems.

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References [1] P.D. Lax, Weak solutions of nonlinear hyperbolic equation and their numerical computation, Comm. Pure and Applied Math.7, (1954), 159-193 [2] H. Nessyahu and E. Tadmor, Non-oscillatory central differencing for hyperbolic conservation laws, J. Comp. Phys., 87, 2, (1990), 408-463.

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Paolo Tricerri CMCS-EPFL - CEMAT-IST, CH Fluid-Structure Interaction simulation of cerebral aneurysm using anisotropic model for the arterial wall Minisymposium Session NFSI: Thursday, 15:00 - 15:30, CO122 Brain aneurysms are abnormal dilatations of the cerebral arterial wall originated from a localized weakening of the arterial tissue. It is estimated that million people around the world are affected by cerebral aneurysm for which the incidence of patient death or serious morbidity following aneurysm rupture justifies the increasing attention that this disease is receiving, similarly to other cardiovascular disorders. The coupling of the arterial tissue models (i.e. the structure) with blood flow models (i.e. the fluid) together with the simulations of the coupled Fluid-Structure Interaction (FSI) system aims at providing a better understanding of the physiological phenomena and, moreover, it can provide a flexible, reliable, and noninvasive predictive tool for medical decisions. Different aspects of this illness have been addressed trying to correlate the onset or the evolution of cerebral aneurysms to specific heamodynamics or morphological conditions [1]. In literature, many studies have analyzed the influence on the numerical results of the coupled FSI problem for different modeling choices (ranging from the blood flow model [6] to parametric studies on the boundary conditions applied on the external wall of the aneurysm to simulate the surrounding tissues [7]). However, only limited investigations have been focused on the discussion and choice of the arterial wall model [4, 5]. Indeed, typically the arterial wall is described as an isotropic material even though the mechanical response of the tissue is strongly anisotropic as experimentally observed. This work aims at the numerical simulation of the coupled FSI system in the case of cerebral aneurysms when considering anistropic models for the arterial wall. More precisely, the arterial tissue will be described by an anisotropic constitutive law [2, 3] in order to model the highly nonlinear and anisotropic mechanical response of the tissue. The blood flow will be described by the Navier-Stokes equations and the coupled FSI problem will be solved using a monolithic approach. Physiological boundary conditions will be applied at the inlet and outlet of the fluid domain in order to properly describe the blood flows [7]. Idealized geometries that mimic anatomically realistic geometries of cerebral aneurysms are considered. Indeed, when idealized geometries of aneurysms are used, the spatial distribution of the collagen fibers can be analytically prescribed and the containment effect on the deformation field due to the collagen fibers can be analyzed. We will investigate the role of the structural models by analyzing the spatial distribution of quantities of interests that are typically associated with the development of cerebral aneurysms (e.g. wall shear stress, wall shear stress gradient, wall stresses, flow impingement). Keywords: arterial tissue structural models, fluid-structure interaction, numerical simulations, anisotropic constitutive law.

References [1] Sforza D., Putman C.M., Cebral J.R., Haemodynamics of cerebral aneurysms, Annual Reviews of Fluid Mechanics, 41 (2009), 91-107.

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[2] Dalong L., Robertson A.M., A structural multi-mechanism constitutive equation for cerebral arterial tissue, International Journal of Solids and Structures 46 (2009), 2920-2928. [3] Holzapfel G.A., Ogden R.W., Constitutive modelling of arteries. Proceedings of the Royal Society A, 2010 466:1551-1597. [4] Torii R., Oshima M., Kobayashi T., Takagi K., Tezduyar T.E., FluidStructure Interaction modelling of a patient-specific cerebral aneurysm: influence of structural modelling, Computational Mechanics, 43 (2008), 151-159. [5] Torii R., Oshima M., Kobayashi T., Takagi K., Tezduyar T.E., Influence of wall thickness on fluid-structure interaction computations of cerebral aneurysms, International Journal for Numerical Methods in Biomedical Engineering, 26 (2010), 336-347. [6] Cebral J.R., Mut F., Weir J., Putman C.M., Association of haemodynamic characteristics and cerebral aneurysm rupture, American Journal of Neuroradiology, 32 (2011), 264-270. [7] Malossi A.C.I., Partitioned solution of geometrical multiscale problems for the cardiovascular system: models, algorithms, and applications, PhD Thesis, École Polytechnique Fédérale de Lausanne, Switzerland, 2012. Joint work with Luca Dedè, Adélia Sequeira, and Alfio Quarteroni.

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Julie Tryoen INRIA Bordeaux Sud-Ouest, FR A semi-intrusive stochastic inverse method for uncertainty characterization and propagation in hyperbolic problems Contributed Session CT2.7: Tuesday, 15:00 - 15:30, CO122 Let U ≡ U (x, t, D) be the solution of a forward hyperbolic model M depending on space, time and uncertain data vector D = {D1 , . . . , DN } ∈ Ξ ∈ RN . Relying on nobs observations of the solution {y 1 , . . . , y nobs } ∈ (Rm )nobs corresponding to measurement points (x1 , t1 ), . . . , (xnobs , tnobs ), our interest is to build a probability description of D which can be used in an efficient way for uncertainty propagation. To this end, we rely on a bayesian setting, which provides rigorous tools to solve such problems, namely stochastic inverse ones, taking into account measurement and/or model uncertainty [Kaipo and Somersalo 2010]. The difference between the predicted solution and the observed one is supposed to be described by the following additive relation y k = U (xk , tk , D) + ek ,

k = 1, . . . , nobs ,

(1)

where the measurement/model error ek ∈ Rm is a realization of a random vector with probability distribution pe (commonly taken as multinormal). Let ppr be a prior probability distribution for D, non-informative in the case of very limited prior knowledge on D. Supposing independent measurements and applying Bayes’ theorem, posterior probability distribution for D follows : ppost (D) ∝ ppr (D)

n obs Y k=1

pe (y k − U (xk , tk , D)).

(2)

To avoid complex numerical integrations, Monte Carlo Markov chains (MCMC) are used to generate iterative samples that behave asymptotically as ppost [Gilks et al. 1996]. From a MCMC sample, one can then estimate moments of D, marginal distributions of its components from density kernel estimations, or posterior confidence intervals. We would like to propagate uncertainty on D obtained from bayesian inference into the solution U of the forward model. The classic approach described above supplies samples of posterior distribution of D, from which uncertainty can be propagated by a classic Monte Carlo approach. Despite its robustness, this method presents a very low convergence rate in the computation of statistics of U with respect to the number of realizations. Recently, promising methods have been proposed to deal with uncertainty propagation in hyperbolic problems where stochastic discontinuities can appear in finite time [Lin et al. 2006, Lin et al. 2008, Poette et al. 2009, Tryoen et al. 2010], relying on a stochastic spectral representation of the output [Ghanem and Spanos 2003]. A semi-intrusive method has also been introduced by Abgrall and Congedo [2013], to propagate input data uncertainty of any probability distribution into hyperbolic models. The purpose of this study is to couple the latter approach with the bayesian framework ; to this end, the object to infer is no more the input data vector D, but a description via their conditional expectancies on a partition of the stochastic domain Ω = ∪P j=1 Ωj , where P is the number of stochastic elements. More precisely, we supposed for the time being the components of independent, and we rebuild the probability distributions of R D as −1 E(Di |Ωj ) = Ωj FD (ω)dω, for i = 1, . . . , N and j = 1, . . . , P , where FDi is the i 384

cumulative distribution function of Di . The methodology is assessed on a quasi 1D Euler test case, with subsonic boundary conditions and an uncertainty on the output pressure.

References [Kaipo and Somersalo 2010] Kaipo, J. and Somersalo, E., “Statistical and Computational Inverse Problems”, Applied Mathematical Sciences, Vol. 160, Springer, 2010. [Gilks et al. 1996] Gilks, W., Richardson, S., and Spiegelhalter, D., “Markov Chain Monte Carlo in Practice”, Chapman & Hall, 1996. [Lin et al. 2006] Lin, G., Su, C.-H., and Karniadakis, G., “Predicting shock dynamics in the presence of uncertainties”, J. Comput. Phys. 217, no. 1, p. 260–276, 2006. [Lin et al. 2008] Lin, G., Su, C.-H., and Karniadakis, G., “Stochastic modeling of random roughness in shock scattering problems : theory and simulations”, Comput. Methods Appl. Mech. Engrg. 197, no. 43-44, p. 3420–3434, 2008. [Poette et al. 2009] Poette, G., Després, B., and Lucor, D., “Uncertainty quantification for systems of conservation laws”, J. Comput. Phys. 228, no. 7, p. 2443–2467, 2009. [Tryoen et al. 2010] Tryoen, J., Le Maître, O., Ndjinga M., and Ern, A., “Intrusive Galerkin methods with upwinding for uncertain nonlinear hyperbolic systems”, J. Comput. Phys. 229, no. 18, p. 6485–6511, 2010. [Ghanem and Spanos 2003] Ghanem, R. and Spanos, P., “Stochastic Finite Elements : A Spectral Approach”, Dover, 2nd edition, 2003. [Abgrall and Congedo 2013] Abgrall, R. and Congedo, P.M., “A semi-intrusive deterministic approach to uncertainty quantification in non-linear fluid flow problems”, J. Comput. Phys. 235, p. 828–845, 2013. Joint work with P.M. Congedo, and R. Abgrall.

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Stefan Turek TU Dortmund, Applied Mathematics and Numerics, DE 3D Level Set FEM techniques for (non-Newtonian) multiphase flow problems with application to pneumatic extension nozzles and micro-encapsulation Minisymposium Session FREE: Tuesday, 11:30 - 12:00, CO2 Multiphase flow problems are very important in many industrial applications, and their accurate, robust and efficient numerical simulation is object of numerous research and simulation projects since many years. Particularly in the case of pneumatic extension nozzles which are often used for the generation of droplets the accurate description of the interaction between the dispersed liquid phase and the surrounding gas phase is essential, especially, if uniform droplet sizes are required. In this work implementation details of the Level Set approach into the 3D parallel FEM based open source software package FeatFlow will be shown, whereas special emphasis will be placed on the surface tension effects and the interface reconstruction which on the one hand guarantees the exact identification of the interface and on the other hand offers the advantages to exploit the underlying multilevel structures to perform an efficient, octree fashioned reinitialization of the Level Set field. Validation of the corresponding 3D code is to be presented with respect to numerical test cases and experimental data. The corresponding applications involve the classical rising bubble problem for various parameters and the generation of droplets from a viscous liquid jet in a coflowing surrounding fluid. Moreover, numerical simulations involving different regulation strategies are to be presented in order to reveal the possibilities of regulation of the underlying droplet generation process in terms of the resulting monodisperse droplet sizes by means of periodic flow rate modulations of the dispersed phase. Preliminary results of additional extensions to the developed 3D multiphase flow solver such as non-Newtonian (shear-dependent) rheological models and the Fictitious Boundary Method (FBM) based particulate flow module are to be presented in the context of particle encapsulation processes. Joint work with Otto Mierka.

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Eugene Tyrtyshnikov Institute of Numerical Mathematics of Russian Academy of Sciences, RU Tensor decompositions in the drug design optimization problems Minisymposium Session LRTT: Tuesday, 11:30 - 12:00, CO3 Tensor decompositions, especially Tensor Train (TT) and Hierarchical Tucker (HT), are fastly becoming useful and widely used computational instruments in numerical analysis and numerous applications. As soon as the input vectors are presented in the TT (HT) format, basic algebraic operations can be efficiently implemented in the same format, at least in a good lot of practical problems. A crucial thing is, however, to acquire the input vectors in this format. In many cases this can be accomplished via the TT-CROSS algorithm, which is a far-reaching extension of the matrix cross interpolation algorithms. We discuss some properties of the TT-CROSS that allow us to adopt it for the needs of solving a global optimization problem. After that, we present a new global optimization method based on special transformations of the scoring functional and TT decompositions of multi-index arrays of values of the scoring functional. We show how this new method works in the direct docking problem, which is a problem of accommodating a ligand molecule into a larger target protein so that the interaction energy is minimized. The degrees of freedom in this problem amount to several tens. Most popular techniques are genetic algorithms, Monte Carlo and molecular dynamic approach. We have found that the new method can be up to one hundred times faster on typical protein-ligand complexes. Joint work with Dmitry Zheltkov.

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André Uschmajew EPFL, ANCHP, CH On asymtotic complexity of hierarchical Tucker approximation in L2 Sobolev classes Minisymposium Session LRTT: Monday, 12:10 - 12:40, CO1 In this talk we would like to bring to attention the asymptic convergence rate of hierachical Tucker approximations of functions from unit balls in certain periodic Sobolev classes in L2 with respect to the hierarchical rank. In particular, we consider the isotropic spaces and the spaces of bounded mixed derivatives. The (almost) exact rates can be determined straightfowardly from the quasi-optimality of the high-order SVD approximation and known results on bilinear approximation rates. These latter results are due to Temlyakov. Based on the convergence rate, the asymptotic complexity to achieve accuracy  can be estimated. When d ≥ 3, the storage complexity is dominated by the storage cost of the transfer tensors. This is different from the case d = 2 recently discussed by Griebel and Harbrecht, where the storage of the HOSVD bases in the leaves is most expensive. In any case, for functions with dominated mixed smoothness the estimates are worse than using sparse grids, which however has to be expected. Joint work with Reinhold Schneider.

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Kristoffer Van der Zee EINDHOVEN University of Technology, NL Adaptive Modeling for Partitioned-Domain Concurrent Continuum Models Minisymposium Session SMAP: Monday, 15:30 - 16:00, CO015 In this contribution adaptive modeling strategies are considered for the control of modeling errors in so-called partitioned-domain concurrent multiscale models. In these models, the exact fine model is considered intractable to solve throughout the entire domain. It is therefore replaced by an approximate multiscale model where the fine model is only solved in a small subdomain, and a coarse model is employed in the remainder. We review two approaches to adaptively improve the approximate model in a general framework assuming that the fine and coarse model are described by (local) continuum models separated by a sharp interface; see [1]. In the classical approach [2] an a posteriori error estimate is computed, and the model is improved in those regions with the largest contributions to this estimate. In the recent shape-derivative approach [3] the interface between the fine and coarse model is perturbed so as to decrease a shape functional associated with the error. Several numerical experiments illustrate the strategies. [1] K.G. van der Zee, S. Prudhomme and J.T. Oden, Adaptive modeling for partitioned-domain multiscale continuum models: A posteriori estimates and shapederivative strategies, submitted [2] J. T. Oden and S. Prudhomme. Estimation of modeling error in computational mechanics. J. Comput. Phys., 182:496–515, 2002. [3] H. Ben Dhia, L. Chamoin, J. T. Oden, and S. Prudhomme. A new adaptive modeling strategy based on optimal control for atomic-to-continuum coupling simulations. Comput. Methods Appl. Mech. Engrg., 200:2675–2696, 2011. Joint work with Kristoffer G. van der Zee, Serge Prudhomme, and J. Tinsley Oden.

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Pierre Vandergheynst EPFL, CH Compressive Source Separation: an efficient model for large scale multichannel data processing Minisymposium Session ACDA: Monday, 16:00 - 16:30, CO122 Hyperspectral imaging (HSI) systems produce large amounts of data and efficient compression is therefore crucial in their design. However current HSI compression techniques, such as those based on 3D wavelets, rely on computationally costly encoding algorithms that are challenging, indeed often impossible, to implement on embedded sensor systems. Recently, Compressive Sensing (CS) has provided an efficient alternative to traditional transform coding, allowing the use of very simple encoders and moving the computational burden to the decoder. The efficiency of CS crucially depends on well-designed sparse signal models as well as provably correct decoding algorithms. While the literature describes several applications of CS to HSI using direct extensions of 2D image sparse models, few works attempt to exploit the strong joint spatial and spectral correlations typical to HSI. We propose and analyze a new model based on the assumption that the whole hyperspectral signal is composed of a linear combination of few sources, each of which has a specific spectral signature, and that the spatial abundance maps of these sources are themselves piecewise smooth and therefore efficiently encoded via typical sparse models. We derive new sampling schemes exploiting this assumption and give theoretical lower bounds on the number of measurements required to reconstruct the HSI and recover its source model parameters. This allows us to segment HSI data into their source abundance maps directly from compressed measurements. We also propose efficient optimization algorithms and perform extensive experimentation on synthetic and real datasets, which reveals that our approach can be used to encode HSI with far less measurements and computational effort than traditional CS methods. Finally we illustrate how our model can be used for various other applications, for instance molecular spectroscopy or functional brain data processing.

Figure 1: Abundance maps estimated from measuring only 3 percent of the total data in a hyperspectral imaging application. Joint work with Mohammad Golbabaee.

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Nick Vannieuwenhoven KU Leuven, BE Parallel tensor-vector multiplication using blocking Contributed Session CT3.3: Thursday, 18:00 - 18:30, CO3 Computing the product of a dense order-d tensor with d − 1 vectors, i.e., vd = (v1 , v2 , · · · , vd−1 , I)T · A, is the key or most costly operation in several tensor decomposition algorithms; it is vital for computing an orthogonal Tucker decomposition (Tucker, 1966) using the tensor-Krylov method (Savas and Eldén, 2013); for computing the CANDECOMP / PARAFAC (CP) decomposition (Carroll and Chang, 1970; Harschman, 1970) using alternating least squares (ALS) algorithms; and for computing the largest tensor singular value (Chang, Qi, and Zhou, 2010). As we push the boundaries to tackle larger problems still, limiting memory consumption and exploiting parallelism becomes unavoidable. In this presentation, we develop a memory-efficient parallel implementation of the tensor-vector product well-suited to shared-memory architectures. Our approach is founded on two crucial observations: first, explicitly computing unfoldings, or matricizations, of the input tensor can be avoided completely; and, second, subdividing the tensor into subtensors and casting the familiar computations into block-form enables data-level parallelism. We illustrate the performance of the proposed method through two key algorithms: the ALS algorithm for computing a CP decomposition and the tensorKrylov method for computing an orthogonal Tucker decomposition. The code was implemented in C++ using Eigen and Intel Threading Building Blocks, and our preliminary experiments indicate excellent sequential and good parallel performance. We also illustrate how the techniques covered in this presentation can be utilized to improve the performance of the Tensor Toolbox v2.5’s ttv(T,v,-k) routine by roughly one order of magnitude. Joint work with N. Vanbaelen, K. Meerbergen, and R. Vandebril.

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Maria Varygina Institute of Computational Modeling SB RAS, RU Numerical Modeling of Elastic Waves Propagation in Block Media with Thin Interlayers Contributed Session CT1.9: Monday, 18:00 - 18:30, CO124 Several nature materials such as rock have distinct structurally inhomogeneous block-hierarchical structure. Block structure appears on different scale levels from the size of crystal grains to the blocks of rock. Blocks are connected to each other with thin interlayers of rock with significantly weaker mechanical properties. Analysis of experimental data of wave propagation in layered media shows that the interlayers behave non-elastically even under small wave amplitudes. The models of interlayer material of various levels of complexity taking into account natural dissipation processes in interlayers based on the rheological method are built [1]. The numerical solution is based on the two-cyclic space-variable decomposition method in combination with monotone grid-characteristic schemes with balanced time steps in layers and interlayers. The scheme in the layer does not possess an artificial energy dissipation and significantly reduces the effect of smoothing the numerical solution peaks with corresponding refinement of the obtained results. Parallel algorithms are implemented as complex of programs for supercomputers with graphics processing units with CUDA technology (Compute Unified Device Architecture). The characteristics of wave propagation processes in layered and block media related to the structural inhomogeneity in rocks were studied. The calculations of planar waves induced by short and long Λ− and Π− impulses on the boundary of layered medium and the Lamb problem of instant concentrated load on a surface of half space in planar case were performed. Fig. 1 and Fig. 2 show the dependencies of the velocity vector on spatial coordinate in a problem of Λ−impulse load. The impulse with a unit amplitude was induced on the left boundary of computational domain, the right boundary was fixed. The impulse duration equal to the time that elastic wave passes through two and a half layers. The numerical results demonstrate a qualitative difference between the wave pattern in layered medium as compared to a homogeneous medium. This difference at the initial stage is revealed in the appearance of waves reflected from the interlayers, i.e. the characteristic oscillations behind the loading wave front as it passes through the interlayer. Eventually stationary wave pattern appears after multiple reflections behind the head wave front, i.e. the so-called pendulum wave expermintaly discovered in [2, 3]. Fourier analysis of the displacement of layers seismograms allows to identify the characteristic frequency of the pendulum wave due to the compliances of interlayers and their thickness. This work was supported by the Russian Foundation for Basic Research (grant no. 11–01–00053) and the Complex Fundamental Research Program no. 18 "Algorithms and Software for Computational Systems of Superhigh Productivity" of the Presidium of RAS.

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References [1] Varygina M.P., Pohabova M.A., Sadovskaya O.V., Sadovskii V.M. Numerical algorithms for the analysis of elastic waves in block media with thin interlayers // Numerical methods and programming. – 2011. – T. 12. (In Russian) [2] Kurlenya M.V., Oparin V.N., Vostrikov V.I. On generation of elastic wave packet under impulse load in block media. Pendulum waves // Reports of the Academy of sciences USSR. 1993. T. 333 No. 4. PP. 3-13. (In Russian) [3] Aleksandrova N.I., Sher E.N., Chernikova A.G. The influence of viscosity of interlayers on the propagation of low frequent pendulum waves in block hierarchical media // Physical and technical problems of developments of mineral resources. 2008. No. 3. PP. 3-13. (In Russian)

Figure 1: Velocity behind front wave of incident wave induced by Λ-impulse in layered medium

Figure 2: Velocity behind front wave of reflected wave induced by Λ-impulse in layered medium

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Yuri Vassilevski Institute of Numerical Mathematics, Russian Academy of Sciences, RU A numerical approach to Newtonian and viscoplastic free surface flows using dynamic octree meshes Minisymposium Session FREE: Monday, 14:30 - 15:00, CO2 We present an approach for numerical simulation of free surface flows of Newtonian and viscoplastic incompressible fluids. The approach is based on the level set method for capturing free surface evolution and features compact finite difference approximations of fluid and level set equations on locally refined and dynamically adapted staggered octree grids. A discretization, constitutive relations, a surface reconstruction, a surface tension forces evaluation: these and other building blocks of the numerical method providing predictive and efficient simulations will be discussed in the talk. In particular, we shall address a finite difference approximation of the advective terms on staggered grids which is stable and low dissipative alternative to semi-Lagrangian methods to treat the transport part of the equations. Numerical examples will demonstrate the performance of the approach for several benchmark and complex 3D Newtonian and viscoplastic fluids with free surfaces.

References [1] K.Nikitin, M.Olshanskii, K.Terekhov, Yu.Vassilevski. A numerical method for the simulation of free surface flows of viscoplastic fluid in 3D. Journal of Computational Mathematics, 29(6), (2011), 605-622. [2] M.Olshanskii, K.Terekhov, Yu.Vassilevski. An octree-based solver for the incompressible Navier-Stokes equations with enhanced stability and low dissipation. Computers and Fluids, to appear. Joint work with M.Olshanskii (University of Houston, Moscow State University), and K.Terekhov (Institute of Numerical Mathematics RAS).

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Marco Verani MOX-Department of Mathematics, Politecnico di Milano, IT Mimetic finite differences for quasi-linear elliptic equations Contributed Session CT2.5: Tuesday, 15:30 - 16:00, CO016 Nowadays, the mimetic finite difference (MFD) method has become a very popular numerical approach to successfully solve a wide range of problems. This is undoubtedly connected to its great flexibility in dealing with very general polygonal meshes (see Figure 1 for an example) and its capability of preserving the fundamental properties of the underlying physical and mathematical models. In this talk, we approximate the solution of a quasilinear elliptic problem of monotone type by using the MFD method and we prove that the MFD approximate solution converges, with optimal rate, to the exact solution in a mesh-dependent energy norm. The resulting nonlinear discrete problem is then solved iteratively via linearization by applying the Kacanov method. The convergence of the Kacanov algorithm in the discrete mimetic framework is also proved. Several numerical experiments confirm the theoretical analysis.

Figure 1: Example of poligonal (exagons) decomposition of the square Ω = (0, 1)2 . Joint work with Paola F. Antonietti (Politecnico di Milano), and Nadia Bigoni (Politecnico di Milano).

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Karen Veroy-Grepl AICES - RWTH Aachen, DE On Synergies between the Reduced Basis Method, Proper Orthogonal Decomposition, and Balanced Truncation Minisymposium Session ROMY: Thursday, 14:30 - 15:00, CO016 In this talk, we present a new method for constructing balanced reduced order models for parametrized systems. The technique is based on synergies between three commonly used model order reduction methods: (i ) the Reduced Basis Method, which provides certified predictions of outputs of parametrized PDEs through Galerkin projection onto a space of solutions at selected parameter values; (ii ) Proper Orthogonal Decomposition, which effectively derives reduced order models from the singular value decomposition of the snapshot matrix; and (iii ) Balanced Truncation, which constructs reduced order models that balance observability and controllability. The proposed method constructs the reduced order model using the most essential aspects of the three methods: the greedy technique and a posteriori error estimation of the Reduced Basis Method, the method of snapshots from Proper Orthogonal Decomposition, and the balancing approach of Balanced Truncation. Joint work with Martin Grepl.

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Martin Vetterli EPFL/IC/LCAV, CH Inverse Problems Regularized by Sparsity Public Lecture: Tuesday, 16:30 - 17:30, Rolex Learning Center Auditorium Sparsity as a modeling principle has been part of signal processing for a long time, for example, parametric methods are sparse models. Sparsity plays a key role in non-linear approximation methods, in particular using wavelets and related constructions. And recently, compressed sensing and finite rate of innovation sampling have shown how to sample sparse signals close to their sparsity levels. In this talk, we first recall that signal processing lives on the edge of continuousand discrete-time/space processing. That duality of the continuum versus the discrete is also inherent in inverse problems. We then review how sparsity can be used in solving inverse problems. This can be done when the setting is naturally sparse, e.g. in source localization, or for solutions that have low-dimensionality in some basis. After an overview of essential techniques for sparse regularization, we present several examples where concrete, real life inverse problems are solved using sparsity ideas. First, we answer the question “can one hear the shape of a room”, a classic inverse problem from acoustics. We show a positive answer, and a constructive algorithm to recover room shape from only a few room impulse responses. Second, we address the problem of source localization in a graph. Assume a disease or a rumor spreading on a social graph, can one find the source efficiently with a small set of observers? A constructive and efficient algorithm is described, together with several practical scenarios. Third, we consider the question of sensor placement for monitoring and inversion of diffusion processes. We present a solution for monitoring temperature using low dimensional modeling and placing a small set of sensors. The ideas of sparse, regularized inversion are finally applied to the problem of trying to recover the amount of nuclear release from the Fukushima nuclear accident. We show that using a transport model and the very limited available measurements, we are able to correctly recover Xenon emission, while the Cesium release remains a challenge.

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Pedro Vilanova KAUST, SA Chernoff-based Hybrid Tau-leap Contributed Session CT3.2: Thursday, 17:30 - 18:00, CO2 Markovian pure jump processes are used to model many phenomena, for example biochemical reactions at molecular level, dynamics of wireless communication networks, spread of epidemic diseases in small populations, among others. There exist algorithms like SSA by Gillespie or Modified Next Reaction Method by Anderson that simulate a single trajectory exactly, but can be time consuming when many reactions take place during a short time interval. The approximated Gillespie’s tau-leap method, on the other hand, can be used to reduce computational time, but introduces a time discretization error that may lead to non-physical values. This talk presents a hybrid algorithm for simulating individual trajectories, which adaptively switches between the SSA and the tau-leap method. The switching strategy is based on the comparison of the expected inter-arrival time of the SSA and an adaptive time step size derived from a Chernoff-type bound for estimating the one-step exit probability. Since this bound is non-asymptotic we do not need to make any distributional approximation for the tau-leap increments. This hybrid method allows to control the global exit probability of a simulated trajectory, and to obtain accurate and computable estimates for the expected value of any smooth observable of the process with low computational work. We present numerical examples that confirm the theory and show the advantages of this approach over both, the exact methods and the tau-leap ones that uses pre-leap checks based on gaussian approximations for the increments. Finally, we will discuss about possible extensions to this method. Joint work with Alvaro Moraes, and Raul Tempone.

398

Gilles Vilmart ENS Rennes and INRIA Rennes, FR Numerical homogenization methods for multiscale nonlinear elliptic problems of nonmonotone type Minisymposium Session MSMA: Monday, 15:00 - 15:30, CO3 We study the effect of numerical quadrature in finite element methods for a class of nonlinear elliptic problems of nonmonotone type. This is a key ingredient to analyze the so-called finite element heterogeneous multiscale method (FE-HMM) applied to nonmonotone homogenization problems. We obtain optimal convergence results for the H 1 and L2 norms in dimension d ≤ 3 and for a fully discrete method taking into account both macro and micro discretizations. We also prove for sufficiently fine meshes the uniqueness of the numerical solution and the convergence of the Newton method needed in the implementation. In addition, we show that the coupling of the nonlinear multiscale method with the reduced basis technique (RB-FE-HMM) considerably improves the efficiency by drastically reducing the number of degrees of freedom.

References [1] A. Abdulle, Y. Bai, and G. Vilmart, Reduced basis finite element heterogeneous multiscale method for quasilinear homogenization problems, preprint (2013), 26 pages. [2] A. Abdulle, Y. Bai, and G. Vilmart, An offline-online homogenization strategy to solve quasilinear two-scale problems at the cost of one-scale problems, preprint (2013), 13 pages. [3] A. Abdulle and G. Vilmart, Fully discrete analysis of the finite element heterogeneous multiscale method for quasilinear elliptic homogenization problems, to appear in Mathematics of Computation (2013), 21 pages. [4] A. Abdulle and G. Vilmart, A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems, Numer. Math. 121 (2012), 397-431. [5] W. E, P. Ming and P. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc. 18 (2005), no. 1, 121–156. Joint work with Assyr Abdulle and Yun Bai.

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Gilles Vilmart ENS Rennes and INRIA Rennes, FR Multi-revolution composition methods for highly oscillatory problems Minisymposium Session ASHO: Tuesday, 10:30 - 11:00, CO123 We introduce a new class of multi-revolution composition methods (MRCM) for the approximation of the N th-iterate of a given near-identity map. When applied to the numerical integration of highly oscillatory systems of differential equations, this numerical homogenization technique benefits from the properties of standard composition methods: it is intrinsically geometric and well-suited for Hamiltonian or divergence-free equations for instance. We prove error estimates with error constants that are independent of the oscillatory frequency. Numerical experiments, in particular for the nonlinear Schrödinger equation, illustrate the theoretical results, as well as the efficiency and versatility of the methods. Joint work with P. Chartier (Rennes), J. Makazaga, and A. Murua (San Sebastian).

400

Martin Vohralik INRIA Paris-Rocquencourt, FR Adaptive regularization, linearization, and algebraic solution in unsteady nonlinear problems Minisymposium Session STOP: Thursday, 15:30 - 16:00, CO1 We show how computable a posteriori error estimates can be obtained for two model nonlinear unsteady problems, namely the Stefan problem and the two-phase porous media flow problem. Regularization of the nonlinear functions, iterative linearizations, and iterative solutions of the arising linear systems are typically involved in the numerical approximation procedure. We show how the corresponding error components can be distinguished and estimated separately. A fully adaptive algorithm, with adaptive choices of the regularization parameter, the number of nonlinear and linear solver steps, the time step size, and the computational mesh, is presented. Numerical experiments confirm tight error control and important computational savings. We present two examples for the two-phase flow in porous media. In the left part of Figure 1, we plot our estimators of the different error components as a function of GMRes iterations for a fixed time and Newton step. We see that our stopping criteria enable to economize an important number of iterations with respect to the classical criterion requiring the relative algebraic residual to be smaller than 1e-13. In the right part of Figure 1, we track the dependence this time with respect to the Newton iterations. We compare our criteria with the classical one requiring the difference between two consecutive pressure and saturation approximations to be smaller than 1e-11. The overall gains achievable thanks to our approach are then illustrated in Figure 2. In its left part, we plot the number of necessary Newton iterations on each time step for both the adaptive and classical stopping criteria. In its right part, the cumulative number of GMRes iterations is given as function of time. From this last graph, we can conclude that in the adaptive approach the number of cumulative GMRes iterations is approximately 12-times smaller compared to that in the classical one. Details can be found in the references [1] and [2]. [1] Di Pietro D. A., Vohralík M., and Yousef S. Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem. HAL Preprint 00690862, submitted for publication, 2012. [2] Vohralík M. and Wheeler M. F. A posteriori error estimates, stopping criteria, and adaptivity for two-phase flows. HAL Preprint 00633594 v2, submitted for publication, 2013.

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Figure 1: Spatial, temporal, linearization, and algebraic estimators and their sum as function of the GMRes iterations (left) and of the Newton iterations (right)

Figure 2: Number of Newton iterations on each time step (left) and cumulative number of GMRes iterations as a function of time (right) Joint work with D. A. Di Pietro, M. F. Wheeler, and S. Yousef.

402

Heinrich Voss Hamburg University of Technology, DE Variational Principles for Nonlinear Eigenvalue Problems Minisymposium Session NEIG: Thursday, 10:30 - 11:30, CO2 Variational principles are powerful tools when studying the qualitative behavior and numerical methods for linear self-adjoint operators. Bounds for eigenvalues, comparison results, interlacing properties, and monotonicity of eigenvalues can be proved easily with variational characterizations of eigenvalues, to name just a few. If A is a self-adjoint operator on a Hilbert space H with domain of definition D and λ1 ≤ λ2 ≤ . . . are the eigenvalues of A below the essential spectrum of A, then they can be characterized by a minmax principle of Poincaré type λn =

min

max

V ⊂D, dim V =n

x∈V, x6=0

hAx, xi hx, xi

or by a maxmin principle of Courant–Fischer–Weyl type λn =

max

V ⊂H, dim V =n−1

min

x∈D, x⊥V, x6=0

hAx, xi . hx, xi

In this talk we discuss generalizations of these variational principles to families of linear operators depending continuously on an eigenparameter λ. References H. Voss, B. Werner. A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems. Math. Meth. Appl. Sci. 4, 415 – 424 (1982) H. Voss. A maxmin principle for nonlinear eigenvalue problems with application to a rational spectral problem in fluid–solid vibration. Applications of Mathematics 48, 607 – 622 (2003) H. Voss A minmax principle for nonlinear eigenproblems depending continuously on the eigenparameter. Numer. Lin. Algebra Appl. 16, 899 – 913 (2009) M. Stammberger, H. Voss Variational characterization of eigenvalues of a nonsymmetric eigenvalue problem in fluid–solid vibrations. Submitted to Applications of Mathematics

403

Benjamin Wacker Institute for Numerical and Applied Mathematics, University of Göttingen, DE A local projection stabilization method for finite element approximation of a magnetohydrodynamic model Minisymposium Session MMHD: Thursday, 12:00 - 12:30, CO017 In this talk, we consider the equations of incompressible resistive magnetohydrodynamics. Based on a stabilized finite element formulation by S. Badia, R. Codina and R. Planas for the linearized equations [1], we propose a modification of this technique by a local projection stabilization finite element method for the approximation of this problem. The introduced stabilization technique is then discussed by investigating the stability and convergence analysis for the problem’s formulation thoroughly. We finally compare our numerical analysis with other approximations presented in the literature.

References [1] S. Badia, R. Codina and R. Planas. On an unconditionally convergent stabilized finite element approximation of resistive magnetohydrodynamics, Journal of Computational Physics, 234:399-416, 2013. Joint work with Gert Lube.

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Shawn Walker Louisiana State University, US A New Mixed Formulation For a Sharp Interface Model of Stokes Flow and Moving Contact Lines Minisymposium Session GEOP: Wednesday, 11:00 - 11:30, CO122 Two phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension (see Falk & Walker in the context of Hele-Shaw flow) that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking level) and allows for moving contact lines and contact angle hysteresis through a variational inequality. We prove the well-posedness of the time semi-discrete and fully discrete (finite element) model and discuss error estimates. Simulation movies will be presented to illustrate the method. We conclude with some discussion of a 3-D version of the problem as well as future work on optimal control of these types of flows.

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Mirjam Walloth Institute of Computational Science, University of Lugano, CH An efficient and reliable residual-type a posteriori error estimator for the Signorini problem Contributed Session CT4.7: Friday, 09:20 - 09:50, CO122 Often, in the numerical simulation of real world problems as, e.g., arising from mechanics or biomechanics, precise information about the regularity of the solution cannot be obtained easily a priori. In fact, the solution may be more or less regular in different regions of the computational domain and even singularities may occur. In this case, increasing the number of degrees of freedom within or close to a critical region of low regularity can improve the overall accuracy of the numerically obtained approximation. The detection of such a critical region can be made feasible by using a posteriori error estimators which do not rely on any additional regularity assumptions. One of the most common a posteriori error estimators is the standard residual estimator which is directly derived from the equivalence of the norm of the error and the dual norm of the residual. For contact problems this relation is disturbed due to the non-linearity. Thus, additional effort is required to derive an a posteriori error estimator for contact problems. Here, we present a new a posteriori error estimator for the linear finite element solution of the Signorini problem in linear elasticity [2]. Inspired by a posteriori error estimators for the closely related obstacle problem, see e. g. [4, 1, 3] the estimator is designed for controlling the H 1 -error of the displacements and the H −1 -error of a suitable approximation of the Lagrange multiplier. The estimator reduces to the standard residual estimator for linear elasticity, if no contact occurs. The estimator contributions addressing the nonlinearity are related to the contact stresses, the complementarity condition, and the approximation of the gap function. Remarkably, the first two terms do not contribute in the case of so-called full-contact. We prove reliability and efficiency of the estimator for two- and three-dimensional simplicial meshes, ensuring the equivalence with the error up to oscillation terms. Our theoretical findings are supported by intensive numerical studies. The adaptively refined grids and the relevance of the different error estimator contributions are analyzed by means of different illustrative numerical experiments in 3D. In our numerical studies, we quantitatively investigate the convergence of the error estimator by comparing to the case of uniformly refined grids. Furthermore, for selected examples in 2D and even in 3D where the contact stresses are known analytically, we compare the numerically computed contact stresses on adaptively refined grids with the exact contact stresses. Interestingly, although the proofs of upper and lower bound are given for meshes of simplices, the numerical studies show also very good performance of the new residual-type a posteriori error estimator for unstructured meshes consisting of hexahedra, tetrahedra, prisms, and pyramids.

References: [1] Fierro, F., Veeser, A.: A posteriori error estimators for regularized total variation of characteristic functions. SIAM J. Numer. Anal. 41, 2032–2055 (2003) [2] Krause, R., Veeser, A., Walloth, M.: An efficient and reliable residual-type 406

a posteriori error estimator for the Signorini problem. Preprint, Institute of Computational Science, University of Lugano, 2012. [3] Moon, K., Nochetto, R., von Petersdorff, T., Zhang, C.: A posteriori error analysis for parabolic variational inequalities. M2AN Math. Model. Numer. Anal. 41, 485–511 (2007) [4] Veeser, A.: Efficient and reliable a posteriori error estimators for elliptic obstacle problems. SIAM J. Numer. Anal. 39, 146–167 (2001)

Joint work with Rolf Krause, and Andreas Veeser.

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Andreas Weinmann Helmholtz Zentrum München and TU München, DE Jump-sparse reconstruction by the minimization of Potts functionals Minisymposium Session ACDA: Monday, 15:00 - 15:30, CO122 This talk is on our recent work concerning Potts and Blake-Zisserman functionals. We start with L1 Potts functionals Pγ (u) = γ · k∇uk0 + ku − f k1 . Here f are given (univariate) data, k∇uk0 counts the number of jumps of u and γ is a parameter controlling the trade-off between data fidelity and regularity. We develop a fast algorithm for minimizing discrete L1 Potts functionals. Furthermore, we obtain convergence results for discrete Potts functionals and their respective minimizers towards their continuous time counterparts. In addition, we show a nice blind deconvolution property of L1 Potts functionals: Mildly blurred jump-sparse signals are reconstructed by minimizing the functional. In the second part of the talk we consider (inverse) Potts problems of the form P¯γ (u) = γ · k∇uk0 + kAu − f kpp → min . Here A is a not necessarily square matrix. We present an ADMM based approach which works very well in practice. Furthermore, we consider a Douglas-Rachford like splitting approach to the above inverse Potts problem for p = 2 and the (inverse) Blake-Zisserman problem X ¯γs,q (u) = γ · B min(|ui − ui−1 |q , sq ) + kAu − f k22 → min . i

Here s is a positive number and q ≥ 1. For the inverse Blake-Zisserman functionals ¯γs,q and the inverse Potts functionals F¯ = P¯γ we consider the corresponding F¯ = B surrogate functional F¯ (u, v) = F¯ (u) −kAu−Avk22 +ku−vk22 . The iteration un+1 = argminu F¯ (u, un ) leads to un+1 = argminu γk∇uk0 + ku − A∗ (Aun − f )k22 in the Potts case. The iteration for the Blake-Zisserman case is obtained by reP placing the regularity term k∇uk0 by the sum i min(|ui −ui−1 |q , sq ). This means that we have to solve (ordinary) Potts or Blake-Zisserman problems with A = id for data A∗ (Aun − f ). This can be done fast by using dynamic programming. In contrast to a recent approach of M. Fornasier and R. Ward to the Blake-Zisserman problem we directly work on the inverse Blake-Zisserman functionals. We show that for inverse Blake-Zisserman functionals as well as for inverse Potts functionals, the above iterative algorithm converges towards a local minimum of the respective functional. Joint work with Laurent Demaret, and Martin Storath.

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Steffen Weißer Saarland University, DE Challenges in BEM-based Finite Element Methods on general meshes Contributed Session CT1.8: Monday, 18:30 - 19:00, CO123 In the field of numerical methods for partial differential equations there is an increasing interest for non-simplicial meshes. Several applications in solid mechanics, biomechanics as well as geological science show the need for general elements within a finite element simulation. Discontinuous Galerkin methods and mimetic finite difference methods are able to handle such kind of meshes. Nevertheless, these two strategies yield non-conforming approximations. Recent developments like the virtual element method [1] overcome these difficulties. Another new kind of conforming finite element method on general meshes has been proposed in [2]. This method uses basis functions that fulfill the differential equation locally. In the local problems constant material parameters and vanishing right hand side are prescribed. Due to this implicit construction, the basis functions are applicable on polygonal and polyhedral elements, respectively. Let Ω ⊂ R2 be a polygonal domain and a ∈ L∞ (Ω), f ∈ L2 (Ω), g ∈ H 1/2 (∂Ω). For the model problem −div(a∇u) = f

in Ω,

u=g

on ∂Ω

the lowest order basis functions ψz which are dedicated to the nodes z ∈ Nh of the mesh Kh with elements K are uniquely defined by −∆ψz = 0 in K for all K ∈ Kh ,  1 z=x ψz (x) = , 0 z= 6 x ∈ Nh ψz linear on each edge of K.

If the material coefficient is approximated by a piecewise constant function, a ≈ aK on K for K ∈ Kh , the standard bilinear form of the variational formulation can be rewritten by the use of Green’s first identity over each element such that Z X ∂ψz ds. aΩ (ψz , ψx ) = aK ψx ∂nK ∂K K∈Kh

Consequently, the integration is reduced to the boundaries of the elements where the trace of the basis functions is known explicitly. The normal derivative can be expressed by means of boundary integral operators. In the numerics these operators are approximated by the use of boundary element methods. Therefore, the global method is called BEM-based FEM. This strategy has been studied in several articles concerning convergence [3] as well as residual error estimates for adaptive mesh refinement [4], for example. The aim of current research is to extend the ideas for the definition of trial functions to three space dimensions such that the method can handle polyhedral meshes, see Figure 1. In the case that the polyhedral elements have triangulated surfaces there exist already straight forward generalizations. But the challenging part is to manage the polygonal faces of the polyhedral elements directly. Furthermore, the question of arbitrary order basis functions is addressed. Following the ideas of [3], an extended ansatz space Vh is defined which admits arbitrary 409

order convergence. With the help of interpolation operators on polygonal meshes the error estimate ku − uh kH 1 (Ω) ≤ c hk |u|H k+1 (Ω) is proven for an exact solution u ∈ H k+1 (Ω) and its approximation uh ∈ Vh . Finally, all theoretical results are confirmed by several numerical experiments.

References [1] L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, A. Russo: Basic principles of virtual element methods. Mathematical Models and Methods in Applied Sciences 23, 199, 2013 [2] D. Copeland, U. Langer, D. Pusch: From the boundary element domain decomposition methods to local Trefftz finite element methods on polyhedral meshes. Domain Decomposition Methods in Science and Engineering XVIII, 315–322, 2009 [3] S. Rjasanow, S. Weißer: Higher order BEM-based FEM on polygonal meshes. SIAM Journal on Numerical Analysis, 50(5):2379–2399, 2012 [4] S. Weißer: Residual error estimate for BEM-based FEM on polygonal meshes. Numerische Mathematik, 118:765-788, 2011

Figure 1: Polyhedral mesh of the unite cube Joint work with Prof. Dr. Sergej Rjasanow.

410

Garth Wells University of Cambridge, GB Domain-specific languages and code generation for solving PDEs using specialised hardware Minisymposium Session PARA: Monday, 16:00 - 16:30, CO016 The development and use of a domain-specific language coupled with code generation has proved to be very successful for creating high-level, high-performance finite element solvers. The use of a domain-specific language allows problems to be expressed compactly in near-mathematical notation, and facilitates the preservation of mathematical abstractions. The latter point is invaluable for automating the creation of auxiliary problems, such as linearisations or adjoint equations. The generation of low-level code from expressive, high-level input can offer performance beyond what one could reasonably achieve using conventional programming techniques. Important in this respect is leveraging domain knowledge that cannot be provided by a general purpose compiler. The generation of low-level code from expressive, high-level input has great appeal for specialised hardware, such as now wide-spread co-processor technology. Recent hardware shifts the burden onto the developer and demands a high level of software expertise. To address this, recent and ongoing investigations into generating target-specific code for solving PDEs using the FEniCS Project toolchain are presented. GPU code is generated from Unified Form Language input, and it is shown how different strategies differ dramatically in performance depending on the equation type and finite element type. To counter this, a formulation that is parameterised over the equation and finite element type is presented. In this way, a code generator can narrow the search space for efficient formulations and strategies. It also offers solver level shielding against future hardware and programming model changes.

411

Thomas Wick The University of Texas at Austin, ICES, US A fluid-structure interaction framework for reactive flows in thin channels Minisymposium Session NFSI: Thursday, 12:00 - 12:30, CO122 We study the reactive flow in a thin strip where the geometry changes take place due to reactions. Specifically, we consider precipitation dissolution processes taking place at the lateral boundaries of the strip. The geometry changes depend on the concentration of the solute in the bulk (trace of the concentration) which makes the problem a free-moving boundary problem. The numerical computations are challenging in view of the nonlinearities in the description of the reaction rates. In addition to this, the movement of the boundary depends on the unknown concentration (and hence part of the solution) and the computation of the coupled model remains a delicate issue. Our aim is to develop appropriate numerical techniques for the computation of the solutions of the coupled convection-diffusion equation and equation describing the geometry changes. The key idea at this point consists in using a fluid-structure interaction framework to formulate and to solve the problem at hand. We use the arbitrary Lagrangian-Eulerian framework and a monolithic solution algorithm for the numerical treatment. Temporal discretization is based on finite differences whereas spatial discretization makes use of a Galerkin finite element scheme. The nonlinear problem is treated with Newton’s method. The performance is demonstrated with the help of several interesting numerical tests. Joint work with Kundan Kumar, and Mary F. Wheeler.

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Olof B. Widlund Courant Institute, US, US Two-level overlapping Schwarz methods for some saddle point problems Minisymposium Session PSPP: Thursday, 10:30 - 11:30, CO3 About fifteen years ago, Axel Klawonn and Luca Pavarino explored the possibility of using two-level overlapping Schwarz methods for a variety of saddle-point problems. It is the purpose of this contribution to provide a theoretical justification for some of these results. Our work is inspired by earlier joint work with Clark R. Dohrmann on almost incompressible elasticity solved by mixed finite elements with discontinuous pressure approximations. A report on some recent numerical experiments will also be given. Joint work with Luca F. Pavarino.

413

Tobias Wiesner Technische Universität München, DE Algebraic multigrid (AMG) methods for saddle-point problems arising from mortarbased finite element discretizations Minisymposium Session PSPP: Thursday, 11:30 - 12:00, CO3 The development of novel discretization schemes and solution algorithms for fully nonlinear contact mechanics problems, i.e. including finite deformations, finite frictional sliding and possibly nonlinear material behavior, has seen a great thrust of research progress over the last decade. With regard to discretization schemes, mortar finite element methods have proven to outperform traditional collocation methods (e.g. node-to-segment) in terms of both robustness and accuracy. While penalty and related methods remain a generally accepted and widely used choice for constraint enforcement, the numerical efficiency of Lagrange multiplier methods has been significantly improved in recent years. In general, the Lagrange multiplier based formulation of contact mechanics problems leads to saddle-point type systems, with the additional Lagrange multiplier degrees of freedom enforcing typical contact conditions (such as tied contact, unilateral contact and frictional sliding). Although there exist discretization strategies such as the so-called dual Lagrange multiplier approach ([1],[2]), which allow for simple algebraic condensation procedures and thus circumvent the saddle-point structure of linear systems, it is still worth to deal with the more challenging saddle-point type problems. For the sake of simplicity, so-called mesh tying (or tied contact) problems are considered first ([3]). Here, a mortar finite element discretization generates algebraic equations of the form      KN1 N1 KN1 M 0 0 0 dN1 fN1 T  KMN1 KMM   fM  0 0 −M d M       0     0 KSS KSN2 DT     dS  =  fS   0   dN2  fN2  0 KN2 S KN2 N2 0 λ 0 0 −M D 0 0 with d the displacement and f the force vector. Obviously the Lagrange multipliers λ couple the two distinct blocks at the mesh tying interface where M denotes the master side and S the slave side degrees of freedom with the corresponding mortar matrix blocks M and D. The big advantage of the saddle-point formulation is the clean distinction between the different physical variables (e.g. structural displacements) and constraint variables (Lagrange multipliers) both in the structure of the block matrix and in the solution vector. This makes it rather easy to consider the underlaying physics when developing special saddle-point precondtioners for these type of problems, especially if the different physically and mathematically motivated variables are formulated in different coordinate systems. So, in case of mesh tying and structural contact problems the structural displacements are usually formulated in Cartesian coordinates whereas the contact constraints are formulated in tangential and normal coordinates relative to the contact interface. This talk is devoted to the design of robust Algebraic Multigrid (AMG) preconditioners that can be used within iterative solvers for linear systems arising from contact and mesh tying problems in saddle-point formulation. Multigrid methods are known to be among the best preconditioning techniques at least for symmetric 414

positive definite systems [4]. However, linear systems with saddle-point structure are still challenging for multigrid methods and make special modifications necessary [5]. The idea is to use multigrid methods to build coarse level representations of the fine-level problem which preserve the saddle-point structure. Then, on each multigrid level basic Schur-complement based multigrid smoothers are applied. The focus of this talk will be on aggregation strategies and multigrid transfer operators for the Lagrange multipliers. We compare different variants of transfer operators and Schur-complement based multigrid smoothing methods by means of examples arising from mortar-based finite element discretizations in contact mechanics.

References [1] Popp, A., Gitterle, M., Gee, M.W. and Wall, W.A.: "A dual mortar appraoch for 3D finite deformation contact with consistent linearization", Internatio-nal Journal for Numerical Methods in Engineering, 84, 543-571 (2010). [2] Popp, A., Wohlmuth, B.I., Gee, M.W. and Wall, W.A.: "Dual quadratic mortar finite element methods for 3D finite deformation contact", SIAM Journal on Scientific Computing, 34, B421-B446 (2012). [3] Puso, M.A.: "A 3D mortar method for solid mechanics", International Journal for Numerical Methods in Engineering, 59, 315-336 (2004) [4] Vanek, P., Mandel, J. and Brezina, M.: "Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic problems", Computing, 56, 179-196 (1996) [5] Adams, M.F.: "Algebraic multigrid methods for constrained linear systems with applications to contact problems in solid mechanics", Numerical Linear Algebra with Applications, 11(2-3), 141-153 (2004) Joint work with A. Popp, W.A. Wall, and M.W. Gee.

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Barbara Wohlmuth Technische Universität München, DE Interfaces, corners and point sources Plenary Session: Monday, 09:00 - 09:50, Rolex Learning Center Auditorium In this talk, we address the convergence of finite element approximations for cases where the solution is locally smooth but has possibly globally a reduced regularity. Typical examples are transmission problems, domains with re-entrant corners, heterogeneous coefficients and right hand sides being not in the H 1 -dual space. These situations occur quite often in the mathematical modelling of multi-physics applications. As examples for interface problems we name structure-acoustic interaction or the coupling between free flow and porous media equations. The permeability in the Darcy equation of porous media models is often assumed to be piecewise constant but involves highly heterogeneous coefficients. Non-convex domains with re-entrant corners occur in the numerical simulation of technical applications. Finally, dimension reduced partial differential equation systems play an important role in the mathematical modelling of physical effects on different scales, e.g., fractures of co-dimension one in porous media systems, networks of co-dimension two. Although these simplified models seem to be very attractive from the computational point of view, for coupled problems they result in a solution of reduced regularity. From the mathematical point of view, transmission problems with piecewise smooth solutions or pdes in a distributional sense with singular solutions arise. Standard remedies to handle these type of problems are graded meshes or enrichment, both techniques result in extra implementational work and computational cost. Here we provide optimality results for interface quantities such as the flux and show that globally no pollution occurs in case of point loads. Although the solution is globally not in H 1 , we observe on a sequence of uniformly refined meshes optimal L2 -a priori convergence on subdomains excluding the point sources. In the case of heterogeneous coefficients or re-entrant corners, the situation is more complex. Then the well-known pollution effect is observed, and the convergence can be extremely poor. This also holds true not only for the L2 norm on subdomains excluding the cross-points and corners but also for other quantities of interests such as the stress intensity factors or eigenvalues. Here we introduce a purely local energy correction function and modify locally the bilinear form. We provide a multi-level algorithm to pre-compute the modification parameters, existence and optimal a priori results. Numerical examples in 2D illustrate that we can recover full optimality in case of uniform meshes for re-entrant corners, heterogeneous coefficients and linear elasticity. As quantities of interest we select the convergence of eigenvalues, the flux across an interface and the stress intensity factor. [1] M. Melenk, H. Rezaijafari, B. Wohlmuth: Quasi-optimal a priori estimates for fluxes in mixed finite element methods and applications to the Stokes–Darcy coupling, IMA J. Numer. Anal., 2013 doi:10.1093/imanum/drs048 [2] H. Egger, U. Rüde, B. Wohlmuth: Energy-corrected finite element methods for corner singularities, to appear in SIAM J. Numer. Anal. [3] T. Köppl, B. Wohlmuth: Optimal a priori error estimates for an elliptic problem with Dirac right-hand side, submitted 2013

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[4] C. Waluga, B. Wohlmuth: Quasi-optimal trace error estimates and a posteriori error estimation for the interior penalty method, submitted 2012

Joint work with M. Melenk (TU Wien), H. Egger (TU Darmstadt), U. Rüde (FAU) and with F. Benesch, T. Horger, M. Huber, T. Köppl, H. Rezaijafari, and C. Waluga (TU München).

417

Winnifried Wollner University of Hamburg, DE Adjoint Consistent Gradient Computation with the Damped Crank-Nicolson Method Minisymposium Session FEPD: Monday, 15:00 - 15:30, CO017 The talk is concerned with a damped version of the Crank-Nicolson (CN) Method for the solution of parabolic partial differential equations. As it is well known, the CN-Methods needs to be damped in order to cope with irregular initial data, due to the missing smoothing property. Since the adjoint of the CN-Method is again a CN-Method, shifted by half a timestep, it is not surprising that a similar problem occurs for the adjoint time stepping scheme. In this talk, we will derive an adjoint consistent damped CN-Scheme that ensures sufficient damping of the dual problem. The necessity of these modifications will be discussed along the use of the dual-weighted residual (DWR) method for the adaptive solution of the Black-Scholes equation. The consequences for optimization problems with parabolic PDEs will be discussed. Joint work with C. Goll, and R. Rannacher.

418

Olaf Wünsch University of Kassel, Institute of Mechanics, DE Numerical simulation of viscoelastic fluid flow in confined geometries Minisymposium Session MANT: Tuesday, 11:00 - 11:30, CO017 The flow of highly viscous fluids in many technical apparatus is dominated through the wall influence. The fluid adheres at the wall and the shear stresses of the high viscosity transport the information of the wall far into the fluid when the Reynolds number is small. Most of these technical geometries show symmetrical behavior like centerlines. In the case of Newtonian material the simulation can apply the geometrical symmetries in order to reduce the numerical costs. The calculation domain ends at the symmetrical line and special boundary conditions must be used. For viscoelastic fluids this procedure is venturous. In dependence of the chosen viscoelastic material model and the parameters the symmetry of the velocity field can be broken. An important quantity is the elongation viscosity and its behavior for high elongation rates. If the critical value of the determining number is exceeded, the flow becomes unsymmetrical. Experimental investigations in different geometries are documented in literature. This paper deals with numerical simulations techniques to calculate such viscoelastic fluid flows in confined geometries. Basis of the calculations are the balance of momentum in connection with the balance of continuity. The fluid is modeled by a modified Maxwell-typed differential equation for the stress tensor T with an anisotropic molecule mobility tensor Q 

O

T + λM T + Q(T)T = 2ηP D + 2λN ηP D. Here D is the rate of strain tensor, λM , λN , ηP denote material parameter and , O are special time derivatives. The numerical treatment of viscoelastic, incompressible fluid flow differs from Newtonian flow [1]. In the latter case the type of the differential equations are elliptical, or for instationary flow even parabolic. For Maxwell-typed equations the classification is complex and depends on the solution. On this account the use of stabilization techniques are necessary in order to get significant solutions. Different methods are reported in the literature. In the viscous formulation the viscoelastic stress tensor is expanded by a Newtonian part in order to increase the elliptical behavior. This is useful for low numbers of the characteristic dimensionless number, the Deborah number. Other methods, like EVSS, DEVSS or the Log-Conformation-Tensor method are more successful. The set of differential equations are discretized by finite-volume-method. We use the opensource library OpenFoam in order to develop a robust solving algorithm. Numerical simulations are presented for different stabilization techniques and for different viscoelastic models with the focus on asymmetrical fluid flows in symmetrical geometries. The influence of increasing Deborah number is shown exemplarily in figure 1 and figure 2. In the so-called cross slot geometry the fluid flows from left and right to top and bottom. In the case of Newtonian fluid the flow is exactly symmetrical in contrast to the viscoelastic fluid. [1] A. Al-Baldawi, Berichte des Instituts für Mechanik, kassel university press GmbH, Kassel, 2012

419

Figure 1: Symmetrical velocity field of a Newtonian fluid in a cross slot geometry

Figure 2: Unsymmetrical velocity field of a Giesekus fluid in a cross slot geometry Joint work with A. Al-Baldawi.

420

Huidong Yang Postdoc, AT Numerical Methods for Fluid-Structure Interaction Problems with a Mixed Elasticity Form in Hemodynamics Minisymposium Session NFSI: Thursday, 14:30 - 15:00, CO122 In this talk, a nearly incompressible elasticity model coupled with an incompressible fluid model for some fluid-structure interaction (FSI) problems in hemodynamics under a three dimensional (3D) configuration is presented. A mixed displacement-pressure formulation is employed in modeling the structure, that overcomes a possible Poisson locking phenomenon. The fluid is modeled by the incompressible Navier-Stokes equations. Implicit first order methods are employed for discretizing the fluid and structure sub-problems in time: a semi-implicit Euler scheme for the fluid and a Newmark-β scheme for the structure. A proper least-square finite element stabilization parameter for the elasticity formulation depending on time discretization parameters, mesh discretization parameters and material coefficients is designed. In such a framework, an extention of the FSI problems within two-layer composite vessels is investigated. Such vessels are assumed to possess jumping material coefficients (e.g., density, Young’s modulus and Poisson ratio) and to vary thicknesses from one layer to another. Numerical experiments demonstrate sensitivities of FSI solutions with respect to the material coefficients, the thicknesses of two layers, and the time discretization parameters. For solving the coupled FSI system, we employ a class of partitioned solvers which are interpreted as Gauss-Seidel iterations applied to a reduced system with fluid velocity and structure displacement unknowns on the interface only. The performance of the algorithm relying on robust and efficient algebraic multigrid methods for the fluid and structure sub-problems is studied.

421

Hamdullah Yücel Max Planck Institute for Dynamics of Complex Technical Systems, DE Distributed Optimal Control Problems Governed by Coupled Convection Dominated PDEs with Control Constraints Contributed Session CT2.9: Tuesday, 15:30 - 16:00, CO124 Many real-life applications such as the fluid dynamic problems in the presence of body forces, chemically reactive flows and electrochemical interaction flows lead to optimization problems governed by systems of convection diffusion partial differential equations (PDEs) with nonlinear reaction mechanisms. Such problems are strongly coupled as inaccuracies in one unknown directly affect all other unknowns. Prediction of these unknowns is very important for the safe and economical operation of biochemical and chemical engineering processes. In addition, when convection dominates diffusion, the solutions of these PDEs typically exhibit layers on small regions where the solution has large gradients. Hence, special numerical techniques are required, which take the structure of the convection into. The integration of discretization and optimization is important for the overall efficiency of the solution process. Recently, discontinuous Galerkin (DG) methods became an alternative to the finite difference, finite volume and continuous finite element methods for solving wave dominated problems like convection diffusion equations since they possess higher accuracy. This talk will focus on an application of DG methods for distributed optimal control problems governed by coupled convection dominated PDEs with control constraints. We use a residual based a posteriori error estimator to reduce these oscillations around the boundary and/or interior layers. The matrix system generated by Newton iterations is solved with an appropriate preconditioner. Joint work with Martin Stoll, and Peter Benner.

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Yongjin Zhang Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, DE Reduced-order modeling and ROM-based optimization of batch chromatography Contributed Session CT4.4: Friday, 08:50 - 09:20, CO015 Batch chromatography, as a crucial separation and purification tool, is widely employed in food, fine chemical and pharmaceutical industries. The optimal operation of batch chromatography is of practical importance since it allows to exploit the full economic potential of the process and to reduce the separation cost. The dynamic behavior of the chromatographic process is described by a complex system of partial differential equations (PDEs), ( ∂cz ∂ 2 cz ∂cz 1− ∂qz 0 < x < L, ∂t +  ∂t = −v ∂x + Dz ∂x2 , (1) ∂qz Eq = κ (q − q ), 0 ≤ x ≤ L, z z z ∂t where cz , qz are the concentrations of the component z (z = A, B) in the liquid and Q solid phase, respectively, v = A the convection velocity, Q the feed volumetric c flow-rate, Ac the cross-sectional area of the column,  the column porosity, t the time coordinate, x the axial coordinate along the column with the length L, κz the mass-transfer coefficient, Dz = PvLe the axial dispersion coefficient and P e the Péclet number. The adsorption equilibrium qzEq is described by the isotherm equations with the bi-Langmuir type, qzEq = fz (cA , cB ) := (

Hz1 Hz2 + )cz , 1 + KA1 cA + KB1 cB 1 + KA2 cA + KB2 cB

(2)

where Hzj and Kzj are the Henry constants and thermodynamic coefficients, respectively. The initial and boundary conditions are given as,   cz (t = 0, x) = 0, qz (t = 0, x) = 0, 0 ≤ x ≤ L, ∂cz v F (3) ∂x |x=0 = Dz (cz (t, x = 0) − cz χ[0,tin ] (t)),  ∂c z | = 0, ∂x x=L where cF z are the feed concentrations of component z, tin is the injection time, and χ[0,tin ] is the characteristic function,  1, if t ∈ [0, tin ], χ[0,tin ] (t) = (4) 0, otherwise. Note that Q and tin are often considered as the operating variables, denoted as µ := (Q, tin ), which play the role of parameters in this system. Consequently, the system is nonlinear, time-dependent and with non-affine parameter dependency. The nonlinearity of the system is reflected by (2). To capture the system dynamics precisely, a large number of degrees of freedom must be introduced for the discretization of the PDEs. Many efforts have been made for the optimization of batch chromatography over the past several decades. They are usually based on the finely discretized full order model (FOM). Such an expensive FOM must be repeatedly solved during the optimization process. As a result, addressing an optimization problem is often time-consuming due to such a multi-query context. Parametric model order reduction (PMOR) is an efficient tool for reducing a large parametric system to a small one, while preserving the dominant dynamics of 423

the FOM and the accuracy of the input-output relationship. The reduced basis method is a robust PMOR technique and has been widely used in many applications. In this work, the reduced basis method [1, 2] is introduced and applied to the simulation of batch chromatography. An adaptive technique is proposed to speed up the generation of the reduced basis. The FOM is derived by using the finite volume discretization, whereby the conservation property of the system is preserved. The resulting reduced-order model (ROM) is used to get a rapid evaluation of the input-output relationships for the underlying optimization. The construction of the ROM is automatically managed by a greedy algorithm with an inexpensive error estimator. Numerical experiments are carried out to show the efficiency and reliability of the ROM. Table 1 shows the average run-time of the simulation over 100 random sample points of µ, and the maximal error defined as Max.error = maxµ∈Pval ||Y F (µ)− Y R (µ)||. Y F (µ) and Y R (µ) are the outputs evaluated by using the FOM and the ROM respectively, at a given parameter µ. The average run-time of detailed simulation is sped up by a factor of 52. The optimization results are shown in Table 2. The optimal solution of the ROM-based optimization converges to that of the FOM-based optimization. Furthermore, the run-time of solving the FOM-based optimization is significantly reduced. The factor of speedup is 64. Table 1: Run-time comparison of the detailed simulation and the reduced simulation over a test set Pval with 100 random sample points. Methods Max. error Average run-time [s] Detailed simulation (N = 1000) – 69.5446 Reduced simulation (N = 40, M = 61) 2.5 × 10−4 1.3359 Table 2: Comparison of the optimization based on the ROM and the FOM. Methods Obj. (P r) Opt. solution (µ) Iterat. Run-time [h] FOM-based Opt. 0.02032 (0.07983, 1.05544) 211 8.5371 ROM-based Opt. 0.02028 (0.07982, 1.05247) 211 0.1332

References [1] A. T. Patera and G. Rozza. Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. Version 1.0, Copyright MIT 2006, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering. [2] M. Drohmann, B. Haasdonk, and M. Ohlberger. Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput., 34(2):937–969, 2012. Joint work with Peter Benner, Lihong Feng, and Suzhou Li.

424

Alexander Zlotnik National Research University Higher School of Economics, RU The Crank-Nicolson scheme with splitting and discrete transparent boundary conditions for the Schrödinger equation on an infinite strip Minisymposium Session TIME: Thursday, 12:00 - 12:30, CO015 The time-dependent Schrödinger equation with several variables is important in quantum mechanics, atomic and nuclear physics, wave physics, nanotechnologies, etc. Often it should be solved in unbounded space domains. In particular, the generalized 2D time-dependent Schrödinger equation with variable coefficients on a semi-infinite strip appears in microscopic description of low-energy nuclear fission dynamics. Several approaches are developed and studied to solve problems of such kind, in particular, see [1]. One of them exploits the so-called discrete transparent boundary conditions (TBCs) at artificial boundaries [2]. Its advantages are the complete absence of spurious reflections, reliable computational stability, clear mathematical background and rigorous stability theory. The Crank-Nicolson finite-difference scheme with the discrete TBCs in the case of a strip or semi-infinite strip was studied in detail in [3, 4, 5]. But the scheme is implicit so that solving of a specific complex system of linear algebraic equations is required at each time level. Efficient methods to solve such systems are well developed by the moment in real but not complex situation. The splitting technique is widely used to simplify solving of the time-dependent Schrödinger and related equations, in particular, see [6]. We apply the Strang-type splitting with respect to the potential to the Crank-Nicolson scheme with rather general approximate TBC in the form of the Dirichlet-to-Neumann map. For the resulting method, we prove the uniform in time L2 -stability under a condition on an operator S in the approximate TBC. To construct the discrete TBC, we consider the splitting scheme on an infinite mesh in the infinite strip. Its uniform in time L2 -stability together with the mass conservation law are proved. We find that an operator Sref in the discrete TBC is the same as for the original Crank-Nicolson scheme in [4], and it satisfies above mentioned condition so that the uniform in time L2 -stability of the resulting method is guaranteed. The operator Sref is written in terms of the discrete convolution in time and the discrete Fourier expansion in direction y perpendicular to the strip. Due to the splitting, an effective direct algorithm using FFT in y is developed to implement the method with the discrete TBC for general potential (while other coefficients are y-independent). The corresponding numerical results on the tunnel effect for rectangular barriers are presented together with the practical error analysis in C and L2 norms confirming the good error properties of the splitting scheme. Notice that the results are rather easily generalized to the case of a multidimensional parallelepiped infinite or semi-infinite in one of the directions. This study is accomplished jointly with B. Ducomet (CEA, France) and I. Zlotnik (MPEI, Russia). The results are presented partly in [7].

References [1] X. Antoine, A. Arnold, C. Besse, M. Ehrhardt and A. Schädle, A review of transparent and artificial boundary conditions techniques for linear and 425

nonlinear Schrödinger equations. Commun. Comp. Phys. 4 (4) (2008) 729796. [2] M. Ehrhardt and A. Arnold, Discrete transparent boundary conditions for the Schrödinger equation. Riv. Mat. Univ. Parma. 6 (2001) 57-108. [3] A. Arnold, M. Ehrhardt and I. Sofronov, Discrete transparent boundary conditions for the Schrödinger equation: fast calculations, approximation and stability. Comm. Math. Sci. 1 (2003) 501-556. [4] B. Ducomet and A. Zlotnik, On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. Part I. Comm. Math. Sci. 4 (2006) 741-766. [5] B. Ducomet and A. Zlotnik, On stability of the Crank-Nicolson scheme with approximate transparent boundary conditions for the Schrödinger equation. Part II. Comm. Math. Sci. 5 (2007) 267-298. [6] C. Lubich, From quantum to classical molecular dynamics. Reduced models and numerical analysis. EMS: Zürich, 2008. [7] B. Ducomet, A. Zlotnik and I. Zlotnik, The splitting in potential Crank-Nicolson scheme with discrete transparent boundary conditions for the Schrödinger equation on a semi-infinite strip. (2013), submitted. http://arxiv.org/abs/1303.3471

426

Walter Zulehner Johannes Kepler University Linz, AT Operator Preconditioning for a Mixed Method of Biharmonic Problems on Polygonal Domains Minisymposium Session CTNL: Tuesday, 11:30 - 12:00, CO015 The first and the second boundary value problem of the biharmonic operator are simple model problems in elasticity for the bending of a clamped plate and a simply supported plate, respectively. For these model problems, mixed second-order formulations are considered which are equivalent to the original fourth-order formulation without additional assumptions on the polygonal domain such as convexity. Based on the mapping properties of the involved operators and their discrete counterparts resulting from a mixed finite element method efficient preconditioners are constructed and numerical experiments are shown.

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Index of Speakers af Klinteberg, Ludvig, 5, 17 Aizinger, Vadym, 4, 18 Akman, Tuğba, 14, 19 Algarni, Said, 16, 20 Amsallem, David, 12, 22 Antil, Harbir, 10, 23 Arjmand, Doghonay, 8, 24 Artina, Marco, 7, 26 Augustin, Christoph, 5, 27 Avalishvili, Gia, 14, 28 Azijli, Iliass, 8, 30 Azzimonti, Laura, 8, 32 Aßmann, Ute, 4, 34

Codina, Ramon, 12, 79 Cohen, Albert, 10, 80 Colciago, Claudia, 13, 81 Crestetto, Anais, 10, 83 Crouseilles, Nicolas, 10, 84 D’Ambrosio, Raffaele, 5, 85 Damanik, Hogenrich, 7, 86 Danilov, Alexander, 4, 88 Davenport, Mark, 4, 89 de la Cruz, Raúl, 4, 90 Debrabant, Kristian, 3, 92 Dede, Luca, 10, 93 Dementyeva , Ekaterina, 14, 94 den Ouden, Dennis, 8, 96 Deparis, Simone, 3, 98 Despres, Bruno, 10, 99 Di Pietro, Daniele, 14, 100 Dimitriu, Gabriel, 14, 102 Dolgov, Sergey, 8, 103 Donatelli, Marco, 10, 104

Bachmayr, Markus, 10, 35 Badia, Santiago, 7, 37 Bai, Yun, 4, 38 Baker, Ruth, 7, 39 Ballani, Jonas, 7, 40 Bartels, Soeren, 7, 41 Basting, Steffen, 7, 42 Bause, Markus, 14, 43 Berger, Lorenz, 5, 45 Berrut, Jean-Paul, 5, 46 Billaud Friess, Marie, 8, 47 Blumenthal, Adrian, 16, 48 Bonelle, Jerome, 7, 49 Bonizzoni, Francesca, 14, 50 Börm, Steffen, 5, 52 Braack, Malte, 3, 53 Budac, Ondrej, 8, 54 Burman, Erik, 3, 55, 7, 56 Bustinza, Rommel, 8, 57

Ehler, Martin, 3, 105 Ehrlacher, Virginie, 7, 106, 4, 107 Einkemmer, Lukas, 12, 108 Elfverson, Daniel, 3, 110 Engblom, Stefan, 14, 112 Engwer, Christian, 3, 114 Ern, Alexandre, 13, 116 Falcó, Antonio, 4, 117 Feistauer, Miloslav, 14, 118 Fishelov, Dalia, 14, 119 Flueck, Michel, 13, 121 Freitag, Melina, 13, 122 Frolov, Maxim, 8, 123 Furmanek, Petr, 16, 124

Caboussat, Alexandre, 7, 58, 5, 59 Caiazzo, Alfonso, 7, 60 Cances, Eric, 3, 61 Cancès, Clément, 3, 62 Cancès, Eric, 7, 63 Capatina, Daniela, 13, 64, 10, 65 Cattaneo, Laura, 5, 66 Cecka, Cris, 3, 68 Cervone, Antonio, 3, 69 Chen, Xingyuan, 10, 70 Chen, Peng, 16, 72 Chinesta, Francisco, 12, 73 Chkifa, Moulay Abdellah, 14, 75 Christophe, Alexandra, 5, 76 Chrysafinos, Konstantinos, 3, 78

Gastaldi, Lucia, 7, 125 Gauckler, Ludwig, 13, 127 Gerbeau, Jean-Frédéric, 13, 128 Gergelits, Tomas, 14, 129 Ghattas, Omar, 12, 131 Giraud, Luc, 10, 132 Golbabaee, Mohammad , 3, 133 Gonzalez, Maria, 8, 134 Gorkem, Simsek, 8, 136 Grandchamp, Alexandre, 5, 138 Grandperrin, Gwenol, 13, 139 428

Greff, Isabelle, 5, 140 Gross, Sven, 7, 142 Grote, Marcus, 12, 144 Guglielmi, Nicola, 13, 146

Kleiss, Stefan, 5, 207 Knobloch, Petr, 16, 208 Kolev, Tzanio, 12, 209 Konshin, Igor, 5, 211 Kosík, Adam, 16, 213 Koskela, Antti, 13, 215 Krahmer, Felix, 3, 216 Kramer, Stephan, 13, 218, 7, 219 Kray, Marie, 14, 220 Kreiss, Gunilla, 5, 222 Krendl, Wolfgang, 16, 224 Kressner, Daniel, 12, 225 Kroll, Jochen, 10, 226 Krukier, Lev, 14, 227 Kucera, Vaclav, 8, 229 Kuzmin, Dmitri, 16, 230

Hachem, Elie, 7, 147 Hadrava, Martin, 8, 148 Hairer, Ernst, 3, 150 Haji-Ali, Abdul-Lateef , 16, 151 Harbrecht, Helmut, 10, 153 Hegland, Markus, 10, 154 Heine, Claus-Justus, 5, 155, 7, 156 Henning, Patrick, 7, 157 Herrero, Henar, 14, 158 Hess, Martin, 14, 160 Hesthaven, Jan, 12, 161 Heumann, Holger, 13, 163 Himpe, Christian, 13, 164 Hintermueller, Michael, 10, 165, 7, 166 Hochbruck, Marlis, 12, 167 Hoel, Haakon, 16, 168 Hoffman, Johan, 10, 170 Holman, Jiří, 16, 171 Huckle, Thomas, 10, 173

Lafitte, Pauline, 10, 231 Lakkis, Omar, 3, 232 Lang, Jens, 13, 233 Lassila, Toni, 12, 234 Le Maitre, Olivier, 12, 235 Lee, Sanghyun, 4, 236 Lee, Jeonghun, 16, 238 Lejon, Annelies, 5, 239 Lilienthal, Martin, 16, 241 Lim, Lek-Heng, 3, 242 Lin, Ping, 12, 243 Linke, Alexander, 8, 244 Long, Quan, 14, 245 Louda, Petr, 16, 246 Luce, Robert, 10, 248 Luh, Lin-Tian , 5, 249 Lukin, Vladimir, 5, 251

Icardi, Matteo, 8, 174 Idema, Reijer, 14, 176 Ishizuka, Hiroki, 5, 178 Jannelli, Alessandra, 8, 180 Janssen, Bärbel, 5, 182 Jaraczewski, Manuel, 14, 183 Jarlebring, Elias, 12, 185 Jiranek, Pavel, 7, 186 John, Lorenz, 16, 187 Jolivet, Pierre, 4, 188 Juntunen, Mika, 5, 189

Macedo, Francisco, 5, 253 Madhavan, Pravin, 8, 254 Maier, Immanuel, 16, 255 Makridakis, Charalambos, 4, 257 Mali, Olli, 14, 258 Matthies, Gunar, 8, 259 Mehl, Miriam, 12, 260 Meinecke, Lina, 14, 262 Melis, Ward, 16, 264 Michiels, Wim, 13, 266 Miedlar, Agnieszka, 13, 268 Migliorati, Giovanni, 5, 269 Miyajima, Shinya, 8, 270 Mula, Olga, 12, 272 Murata, Naofumi, 14, 274 Muslu, Gulcin Mihriye, 8, 275

Kadir, Ashraful, 16, 190 Kalise, Dante, 3, 191 Kamijo, Kenichi, 16, 192 Karasozen, Bulent, 14, 193 Kazeev, Vladimir, 10, 194 Keslerova, Radka, 7, 195, 16, 197 Kestler, Sebastian, 4, 199 Khoromskaia, Venera, 3, 200 Khoromskij, Boris, 8, 201 Kieri, Emil, 16, 202 Kirby, Michael, 14, 204 Kirchner, Alana, 3, 205 Klawonn, Axel, 13, 206 429

Sangalli, Giancarlo , 4, 337 Savostyanov, Dmitry, 8, 338 Scheichl, Robert, 10, 339 Schieweck, Friedhelm, 4, 340 Schillings, Claudia, 12, 341 Schnass, Karin, 3, 342 Schneider, Reinhold , 7, 343 Schratz, Katharina, 13, 344 Sepúlveda, Mauricio, 8, 345 Shapeev, Alexander, 3, 346 Sharma, Natasha, 13, 347 Sheng, Zhiqiang, 3, 348 Simian, Corina, 16, 349 Simoncini, Valeria, 10, 350 Skowera, Jonathan, 3, 351 Smears, Iain, 3, 352 Smetana, Kathrin, 4, 353 Smirnova, Alexandra, 13, 354 Stamm, Benjamin, 3, 356 Steinig, Simeon, 4, 357 Stenberg, Rolf, 10, 358 Stohrer, Christian, 8, 359 Stoll, Martin, 7, 361 Strakos, Zdenek, 10, 362 Sumitomo, Hiroaki, 5, 364 Szepessy, Anders, 16, 366

Nadir, Bayramov, 8, 276 Negri, Federico, 16, 277 Nguyen, Thi Trang, 8, 278 Nikitin, Kirill, 3, 280 Nore, Caroline, 13, 281 Ogita, Takeshi, 8, 282 Ohlberger, Mario, 12, 284 Ojala, Rikard, 8, 286 Olshanskii, Maxim, 10, 287, 13, 288 Ortner, Christoph, 3, 289 Ouazzi, Abderrahim, 8, 290 Ozaki, Katsuhisa, 8, 291 Papez, Jan, 5, 293 Pavarino, Luca , 12, 294 Pekmen, Bengisen, 5, 295 Pena, Juan Manuel, 8, 297 Perotto, Simona, 3, 298 Perugia, Ilaria, 10, 299 Peter, Steffen, 4, 300 Pfefferer, Johannes, 4, 301 Picasso, Marco, 10, 302 Pieper, Konstantin, 3, 303 Pořízková, Petra, 16, 304 Possanner, Stefan, 7, 305 Pousin, Jerome, 5, 306 Powell, Catherine, 7, 308 Prokop, Vladimír, 5, 309 Puscas, Maria Adela, 12, 310

Tamellini, Lorenzo, 14, 367 Tani, Mattia, 16, 369 Tempone, Raul, 12, 370 ten Thije Boonkkamp, Jan, 8, 371 Tesei, Francesco, 16, 372 Tews, Benjamin, 4, 374 Tezer-Sezgin, Münevver, 5, 375 Thalhammer, Mechthild, 7, 377 Tobiska, Lutz, 4, 378, 10, 379 Touma, Rony, 8, 380 Tricerri, Paolo, 13, 382 Tryoen, Julie, 8, 384 Turek, Stefan, 7, 386 Tyrtyshnikov, Eugene, 7, 387

Qingguo Hong, Qingguo, 13, 311 Rademacher, Andreas, 16, 312 Reguly, Istvan, 4, 314 Reigstad, Gunhild Allard, 16, 316 Reimer, Knut, 16, 318 Repin, Sergey, 16, 319 Richter, Thomas, 12, 320 Rozgic, Marco, 14, 321 Rozza, Gianluigi, 13, 323 Rupp, Karl, 3, 325 Ruprecht, Daniel, 5, 327

Uschmajew, André, 3, 388 Van der Zee, Kristoffer, 4, 389 Vandergheynst, Pierre, 4, 390 Vannieuwenhoven, Nick, 14, 391 Varygina, Maria, 5, 392 Vassilevski, Yuri, 4, 394 Verani, Marco, 8, 395 Veroy-Grepl, Karen, 13, 396

Sadovskaya, Oxana, 14, 329 Sadovskii, Vladimir, 14, 331 Saffar Shamshirgar, Davood, 5, 333 Sahin, Mehmet, 7, 334 Samaey, Giovanni, 16, 335 Sandberg, Mattias, 16, 336 430

Vetterli, Martin, 8, 397 Vilanova, Pedro, 14, 398 Vilmart, Gilles, 4, 399, 7, 400 Vohralik, Martin, 13, 401 Voss, Heinrich, 12, 403

Widlund, Olof B., 12, 413 Wiesner, Tobias, 12, 414 Wohlmuth, Barbara, 3, 416 Wollner, Winnifried, 4, 418 Wünsch, Olaf, 7, 419

Wacker, Benjamin, 12, 404 Walker, Shawn, 10, 405 Walloth, Mirjam, 16, 406 Weinmann, Andreas , 4, 408 Weißer, Steffen, 5, 409 Wells, Garth, 4, 411 Wick, Thomas, 12, 412

Yang, Huidong, 13, 421 Yücel, Hamdullah, 8, 422 Zhang, Yongjin, 16, 423 Zlotnik, Alexander, 12, 425 Zulehner, Walter, 7, 427

431