Program & Abstracts Swiss Numerics Colloquium April 13 ... · nite di erence methods for hyperbolic...

Program & Abstracts

Swiss Numerics Colloquium

April 13, 2012

University of Bern

Conference Location 1

Conference Location

Swiss Numerics Colloquium 2012Mathematisches Institut, Exakte Wissenschaften,

Universitat Bern, Sidlerstrasse 5, 3012 Bern

How to get there: From the platforms at Bern train station, go downstairs, follow thesigns “Universitat”, and take the elevator to the “Grosse Schanze” (top floor). Walk about150 meters north-east, passing the university main building, to the building of “ExakteWissenschaften”.

Swiss Numerics Colloquium, University of Bern, Mathematisches Institut, Friday, April 13, 2012

2 Conference Location

Local Information

Below, you can find a selection of restaurants and hotels in Bern.

Restaurants

1. Casa d’Italia, Buhlstrasse 57, 031 301 90 74

2. Mappamondo, Langgassstrasse 44, 031 301 30 82

3. Beaulieu, Erlachstrasse 3, 031 331 25 25

4. Zebra, Schwalbenweg 2, 031 301 23 40

5. Cavallo Star, Bubenbergplatz 8, 031 311 76 79

6. La Nonna, Speichergasse 29, 031 312 21 80

7. all restaurants at Barenplatz

Hotels

1. Hotel Arabelle, Mittelstrasse 6, 031 301 03 05

2. Hotel National, Hirschengraben 24, 031 381 19 88

3. Hotel Metropole, Zeughausgasse 28, 031 329 94 94

4. Hotel Restaurant Goldener Schlussel, Rathausgasse 72, 031 311 02 16

You can find an overview on a map via this link: http://g.co/maps/6c7nw.


http://g.co/maps/6c7nw

Time Table 3

Time table

09:00 – 09:25 Registration

09:25 – 09:30 Opening

09:30 – 10:20 Plenary talk I

10:20 – 10:50 Coffee break

10:50 – 12:10 Contributed talks I

12:10 – 13:20 Poster session and lunch break (Sandwich lunch)

13:20 – 15:00 Contributed talks II

15:00 – 15:30 Coffee break

15:30 – 16:20 Plenary talk II

16:20 – 16:30 Closing

Plenary talk I, 09:30 – 10:20, Room A6Alexandre Ern, Universite Paris-Est, CERMICS , ENPC, see page 9.Adaptive inexact Newton methods for discretizations of nonlinear diffusion PDEs

First morning session, Room B5

10:50 – 11:10 Simone Deparis, EPFL, CMCS, see page 8.

Modeling and Simulation of Blood Flow

11:10 – 11:30 Michael Gloor, ETHZ, Institute of Fluid Dynamics, see page 9.

Linking linear stability and acoustic theory

for the investigation of jet noise

11:30 – 11:50 Aymen Laadhari, EPFL, CMCS, see page 11.

An Eulerian Finite Element Method for the

Simulation of Fluid-Structure Interaction

11:50 – 12:10 Marcello Righi, ZHAW, IMES, see page 15.

A finite-volume gas-kinetic method for the

solution of the Navier-Stokes equations


4 Time Table

Second morning session, Room B6

10:50 – 11:10 Paola Console, University of Geneva, Section de Math., see page 7.

Symmetric multistep methods for constrained

Hamiltonian Systems

11:10 – 11:30 Soheil Hajian, University of Geneva, Section de Math., see page 11.

An Energy preserving Discontinuous Galerkin Method

for the Vlasov-Poisson system

11:30 – 11:50 Christophe Zbinden, University of Geneva, Section de Math., see page 17.

On conjugate-symplecticity of implicit Runge-Kutta methods

11:50 – 12:10 Ulrik Skre Fjordholm, ETHZ, SAM, see page 9.

High-order accuracy, entropy stability and convergence for

finite difference methods for hyperbolic conservation laws

Third morning session, Room B7

10:50 – 11:10 Christine Tobler, EPFL SB MATHICSE MATHICSE-GE, see page 16.

Low-Rank Methods for High-Dimensional Eigenvalue Problems

11:10 – 11:30 Luca Dede’, EPFL, CMCS, see page 7.

Isogeometric Analysis for topology optimization

in a phase field approach

11:30 – 11:50 Thomas Dickopf, USI, Institute of Computational Science, see page 8.

Analysis and numerics of nodal interpolation and other

transfer operators between non-nested finite element spaces

11:50 – 12:10 Jerome Michaud, University of Geneva, Section de Math., see page 13.

The IDSA and Boltzmann’s equation: Discretization,

comparison and modeling error

Poster session and sandwich lunch 12:10–13:20, Wandelhalle


Time Table 5

First afternoon session, Room B5

13:20 – 13:40 Hossein Gorji, ETHZ, Institute of Fluid Dynamics, see page 10.

An Asymptotic Preserving Numerical Scheme

for a Fokker-Planck Kinetic Model

13:40 – 14:00 Jonas Sukys, ETHZ, SAM, see page 16.

MLMC-FVM for shallow water equations with

uncertain bottom topography

14:00 – 14:20 Francesca Bonizzoni, EPFL, MATHICSE-GE, see page 6.

Equations for the probabilistic moments of the solution of SPDEs

14:20 – 14:40 Giovanni Migliorati, EPFL, SB MATHICSE CSQI, see page 14.

Approximation of Quantities of Interest in stochastic PDEs by the

discrete L2 projection on polynomial spaces with random evaluations

14:40 – 15:00 Konstantinos Zygalakis, EPFL, ANMC, see page 17.

High order weak methods for SDEs based on modified equations

Second afternoon session, Room B6

13:20 – 13:40 Yun Bai, EPFL, ANMC, see page 6.

Reduced basis finite element heterogeneous multiscale method for

elliptic homogenization problems

13:40 – 14:00 Toni Lassila, EPFL, CMCS, see page 12.

Dimensionality reduction for parameter-dependent PDEs in

variable domains

14:00 – 14:20 Bankim C. Mandal, University of Geneva, Section de Math., see page 13.

Dirichlet-Neumann Waveform Relaxation for the Time Dependent

Heat Equation

14:20 – 14:40 Erwin Veneros, University of Geneva, Section de Math., see page 16.

Optimized Schwarz Methods for Maxwell Equations with

Discontinous Coefficients

14:40 – 15:00 Hui Zhang, University of Geneva, Section de Math., see page 17.

Optimized Schwarz methods with overlap for the Helmholtz equation


6 Abstracts

Plenary talk II, 15:30 – 16:20, Room A6Valeria Simoncini, Universita di Bologna, Dipartimento di Matematica, see page 15.Exploring (un)conventional preconditioning strategies for large saddle point algebraic linearsystems

Abstracts

The talks are listed in alphabetical order of the speakers’ names.

1. Reduced basis finite element heterogeneous multiscale method for elliptichomogenization problems

Yun Bai, EPFL MATHICSE ANMC

13:20–13:40

Multiscale problems characterized by data (e.g., conductivity tensor, applied forces,etc.) varying over a wide range of scales, cannot be easily solved by classical numer-ical methods that need mesh resolution down to the finest scales. Departing fromthe classical approach used in the finite element heterogeneous multiscale methodconsisting in solving micro problems on sampling domains at each quadrature pointsof a macro FEM with numerical integration, in this talk the speaker will introducethat interpolation techniques based on the reduced basis methodology (based on anoffline-online strategy) allow to design an efficient numerical method relying only ona small number of accurately computed micro solutions. The speaker will also showthe numerical results with high order finite element methods and for high dimensionalproblems which illustrate the applicability and efficiency of the numerical method.

2. Equations for the probabilistic moments of the solution of SPDEs

Francesca Bonizzoni, Ecole Polytechnique Federale de Lausanne

14:00–14:20

The boundary value problems for PDEs which model many natural phenomena andengineering applications are affected by uncertainty in the input data. One way toeffectively address this issue is to describe the problem data as random variables orrandom fields, so that the deterministic problem turns into a stochastic differentialequation (SPDE). The solution of a SPDE is itself a random field with values in asuitable function space. The simplest approach is Monte Carlo Method. Generally,its convergence rate is slow, so that this method turns to be costly. An alternativetechnique is to derive the moment equations, that is the deterministic equationssolved by the probabilistic moments of the stochastic solution. We take into account


Abstracts 7

steady state linear stochastic partial differential equations. We consider both casesof elliptic equations with stochastic loads and random coefficients. Given completestatistical information on the random input data, the aim of our work is to computethe statistics of the random solution.

3. Symmetric multistep methods for constrained Hamiltonian Systems

Paola Console, University of Geneva

10:50–11:10

A method of choice for the long-time integration of constrained Hamiltonian systemsis the Rattle algorithm: it is symmetric, symplectic and nearly preserves the Hamil-tonian, but it is only of order two. We prove that certain symmetric linear multistepmethods have the same qualitative behaviour and can achieve an arbitrarily highorder with a complexity comparable to that of the Rattle algorithm.

4. Isogeometric Analysis for topology optimization in a phase field approach

Luca Dede’, Chair of Modeling and Scientific Computing (CMCS), MATHICSE,Ecole Polytechnique Federale de Lausanne,

11:10–11:30

Isogeometric Analyis (IGA) is an approximation method for Partial Differential Equa-tions (PDEs) laying on the isoparametric concept for which the same basis used forthe geometrical representation is then used to approximate the unknown solutionfields of the PDEs [Hughes, Cottrell, Bazilevs, 2005]. A common choice in the frame-work of IGA considers a standard Galerkin formulation with Non-Uniform RationalB-splines (NURBS) as basis functions. This facilitates encapsulating the exact, CAD-based geometrical description in the analysis, other then providing several advantagesin terms of accuracy and computational efficiency.

In this talk, we consider the numerical approximation of topology optimization prob-lems in structural engineering applications by means of IGA. In particular, we refor-mulate the topology optimization problem in a phase field framework with a phasefield variable smoothly describing the material properties and providing geometricalinformation on the interfaces between the material and the void. The optimal de-sign is obtained as the steady state solution of the phase transition model, whichcorresponds to a generalized Cahn-Hilliard equation, a PDE involving a fourth-orderspatial differential operator. The numerical solution of the phase field model is car-ried out by means of NURBS-based IGA for which globally high-order continuousbasis functions are used, resulting in an efficient and accurate approximation method.We present numerical results, obtained in an high-performance computing setting,to highlight the effectiveness of the numerical approach.


8 Abstracts

5. Modeling and Simulation of Blood Flow

Simone Deparis, MATHICSE - EPFL

10:50–11:10

When modeling the hemodynamics in the arterial system it is necessary to considerdifferent level of complexity: three dimensional models which includes fluid-structureinteraction are needed to understand the local behavior of the flow, whilst a networkof one dimensional hyperbolic models is able to roughly reproduce the global arterialsystem. Geometrical multiscale modeling allows to account for both the systemic sys-tem and local three-dimensional features. We present our algorithms for the solutionof the discretized system when dealing with three dimensional problems in parallelor with the coupled multiscale network. We then takle the numerical simulation ofblood-flow in physiological and pathological situations using patient specific data –geometries, flow rate, and blood pressure.

6. Analysis and numerics of nodal interpolation and other transfer operatorsbetween non-nested finite element spaces

Thomas Dickopf, Universita della Svizzera italiana, Institute of Computational Sci-ence

11:30–11:50

The question of how to interpolate functions in finite element spaces is as old as thefinite element method itself. Approximation operators, which map a given function toa finite element space, appear frequently in numerical analysis for a variety of reasons.Often the input function is in an infinite dimensional function space. In this talk,though, we focus on the information transfer between finite element spaces associatedwith non-nested meshes. First, we show that the nodal interpolation operator actingbetween spaces of piecewise linear functions of one variable is uniformly H1-stable(with constant one). This holds true without any assumptions on the mesh sizes oron the relations between the meshes [1]. We also give counterexamples for the nodalinterpolation in higher order finite element spaces. In the second part of the talk, weoutline a series of quantitative studies evaluating several local approximations of theglobal L2-orthogonal projection between non-nested finite element spaces [2]. Thenumerical studies in 3D provide estimates of the quantitative differences between arange of transfer operators. The obtained results seem to be largely independent ofthe underlying computational domain. This is demonstrated by several examples.

References:

1. T. Dickopf. Nodal interpolation between first-order finite element spaces in 1D is uni-formly H1-stable. (accepted by Numerical Mathematics and Advanced Applications,Proceedings of ENUMATH 2011, Springer)

2. T. Dickopf, R. Krause. Evaluating local approximations of the L2-orthogonal projectionbetween non-nested finite element spaces. (submitted)


Abstracts 9

7. Adaptive inexact Newton methods for discretizations of nonlinear diffu-sion PDEs

Alexandre Ern, CERMICS

9:30–10:20

We consider nonlinear algebraic systems resulting from numerical discretizations ofnonlinear partial differential equations of diffusion type. To solve these systems, someiterative nonlinear solver, and, on each step of this solver, some iterative linear solverare used. We derive adaptive stopping criteria for both iterative solvers. Both criteriaare based on an a posteriori error estimate which distinguishes the different errorcomponents, namely the discretization error, the linearization error, and the algebraicerror. We stop the iterations whenever the corresponding error does no longer affectthe overall error significantly. Our estimates also yield a guaranteed upper boundon the overall error at each step of the nonlinear and linear solvers. We prove theefficiency and robustness of the estimates with respect to the size of the nonlinearityowing, in particular, to the error measure involving the dual norm of the residual. Ourdevelopments are carried at an abstract level, yielding a general framework. We showhow to apply this framework to the Crouzeix–Raviart nonconforming finite elementdiscretization, Newton linearization, and conjugate gradient algebraic solution, andwe illustrate on numerical experiments for the p-Laplacian the tight overall errorcontrol and important computational savings achieved in our approach. Finally, weshow how to apply our abstract framework to a broad class of discretization methods,with a special focus on discontinuous Galerkin methods.

8. High-order accuracy, entropy stability and convergence for finite differencemethods for hyperbolic conservation laws

Ulrik Skre Fjordholm, ETH Zurich

11:50–12:10

We present entropy stable, high-order accurate finite difference methods for generalsystems of hyperbolic conservation laws. For scalar equations and for certain or-ders of accuracy, a weak bound on the total variation of the solution ensures strongconvergence to a weak solution.

9. Linking linear stability and acoustic theory for the investigation of jetnoise

Michael Gloor, ETH Zurich, Institute of Fluid Dynamics

11:10–11:30

Linear Stability Theory (LST) has classically been applied to study the spatial ortemporal development of flow disturbances. Often a modal approach is used in LST,in which a flow disturbance is decomposed into its eigenmodes. We study the sound


10 Abstracts

radiation from coaxial jets by means of viscous compressible linear stability theory(LST) which yields an eigenvalue problem for the disturbance frequency (temporalstability problem) or the disturbance wavenumber (spatial stability problem). Thelinearized disturbance equations, which lead to the eigenvalue problem, were dis-cretized by using a Chebyshev collocation method and the full spectrum was thenobtained numerically. For viscous compressible flows at high subsonic Mach num-bers, it is possible to identify a continuous spectrum of weakly damped acousticeigenmodes which support noise propagation to the acoustic far-field. The direc-tivity of the sound radiation can be attributed to the specific spectral shape of thedisturbances inside the flow field . The obtained results from solving the eigenvalueproblem are used to study the sound radiation originating from flow disturbancesin coaxial jet flows: By expressing an arbitrary disturbance as a superposition ofeigenmodes it becomes possible to separate the acoustic from the hydrodynamic dis-turbance development. The directivity of the sound radiation can then be obtainedby analyzing the dominant acoustic eigenmodes. To validate the directivity predic-tions obtained from transforming the prescribed flow disturbance into the space ofthe eigenmodes, we compute the impulse response of wave-packet disturbances toobtain the temporal evolution of the hydrodynamic and the acoustic field .

10. An Asymptotic Preserving Numerical Scheme for a Fokker-Planck KineticModel

Hossein Gorji, ETH Zurich

13:20–13:40

In this work we present a numerical scheme for Ito type stochastic differential equa-tions resulting from the Fokker-Planck kinetic model. The advantage of the newscheme is that the computational cost of the time integration is independent of theKnudsen number (which is the ratio between a collisional length scale and a relevantmacroscopic length scale) and thus the scheme is asymptotic preserving. It is shownthat with the new time integration scheme, the collisional scales should not be nec-essarily resolved. Therefore unlike in the conventional direct simulation Monte Carlo(DSMC) method, the new scheme is applicable even in the limit of small Knudsennumbers. Two different stochastic equations are presented for monatomic gas flows,i.e. one with a linear and the other with a cubic drift term. For the linear case theanalytical solution is available, assuming frozen macroscopic coefficients. Howeverfor the nonlinear stochastic model, certain treatments have been made in order toformulate the scheme in an asymptotic preserving form. The comparisons have beenpreformed between the new scheme and conventional DSMC, where the former isrevealed to save computational costs significantly with respect to the latter, at leastfor small Knudsen numbers.


Abstracts 11

11. An Energy preserving Discontinuous Galerkin Method for the Vlasov-Poisson system

Soheil Hajian, Universite de Geneve

11:10–11:30

One of the simplest model problems in the kinetic theory of plasma-physics is theVlasov-Poisson (VP) system with periodic boundary conditions. Such system de-scribes the evolution of a plasma of charged particles (electrons and ions) under theeffects of the transport and self-consistent electric field. We consider a semi-discretenumerical scheme for the approximation of the Vlasov-Poisson system. The methodis based on the coupling of discontinuous Galerkin (DG) approximation to the Vlasovequation and non-conforming finite element approximation for the Poisson problem.We investigate the numerical performance of this scheme in challenging questionssuch as the Landau damping and two stream instability. We study and validate theconservation of physical properties such as Lp-norms, mass and total energy.

12. An Eulerian Finite Element Method for the Simulation of Fluid-StructureInteraction

Aymen Laadhari, CMCS, Ecole Polytechnique Federale de Lausanne

11:30–11:50

Biophysics and biomechanics are two fields where Fluid-Structure Interactions playan important role, both from the modeling and computing points of view. In manyapplications, flow and solid models coexist with biochemical systems. For such prob-lems it is desirable to have an accurate and efficient tool coupling models of differentnature, typically Eulerian for fluids and Lagrangian for solids. In this talk, we aimat providing suitable tools based on an Eulerian Level Set approach for the numer-ical simulation of Fluid-Structure problems arising in biophysics. The first one isconcerned with the modelisation of Red Blood Cells (RBCs). We use the model in-troduced by Canham and Helfrich [2] in which the cost in bending energy is given bythe mean curvature, under the incompressibility and the inextensibility constraints.The second deals with isolated cardiomyocyte rhytmic contraction driven by intracel-lular calcium waves. In the first part we focus our attention on describing the staticequilibrum and the dynamics of a single suspended RBC in a linear shear gradientof a plane flow. Concerning the static equilibrum, a mechanical equilibrium equation(Euler-Lagrange equation) of a RBC membrane under a generalized elastic bendingenergy is obtained [2]. Based on shape optimization tools, this approach is new andmore concise than the tensorial tools used previously for this problem. In dynamics,a saddle-point approach allows us to characterize the solution in a weak formulation,which is discretized using mixed finite elements. Moreover, we focus on the numericalmethod and we present a new algorithm of numerical resolution combining one tech-nique of Lagrange multipliers with an automatic mesh adaptation that ensures the


12 Abstracts

accurate conservation of volumes and surfaces. Thus this algorithm enables to exceedan existing crucial restriction of the Level Set method, that’s to say, the wastes ofmass usually noticed in this kind of problems. Numerical results illustrate the RBCmembrane in the Tumbling and the Tank-Treading well-known regimes. Finally, forthe first time [1], the effect of the inertia terms is elaborated and we show that be-yond a critical value of the number of Reynolds the RBC passes from a Tumbling toa Tank-Treading regime. In the second part, we propose an Eulerian Finite Elementapproximation of a coupled Fluid-Structure Interaction problem arising in the studyof mesoscopic cardiac biomechanics. We simulate the active response of a myocardialcell (here considered as an orthotropic, hyperelastic and incompressible material),its interaction with the propagation of calcium concentrations inside it, and a sur-rounding Newtonian fluid. An orthotropic active strain approach is employed forthe mechanical activation and the deformation of the cell is captured using a LevelSet strategy. Namely, we employ a new algorithm proposed for improving the massconservation of the Level Set method in the Finite Element context. Based on the im-position of additional constraints via Lagrange multipliers (see [3]), the performancesof the proposed method are tested on some academic test cases, and the convergencerate versus the element mesh size are founded to be improved. Moreover, we con-sider an active stra in description of the activation mechanism which is based on theassumption of a multiplicative decomposition of the deformation gradient [4], andwe adapt a model for the calcium-driven mechanical activation. Finally, we reportseveral numerical experiments.

References:

1. A. Laadhari, P. Saramito and C. Misbah, Vesicle tumbling inhibited by inertia, Phys.Fluids 24, 031901 (2012)

2. A. Laadhari, P. Saramito and C. Misbah, On the equilibrium equation for a general-ized biological membrane energy by using a shape optimization approach, Physica D,239:1567-1572 (2010).

3. A. Laadhari, P. Saramito and C. Misbah, Improving the mass conservation of the levelset method, Comptes rendus mathematiques, Ser. I, 348:535-540 (2010).

4. R. Ruiz-Baier, D. Ambrosi, S. Pezzuto, S. Rossi, and A. Quarteroni, Numerical simu-lation of cardiac elemtrome- chanics using active strain models, IUTAM Proceedings,(2012).

13. Dimensionality reduction for parameter-dependent PDEs in variable do-mains

Toni Lassila, CMCS - MATHICSE - EPFL

13:40–14:00

Many problems related to shape optimization, shape identification, uncertainty quan-tification, and inverse problems are formulated as partial differential equations on


Abstracts 13

domains of varying shape. In the case that the topology of the shape is known be-forehand, we use diffeomorphic maps of a fixed reference domain and apply a free-formdeformation based on radial basis functions. We propose a novel approach to rep-resenting shape deformations with low-dimensional parametrizations that proceedsin three steps: (i) adaptive shape parametrization using radial basis functions and agreedy algorithm, (ii) sensitivity analysis to identify the most significant directionsof output variation, and (iii) reparametrization in a low-dimensional linear subspace.

14. Dirichlet-Neumann Waveform Relaxation for the Time Dependent HeatEquation

Bankim C. Mandal, University of Geneva

14:00–14:20

We present a waveform relaxation version of the Dirichlet-Neumann method for thetime dependent heat equation. Like the Dirichlet-Neumann method for steady prob-lems, the method is based on a non-overlapping spatial domain decomposition, andthe iteration involves subdomain solves with Dirichlet boundary conditions followedby subdomain solves with Neumann boundary conditions. However, each subdo-main problem is now in space and time, and the interface conditions are also time-dependent. An analysis using Laplace transforms shows linear convergence for un-bounded spatial domains, except for a very specific choice of the relaxation parameter,for which the method converges in a finite number of steps. A more refined analy-sis on bounded domains reveals then that for this optimal choice of the relaxationparameter, we get superlinear convergence when we consider finite time windows,similar to the case of Schwarz waveform relaxation algorithms. The convergence ratedepends on the length of the subdomains as well as the size of the time window. Forany other choice of the relaxation parameter, convergence is only linear. We illustrateour theoretical results with numerical experiments.

15. The IDSA and Boltzmann’s equation: Discretization, comparison andmodeling error

Jerome Michaud, Universite de Geneve

11:50–12:10

In this talk we present the Boltzmann equation for neutrino transport used in corecollapse supernovae as well as the Isotropic Diffusion Source Approximation (IDSA)of it [1]. The purpose of this talk is to present a numerical treatment of a reducedBoltzmann model problem based on time splitting and finite volumes and revise thediscretization of the IDSA for this problem [2]. Discretization error studies carriedout on the reduced Boltzmann model problem and on the IDSA reveal errors of orderone in both cases. By means of a numerical example, a detailed comparison of thereduced model and the IDSA is performed and interpreted. For this example, the


14 Abstracts

IDSA modeling error with respect to the reduced Boltzmann model is numericallydetermined and localized.

References:

1. M. Liebendorfer, S.C. Whitehouse, and T. Fischer. The Isotropic Diffusion SourceApproximation for Supernova Neutrino Transport. ApJ, 698:1174-1190, 2009.

2. H. Berninger, E. Frenod, M.J. Gander, M. Liebendorfer, J. Michaud, and N. Vasset.A Mathematical Description of the IDSA for Supernova Neutrino Transport, its Dis-cretization and a Comparison with a Finite Volume Scheme for Boltzmann’s Equation.Submitted to: ESAIM Proceedings of CEMRACS 2011.

16. Approximation of Quantities of Interest in stochastic PDEs by the discreteL2 projection on polynomial spaces with random evaluations

Giovanni Migliorati, CSQI-MATHICSE, EPFL, Switzerland

14:20–14:40

This is a joint work with F.Nobile (EPFL), E.von Schwerin (KAUST) and R.Tempone(KAUST). In many PDE models the parameters are not known with enough accu-racy, or they naturally feature randomness and can be treated therefore as randomvariables. The challenge is then to efficiently compute the law of the solution of thePDE or some quantities of interest (outputs), given the probability distribution ofthe random input parameters. We consider cases in which the parameter to solutionmap is smooth, and look for a multivariate polynomial approximation of it (poly-nomial chaos expansion). An approach that has been advocated recently knownunder several names: non-intrusive polynomial chaos expansion, Point Collocation,regression, ... consists in evaluating the solution on randomly chosen parametersand doing a discrete L2 projection on the polynomial space. This problem can beanalyzed in a regression framework with random design. As usual, the regressionfunction minimizes the L2 risk, but here the observations are noise-free evaluationson random points. We consider univariate or multivariate target functions and studythe approximation properties of the random L2 projection with respect to the num-ber of sampling points, the maximum polynomial degree, and the smoothness of thefunction to approximate. We prove optimality estimates (up to a logarithmic factor)when the random points are sampled from bounded random variables with strictlypositive probability density functions. Our analysis of the random projection provesthat the optimal convergence rate is achieved when the number of sampling pointsscales as the square of the dimension of the polynomial space. Moreover, it givesan insight on the role of smoothness and the conditioning of the random projectionoperator in the accuracy and stability of the L2 regression. In this talk we willpresent the application of this methodology to compute Quantities of Interest associ-ated to the solution of stochastic PDEs. We will deal with stochastic coefficients andwith random domains, i.e. domains whose shape is described by random variables.Numerical examples in low and high dimensions will be shown as well.


Abstracts 15

17. A finite-volume gas-kinetic method for the solution of the Navier-Stokesequations

Marcello Righi, Zurich University of Applied Sciences

11:50–12:10

A few gas-kinetic schemes, usually based on the Boltzmann equation modified accord-ing to the BGK collision model, have been proposed in the 1990s as an alternativeto the most popular schemes adopted for the Euler and the Navier-Stokes equations,which invariably assume continuity of the flow or solve a Riemann problem at cellsinterfaces. Higher computational costs have prevented these schemes from acquiringa higher popularity and reaching a higher maturity level. However, the advantageswith respect to more traditional approaches consist in (i) treating discontinuities ina natural way, (ii) coupling the spatial and the temporal evolution of the gas duringa time-step (and for instance trying and reduce the critical role played by limiters),(iii) obtaining combined advective and viscous fluxes, (iv) having a formulation thatclearly separates physical and artificial dissipation and (v) providing a natural non-linear turbulent viscosity based multiscale approach to turbulence modelling. Thisstudy concerns the implementation of the scheme devised by Prof. Xu into a 2Dfinite-volume structured solver in which multigrid acceleration and preconditioningprovide acceptable convergence rate on “stiff” problems, such as those representedby transonic turbulent flow around aircraft airfoils. The work shows two original as-pects: the implementation of a turbulence model and the use of time-accurate fluxes.As a matter of fact, time-dependent terms in fluxes have been so far neglected in theapplication of gas-kinetic schemes to non-trivial benchmarks, being conventional andvalidated (but lower-order) time-stepping schemes preferred. Other original aspectsconcern the reduction of the computational cost of the original model with simplenumerical techniques.

18. Exploring (un)conventional preconditioning strategies for large saddle pointalgebraic linear systems

Valeria Simoncini, Universita di Bologna

15:30–16:20

Symmetric and indefinite block structured matrices often arise after the discretizationof a large variety of application problems, where the block form stems from thepresence of more than one partial differential equation (PDE) in the problem, orfrom an optimization process with constraints, often also involving PDEs. Structure-aware preconditioning strategies have emerged as winning devices for efficiently andoptimally solving the associated large linear systems. In this talk we review variousforms of symmetric definite and indefinite preconditioners. We report on their prosand cons when applied to accelerate the convergence of iterative solvers for linearsystems stemming from certain PDE-constrained optimization problems.


16 Abstracts

19. MLMC-FVM for shallow water equations with uncertain bottom topog-raphy

Jonas Sukys, Seminar for Applied Mathematics, ETH Zurich

13:40–14:00

The initial data and bottom topography, used as inputs in shallow water models,are prone to uncertainty due to measurement errors. We model this uncertaintystatistically in terms of random shallow water equations. We extend the Multi-LevelMonte Carlo (MLMC) algorithm to numerically approximate the random shallowwater equations efficiently. The MLMC algorithm is suitably modified to deal withuncertain (and possibly uncorrelated) data on each node of the underlying topographygrid by the use of a hierarchical topography representation. Numerical experimentsin one and two space dimensions are presented to demonstrate the efficiency of theMLMC algorithm.

20. Low-Rank Methods for High-Dimensional Eigenvalue Problems

Christine Tobler, MATHICSE, EPF Lausanne

10:50–11:10

We consider the symmetric eigenvalue problem Ax = λx, where x ∈ Rn1n2···nd can bereshaped to a tensor X ∈ Rn1×n2×···×nd . Explicit storage of X is not feasible for higherdimensions d. However, in many practical problems, X can be well approximated in alow-rank tensor format, such as our choice of the Hierarchical Tucker decomposition.We propose a low-rank variant of the LOBPCG method, a classical preconditionedeigenvalue solver, and compare it with the DMRG method known from computa-tional quantum physics. Finally, we present a combination of DMRG and low-rankLOBPCG, and illustrate its effectiveness with several numerical experiments.

21. Optimized Schwarz Methods for Maxwell Equations with DiscontinousCoefficients

Erwin Veneros, Universite de Geneve

14:20–14:40

We study non-overlapping Schwarz Methods for solving time-harmonic Maxwell’sequations in heterogeneous media. For this paper we consider Maxwell’s equationsin two dimensions in both the transverse electric and transverse magnetic modeformulations. We first present the classical Schwarz Method for the problem, whichuses characteristic transmission conditions and can therefore be used without overlap.Choosing the interfaces between the subdomains aligned with the discontinuitiesin the coefficients, we prove convergence of the method for a model problem. Wethen define several optimized transmission conditions dependent on the discontinu-ities of the magnetic permeability and the electric permittivity. These conditions are


Posters 17

determined by solving the corresponding min-max problems. We prove asymptoti-cally that the resulting methods converge in certain cases independently of the meshparameter, even though the methods are non-overlapping.

22. On conjugate-symplecticity of implicit Runge-Kutta methods

Christophe Zbinden, University of Geneva

11:30–11:50

The long-time integration of Hamiltonian differential equations requires special nu-merical methods. Symplectic integrators are an excellent choice, but there are situa-tions (e.g., multistep schemes or energy-preserving methods), where symplecticity isnot possible. It is then of interest to study if the methods are conjugate-symplecticand thus have the same long-time behavior as symplectic methods. Algebraic cri-teria for conjugate-symplecticity up to a certain order are presented in terms ofthe coefficients of the B-series. These criteria are then applied to characterize theconjugate-symplecticity of implicit Runge-Kutta methods (Lobatto IIIA and LobattoIIIB) and of energy-preserving collocation methods.

23. Optimized Schwarz methods with overlap for the Helmholtz equation

Hui Zhang, Universite de Geneve

14:40–15:00

Optimized Schwarz methods without overlap for the Helmholtz equation were pro-posed in Gander, Magoules, Nataf, 2002 and further developed in Gander, Halpern,Magoules, 2007. We will present here optimized Schwarz methods with overlap forthe Helmholtz equation. By scaling the mesh size and the wave-number, and usingasymptotic analysis, we are able to give easy to use formulas for calculating the op-timized parameters of the method. The convergence rates are also derived from thecorresponding asymptotic analysis, and we can show an important improvement, dueto the overlap. We also illustrate our theoretical results with numerical experiments.

24. High order weak methods for SDEs based on modified equations

Konstantinos Zygalakis, EPFL

14:40–15:00

In this talk, we describe a new methodology for constructing higher order numericalintegrators for stochastic differential equations, inspired by recent advances in thetheory of modified differential equations. Using this methology we will construct newhigh order weak methods (both implicit and explicit), well suited for stiff stochasticproblems.


18 Posters

Posters

1. Supernova Neutrino Transport by Asymptotic Expansions

Heiko Berninger, Universite de Geneve, Section de Math.

2. Regularity and error estimate for stochastic optimal Robin boundary con-trol problems constrained by advection dominated elliptic equation

Peng Chen, EPFL, CMCS

3. Chebyshev interpolation for nonlinear eigenvalue problems

Cedric Effenberger, EPFL, SB MATHICSE ANCHP

4. Solving the Full Space KKT system: All You Need is Control

Johannes Huber, University of Basel, Math. Inst.

5. Discontinuous Galerkin Finite Element Heterogeneous Multiscale Methodfor Advection-Diffusion Problems with Multiple Scales

Martin Huber, EPFL, ANMC

6. A predictor-corrector Crank-Nicolson method for the reaction-diffusionequation

Felix Kwok, University of Geneva, Section de Math.

7. High-Order Local Time-Stepping with Explicit Runge-Kutta Methods

Teodora Mitkova, University of Fribourg, Dept. of Math.

8. The Parallel QR Algorithm on Distributed Memory Systems

Meiyue Shao, Umea University, Department of Computing Science and HPC2N

9. Parareal Algorithms for Time-Periodic Problem

Bo Song, University of Geneva

10. FE-HMM for the wave equation: Longtime behavior

Christian Stohrer, University of Basel, Math. Inst.

11. The geometry of algorithms using hierarchical Tucker tensors

Bart Vandereycken, EPFL, MATHICSE ANCHP


19

Access to Wireless Internet at University of Bern

Network Adapter Settings

• TCP/IP properties DHCP (get ip address automatically); DNS Server address auto-matically

• supported mode 802.11b/g, 802.11a/n

• SSID: public-unibe

• WEP or WPA (Encryption): allow connection even if not secure

• 803.1x Authentication deactivated

Modes of Network Access and AuthenticationIf you start up your browser, you will automatically be prompted to authenticate. Useyour credentials to log in. Please keep in mind that although the authentication itself isencrypted, the subsequent traffic is NOT, i.e. you should not send passwords or otherconfidential data over this connection (wireless!). Accessible services to destinations insidethe University campus are restricted to HTTP, HTTPS and eMail. Access to other servicesfrom servers residing inside the network of the University require the use of the VPN-Clientand are reserved to members of the University. To the outside world there is no restrictionfor most applications.

Upload of Presentations

In order to avoid tedious switching between different laptops for your talks, we will providea laptop from which all presentations are started. Therefore, we kindly ask you to uploadyour presentation during a break before your session.

Support

We kindly acknowledge the financial support of the Swiss Academy of Sciences and of theStiftung zur Forderung der mathematischen Wissenschaften in der Schweiz.


20 List of Participants

List of Participants

Abdulle Assyr EPFLAmrein Mario University of BernAndreev Roman ETHZ, SAM

Bai Yun EPFL, MATHICSE, ANMCBerninger Heiko University of GenevaBesson Olivier University of NeuchatelBlumenthal Adrian EPFLBonizzoni Francesca EPFLBudac Ondrej EPFL

Chen Peng EPFLCohen David KIT KarlsruheConsole Paola University of GenevaCorti Paolo ETHZ, SAM

Dede’ Luca EPFL, CMCSDeparis Simone EPFL, MATHICSEDickopf Thomas USI, Institute of Computational Science

Effenberger Cedric EPFLErn Alexandre Universite Paris-Est

Fjordholm Ulrik Skre ETHZFlueck Michel EPFL, MATHICSE, ASN

Gander Martin University of GenevaGaudio Loredana University of BaselGauthier Carl-Erik University of NeuchatelGermann Philipp ETHZ, D-BSSEGloor Michael ETHZ, Institute of Fluid DynamicsGorji Hossein ETHZGrella Konstantin ETHZ, SAMGrote Marcus University of BaselGutknecht Martin ETHZ

Hairer Ernst University of GenevaHajian Soheil University of GenevaHarbrecht Helmut University of BaselHiltebrand Andreas ETHZ, SAMHiptmair Ralf ETHZ, SAMHoffmann Christian retired, formerly WSL BirmensdorfHousley Brian GCCSHuber Martin EPFLHuber Johannes University of Basel


List of Participants 21

Janssen Barbel University of Bern

Kiriyanthan Silja University of Basel, Medical Image Analysis CenterKressner Daniel EPFLKwok Felix University of Geneva

Laadhari Aymen EPFL, CMCSLassila Toni EPFL, CMCS, MATHICSELaurmaa Viljami EPFLLiu Xin

Mandal Bankim C. University of GenevaMehlin Michaela University of BaselMichaud Jerome University of GenevaMigliorati Giovanni EPFL, CSQI, MATHICSEMitkova Teodora University of Fribourg, Department of MathematicsMitrou Giannoula University of Basel

Niessner Herbert Independent

Parsania Asieh University of ZurichPeters Michael University of BaselPicasso Marco EPFL

Righi Marcello ZHAW

Schneebeli Hans Rudolf Kantonssschule BadenSchwab Christoph ETHZ, SAMShao Meiyue Umea University, Dept. of Computing Science, HPC2NSiebenmorgen Markus University of BaselSimoncini Valeria Universita di BolognaSong Bo University of GenevaStohrer Christian University of BaselStraubhaar Regis University of NeuchatelSukys Jonas ETHZ, SAM

Tobler Christine EPFL, MATHICSETokareva Svetlana ETHZ, SAMTorabi Parvin University of Zurich, Mathematics Department

Ungricht Heinz ZHAW SoEUtzinger Manuela University of Basel

Vandereycken Bart EPFL, MATHICSE, ANCHPVeneros Erwin University of Geneva

Waldvogel Joerg ETHZ, SAMWeymuth Monika University of ZurichWihler Thomas University of Bern


22 List of Participants

Wirz Marcel University of Bern

Zbinden Christophe University of GenevaZhang Hui University of GenevaZygalakis Konstantinos EPFL


Program & Abstracts Swiss Numerics Colloquium April 13 ... · nite di erence methods for hyperbolic...

Documents

Transcript of Program & Abstracts Swiss Numerics Colloquium April 13 ... · nite di erence methods for hyperbolic...