Zurich Colloquium in Applied and Computational Mathematics

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Archive 2015

Date / Time

Speaker

Title

Location

4 February 2015
16:15-17:15

Prof. Dr. Martin Schanz
TU Graz

Event Details

Speaker invited by

Stefan Sauter

Abstract

Many engineering problems are coupled field problems. These couplings are not only interface couplings but also volumetric coupled problems. An example for the latter is poroelasticity, where the displacement field is strongly coupled to the pore pressure field. Such kind of coupling result in a coupled set of differential equations to be solved. In the presentation, the three field model for partial saturated poroelasticity is presented. The equations may be used to model the dynamic behavior of soil. As well the saturated case will be shown and its analogy to thermoelasticity. Finally, a collocation boundary element formulation for partial saturated poroelasticity is presented with a numerical realization.

Multi field problems in poroelasticity: Governing equations and boundary element solution

HG E 1.2

18 February 2015
16:15-17:15

Prof. Dr. Claude Bardos
Laboratoire Jacques Louis Lions Paris and University of Paris 7, France

Event Details

Speaker invited by

Siddhartha Mishra

Abstract

It seems that it is in presence of boundary effects that the classical issues of turbulence ie loss of regularity, the appearance of a non trivial Reynolds stress tensor and anomalous energy dissipation are the more visible. Up to know the only general result is a theorem of Kato which connect these different effect. I intend to give some observation on these issues using first the notion of wild solution of De Lellis and Sz\'{e}kelyhidi . Then to show the robustness of these problem to compare zero viscosity limit of the Navier Stokes in presence of boundary effects with macroscopic limit of the Boltzmann equation in the incompressible limit. Most of the material of this talk is contained in a joint paper with Francois Golse and Lionel Paillard.

Appearance of turbulence in the Euler limit with Boundary effects

HG E 1.2

10 March 2015
17:15-18:30

Prof. Dr. Holger Rauhut
RWTH Aachen

Event Details

Speaker invited by

Christoph Schwab

Abstract

Compressive sensing enables accurate recovery of approximately sparse vectors from incomplete information. We apply this principle to the numerical solution of parametric operator equations where the parameter domain is high-dimensional. In fact, one can show that the solution of certain parametric operator equations (parametric PDEs) is analytic in the parameters which can be exploited to show convergence rates for nonlinear (sparse) approximation. Building on this fact, we show that methods from compressive sensing can be used to compute approximations from samples (snapshots) of the parametric operator equation for randomly chosen parameters. In order to make our scheme work, we require a weighted version of standard compressive sensing, for which theory and algorithms have been developed recently. Based on the snapshots obtained by Petrov-Galerkin approximation, coefficients of a polynomial chaos expansion of the parametric solution in terms of tensorized Chebyshev polynomials are computed via weighted l1-minimization. We provide theoretical approximation rates for this scheme. Based on joint works with Christoph Schwab and Rachel Ward.

Solving high-dimensional parametric operator equations via compressive sensing

HG D 1.2

11 March 2015
16:15-17:15

Prof. Dr. Philippe Ciarlet
City University of Hong Kong

Event Details

Speaker invited by

Michel Chipot

Abstract

We describe and analyze an ?intrinsic approach? to the pure Neumann problem of two-dimensional or three-dimensional linearized elasticity, whose novelty consists in considering the strain tensor field as the sole unknown, instead of the displacement vector field as is customary. This approach leads to a well-posed minimization problem of a new type, constrained by a weak form of the classical Saint Venant compatibility conditions. We then describe a natural finite element subspace for approximating this minimization problem in dimension two, which possesses the remarkable feature that the Saint-Venant compatibility conditions can be exactly satisfied by the approximate strains obtained over such a subspace. This discretization of the intrinsic approach thus provides a direct finite element approximation of the strain tensor field, or equivalently, by means of the constitutive equation, a direct finite element approximation of the stress tensor field.

Direct Computation Of The Stresses In Elasticity

HG E 1.2

17 March 2015
16:15-17:15

Karoline Köhler
Humboldt-Universität Berlin

Event Details

Speaker invited by

Stefan Sauter

Abstract

This talk studies the obstacle problem for the Laplace operator and its discretisation with non-conforming Crouzeix-Raviart finite element methods. It focuses on a posteriori error analysis. This analysis relies on the discrete Lagrange multiplier, which is not unique and allows for various choices. The resulting a posteriori error estimator is studied with respect to its reliability and efficiency and the analysis focuses on the terms which involve the discrete Lagrange multiplier. Numerical experiments hightlight these results and present an outlook on adaptive mesh-refinement.

Non-conforming FEM for the obstacle problem

Y27 H28

22 April 2015
16:15-17:15

Dr. Ricardo Ruiz-Baier
University of Lausanne

Event Details

Speaker invited by

Ralf Hiptmair

Abstract

We present an overview of the numerical simulation of the interaction between cardiac electrophysiology, sub-cellular activation mechanisms, and macroscopic tissue contraction; that together comprise the essential elements of the electromechanical function of the human heart. We discuss the development of some mathematical models tailored for the simulation of the cardiac excitation-contraction mechanisms, which are primarily based on nonlinear elasticity theory and phenomenological descriptions of the mechano-electrical feedback. Here the link between contraction and the biochemical reactions at microscales is described by an active strain decomposition model. Then we turn to the mathematical analysis of a simplified version of the model problem consisting in a reaction-diffusion system governing the dynamics of ionic quantities, intra and extra-cellular potentials, and the elastodynamics equations describing the motion of an incompressible material. Under the assumption of linearized elastic behavior and a truncation of the updated nonlinear diffusivities, we are able to prove existence of weak solutions to the underlying coupled system and uniqueness of regular solutions. A finite element formulation is also introduced, for which we establish existence of discrete solutions and show convergence to a weak solution of the original problem. We close with some numerical examples illustrating the convergence of the method and some features of the model.

Modelling, numerics, and analysis of cardiac electromechanics

HG E 1.2

29 April 2015
16:15-17:15

Prof. Dr. Zdzislaw Brzezniak
University of York

Event Details

Speaker invited by

Arnulf Jentzen

Abstract

I will discuss finite element based space-time discretisations of the incompressible Navier-Stokes equations with noise. For the 3-d case, our sequence of numerical solutions converges, for vanishing discretisation parameters, to a weak martingale solution. For the 2-d case, our sequence of numerical solutions converges, to the unique strong solution. We will also discuss rates of convergence. This is based on a joint work with Erich Carelli and Andreas Prohl. At the end I will also mention about a result (joint with H Bessaih and A Millet) about a time splitting method for 2-d Stochastic NSEs.

Finite element based discretisations of the incompressible Navier-Stokes equations with multiplicative random forcing

HG E 1.2

6 May 2015
16:15-17:15

Ivan Oseledets
INM RAS and SkolTech, Moscow, Russia

Event Details

Speaker invited by

Christoph Schwab

Abstract

Low-rank factorizations of matrices and tensors attract a lot of attention in recent years in numerical analysis and linear algebra. They help to reduce dimensionality of the problem and in many cases even break the "curse of dimensionality" in the solution of multiparametric and stochastic problems, modelling of biochemical networks, solution of high-dimensional PDEs and many others. However, the more we look into the application areas, the more we find similar concepts in absolutely different areas. To name only few: Sum-of-product networks in machine learning, probabilistic context-free grammars in natural language processing, weighted finite automata, graphical models in machine learning and statistical physics, tensor networks and quantum information theory and solid state physics. Many of the approaches used in these areas are similar, however many of them are different. In this talk, I will present a brief introduction to these new areas (and point out the results that may be interesting for the numerical analysis) and also present several new results about low-rank approximation of matrices and tensors.

Tensor networks: applications and algorithms

HG E 1.2

13 May 2015
16:15-17:15

Prof. Dr. Steffen Börm
Ch.-Albrechts-Universität Kiel, Germany

Event Details

Speaker invited by

Stefan Sauter

Abstract

We consider multigrid solvers for elliptic partial differential equations with axially-aligned singular perturbations, e.g., the equation \begin{displaymath} -\operatorname{div} \begin{pmatrix} r & \\ & 1/r \end{pmatrix} \operatorname{grad} u(r,\psi) = r f(r,\psi) \end{displaymath} arising when expressing Poisson's equation in polar coordinates. Standard multigrid methods do not show mesh-independent convergence for this kind of problem. A well-known approach is to replace the standard isotropically refined mesh hierarchy by a \emph{semi-coarsened} mesh and to replace standard Gauss-Seidel or Jacobi smoothers by \emph{block smoothers} that treat entire rows of indices simultaneously. In this talk, we present an elegant proof for a mesh-independent bound for the resulting convergence rate. This bound is even robust with respect to varying coefficients. The fundamental idea of the proof is to consider an auxiliary eigenvalue problem that gives rise to a decomposition of the trial space into invariant subspaces. In each of these subspaces, the bilinear form is closely related to a simple one-dimensional problem that can be analyzed to obtain robust bounds.

Tensor multigrid

HG E 1.2

20 May 2015
16:15-17:15

Prof. Dr. Manuel Castro Diaz
University of Malaga, Spain

Event Details

Speaker invited by

Siddhartha Mishra

Abstract

In this talk, we focus on models for sedimentation transport consisting of a shallow water system coupled with a so called Exner equation that described the evolution of the topography. We present different parametrizations of the bedload transport rate and we discuss the hiperbolicity of the proposed models. We also present the discretization of the system in the framework of path-conservative schemes and an efficient implementation on GPUs using non-structured meshes. Finally, some numerical results will be presented.

Bedload sediment transport in shallow flows: models and numerics

HG E 1.2

16 September 2015
16:15-17:15

Prof. Dr. Ivan Graham
University of Bath

Event Details

Speaker invited by	Stefan Sauter

On domain decomposition preconditioners for finite element approximations of the Helmholtz equation using absorption

Y27 H 25

23 September 2015
16:15-17:15

Prof. Dr. Eugene Tyrtyshnikov
Russian Academy of Sciences

Event Details

Speaker invited by

Christoph Schwab

Abstract

We consider special decompositions for multi-dimensional matrices (tensors) that are intrinsically based on low-rank decompositions of some associated matrices and chiefly focus on the ``cross methods'' that construct these decompositions using only very small portion of the entries of those matrices. We consider possible extensions of the maximal volume concept and relation with the following classical problem: given a system of n vectors of size m, find a subsystem consisting of k vectors so that the expansion of any other vector over this subsystem has the coefficients sufficiently small in modulus. The maximal volume principle allows one to find a subsystem of k=m vectors with a guarantee that all expansions have the coefficients in modulus bounded by 1. If we increase k, then smaller coefficients could be obtained. We present different settings of the problem and some new results and discuss applications to the problem of construction of low-rank approximations to matrices and tensors.

Advances in low-rank approximation of tensors and matrices

HG E 1.2

14 October 2015
16:15-17:15

Prof. Dr. Jean-Claude Latché
CEA-IRNS

Event Details

Speaker invited by

Remi Abgrall

Abstract

Finite volume schemes using a staggered arrangement of the unknowns are widely used for the computation of incompressible flows. This talk will present extensions of theses algorithms to compressible flows. Space discretization is based either on the classical Marker And Cell (MAC) scheme or on the low-order finite elements of Rannacher&Turek or Crouzeix&Raviart. Time discretization may be implicit or realized by a fractional step algorithm inspired from pressure-correction techniques. The barotropic Euler equations will first be addressed : the schemes will be described, and the asymptotic preserving property for vanishing Mach numbers will be proved ; more precisely speaking, we will show that, for a given mesh and when the Mach number tends to zero, the discrete density converges to a constant, and the pressure and velocity fields converge to a solution of a standard (inf-sup stable) scheme for incompressible flows. Then the approach will be extended to cope with full (i.e. non-barotropic) Euler equations.

Staggered schemes for all Mach flows

Y27 H 25

28 October 2015
16:15-17:15

Prof. Dr. Thomas Wihler
Uni Bern

Event Details

Speaker invited by

Christoph Schwab

Abstract

In this talk we develop an adaptive procedure for the numerical solution of general, semilinear elliptic problems with possible singular perturbations. Our approach combines both prediction-type adaptive Newton methods and a linear adaptive finite element discretization (based on a robust a posteriori error analysis), thereby leading to a fully adaptive Newton-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach for various examples. An outlook to time-dependent problems will be given also.

Fully Adaptive Newton-Galerkin Methods for Semilinear Problems

Y27 H 25

4 November 2015
16:15-17:15

Volker Mehrmann
TU Berlin

Event Details

Speaker invited by

Habib Ammari

Abstract

In optimal control of physical problems governed by non-stationary partial differential-equations, a typical procedure is to first discretize the forward problem, then to carry out a model order reduction and to perform the optimization on the reduced order model. Although this works very well for many parabolic problems, it quickly gets to its limits (in particular for transport dominated problems) due to the high cost of the model reduction process and the difficulty to capture the important phenomena. In classical control engineering, instead one uses a small surrogate model that is produced from measured or simulated input/output data via a realization and applies the optimal control in a local feedback loop. This approach is very successful in almost all areas of science and technology but also reaches its limits when it is difficult to capture highly nonlinear behavior. In this talk we present a concept that uses a direct Galerkin/Petrov Galerkin discretization of the input/output map that can be adapted to the behavior of the transfer operator as well as the state behavior to obtain reduced order models and to use in optimal control of flow problems. We present some analytic results that show that in this way the error between space, time and transfer operator discretization can be balanced. We also present several numerical examples of successful applications of this technique.

Model reduction vs reduced order modeling in optimal feedback control of flow problems

Y27 H 25

11 November 2015
17:30-18:30

Prof. Dr. Christof Schuette
FU Berlin

Event Details

Speaker invited by

Christoph Schwab

Abstract

Molecular dynamics and related computational methods enable the description of biological systems with all-atom detail. However, these approaches are limited regarding simulation times and system sizes. A systematic way to bridge the micro-macro scale range between molecular dynamics and experiments is to apply coarse-graining (CG) techniques. We will discuss Markov State Modelling, a CG technique that has attracted a lot of attention in physical chemistry, biophysics, and computational biology in recent years. First, the key ideas of the mathematical theory and its algorithmic realization behind Markov State Modelling will be explained, next we will discuss the question of how to apply it to understanding molecular function, and last we will ask whether this may help in designing molecules with prescribed function. All of this will be illustrated by telling the story of the design process of a pain relief drug without concealing the potential pitfalls and obstacles.

Computational Science Distinguished Colloquium: Computational Molecular Design - From Mathematical Theory via High Performance Computing to In Vivo Experiments

ML E 12

18 November 2015
16:15-17:15

Prof. Dr. Sonja Cox
University of Amsterdam

Event Details

Speaker invited by

Arnulf Jentzen

Abstract

Suppose one wishes to evaluate some (deterministic!) statistical quantity derived from the solution of a stochastic differential equation (SDE). Examples of such quantities are: the expected energy of the solution at a certain point in time, the expected maximum, etc. If exact simulations of the solution are not available, the desired statistical quantity can be approximated based on numerically schemes for the SDE. Weak convergence rates for the numerical schemes indicate how good the approximation is. A general rule of thumb is that the optimal weak convergence rates of numerical schemes are generally twice as large as the optimal strong convergence rates, but significantly more difficult to establish -- especially for stochastic differential equations in infinite dimensions, i.e., stochastic partial differential equations. In my talk I will give an overview of the known results on weak convergence rates, explain some proof techniques, and explain what results we believe to obtain by new methods.

Weak convergence of numerical approximations to stochastic (partial) differential equations

Y27 H 25

2 December 2015
16:15-17:15

Prof. Dr. Guido Kanschat
University Heidelberg, Germany

Event Details

Speaker invited by

Christoph Schwab

Abstract

While discontinuous Galerkin (DG) methods had been developed and analyzed in the 1970s and 80s with applications in radiative transfer and neutron transport in mind, it was pointed out later in the nuclear engineering community, that the upwind DG discretization by Reed and Hill may fail to produce physically relevant approximations, if the scattering mean free path length is smaller than the mesh size. Mathematical analysis reveals, that in this case, convergence is only achieved in a continuous subspace of the finite element space. Furthermore, if boundary conditions are not chosen isotropically, convergence can only be expected in relatively weak topology. While the latter result is a property of the transport model, asymptotic analysis reveals, that the forcing into a continuous subspace can be avoided. By choosing a weighted upwinding, the conditions on the diffusion limit can be weakened. It has been known for long time, that the so called diffusion limit of radiative transfer is the solution to a diffusion equation; it turns out, that by choosing the stabilization carefully, the DG method can yield either the LDG method or the method by Ern and Guermond in its diffusion limit. Finally, we will discuss an efficient and robust multigrid method for the resulting discrete problems.

Asymptotically Correct Discontinuous Galerkin Methods for Radiation Transport

Y27 H 25

9 December 2015
16:15-17:15

Dr. Larisa Yaroslavtseva
University of Passau

Event Details

Speaker invited by

Arnulf Jentzen

Abstract

In this talk we consider two classical approximation problems for stochastic differential equations (SDEs) – the pathwise approximation of the solution at the final time and the corresponding quadrature problem, i.e. approximation of the expected value of a function of the solution at the final time. While the majority of results for these problems deals with equations that have globally Lipschitz continuous coefficients, such assumptions are typically not met for real world applications. In recent years a range of positive results for these problems has been established under substantially weaker assumptions on the coefficients such as global monotonicity conditions: new types of algorithms have been constructed that are easy to implement and still achieve a polynomial rate of convergence under these weaker assumptions. In our talk we present negative results. First, we show that there exist SDEs with smooth and bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion solves the pathwise approximation problem with a polynomial rate. Second, we present classes of SDEs with smooth and bounded coefficients such that no algorithm based on finitely many evaluations of the coefficients and all their derivatives solves the quadrature problem with a polynomial rate in the worst case sense with respect to the equations. Our results generalize recent results of M.Hairer, M.Hutzenthaler, A.Jentzen, Loss of regularity for Kolmogorov equations, Ann. Probab. 2015, where a non-polynomial rate has been established for the Euler scheme in the case of the pathwise approximation problem and the Euler Monte Carlo method in the case of the quadrature problem.

On very hard approximation problems for stochastic differential equations

Y27 H 25

16 December 2015
16:15-17:15

Prof. Dr. Josselin Garnier
Université Paris-Diderot

Event Details

Speaker invited by

Habib Ammari

Abstract

We analyze wave propagation in random media in the so-called paraxial regime, which is a special high-frequency regime in which the wave propagates along a privileged axis. We show by multiscale analysis how to reduce the problem to the Ito-Schrodinger stochastic partial differential equation. We also show how to close and solve the moment equations for the random wave field. Based on these results we propose to use correlation-based methods for imaging in complex media and consider two examples: virtual source imaging in seismology and ghost imaging in optics.

The random paraxial wave equation and application to correlation-based imaging

Y27 H 25

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