Departement Mathematik

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Zurich colloquium in applied and computational mathematics

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Herbstsemester 2017

Datum / Zeit Referent:in Titel Ort
4. Oktober 2017
16:15-17:15 		Prof. Dr. Bertrand Maury
Universite de Paris Sud
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Zurich Colloquium in Applied and Computational Mathematics

Titel Micro-macro issues in crowd motion modeling
Referent:in, Affiliation Prof. Dr. Bertrand Maury, Universite de Paris Sud
Datum, Zeit 4. Oktober 2017, 16:15-17:15
Ort HG E 1.2
Abstract Modeling collections of living entities like human crowds has become a major challenge among physicists and mathematicians in the last two decades. This domain of investigation shares some features with the modeling of "classical" particle systems, but it also raises new issues from the mathematical standpoint. We shall focus here on very crude models (in terms of individual behavior), assuming everyone tends to selfishly realize a given goal (like exiting a room in fire). The resulting motion is then built as a trade-off between individual tendencies and congestion constraints. This standpoint can be straightforwardly instantiated at both microscopic and macroscopic scales. We shall present how the Wasserstein setting of Optimal Transportation makes it possible to transpose the microscopic framework (differential inclusions in a Hilbert space) at the macroscopic level. Beyond this formal analogy, we shall emphasize the deep differences between the two levels of description. In particular, while the macroscopic model exhibits a good old Laplacian, its microscopic counterpart relies on a bizarre discrete operator which in particular does not verify the maximum principle. We shall explain how the emergence of static jams upstream exits, as well as the so-called "faster is slower effect", can be explained by the very pathological properties of this operator.
Micro-macro issues in crowd motion modelingread_more
HG E 1.2
11. Oktober 2017
16:15-17:15 		Prof. Dr. Dirk Bloemker
University of Augsburg, FRG
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Zurich Colloquium in Applied and Computational Mathematics

Titel Modulation- and Amplitude-Equations for stochastic partial differential equations
Referent:in, Affiliation Prof. Dr. Dirk Bloemker, University of Augsburg, FRG
Datum, Zeit 11. Oktober 2017, 16:15-17:15
Ort HG E 1.2
Abstract Modulation- or Amplitude-Equations are a universal tool to approximate solutions of complicated systems given by partial or stochastic partial differential equations (SPDEs) near a change of stability, when there is no center manifold theory available. Relying on the natural separation of time-scales at the bifurcation thesolution of the original equation is well approximated by the bifurcatingpattern. For the talk we consider for simplicity the one-dimensional stochastic Swift-Hohenberg equation, which acts as a toy model for the convective instability in Rayleigh-Benard convection. On an unbounded spatial domain the amplitude of the dominating pattern is slowly modulated in time and alsoin space. Furthermore, it solves a stochastic Ginzburg-Landau equation perturbed by an additive space-time white noise. Major problems arise due to the weak regularity of solutions and their unboundedness in space, so that the methods from the theory of deterministic modulation equations all fail.
Modulation- and Amplitude-Equations for stochastic partial differential equationsread_more
HG E 1.2
18. Oktober 2017
16:15-17:15 		Prof. Dr. Thomas Kurtz
University of Wisconsin
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Zurich Colloquium in Applied and Computational Mathematics

Titel Particle representations for stochastic partial differential equations
Referent:in, Affiliation Prof. Dr. Thomas Kurtz, University of Wisconsin
Datum, Zeit 18. Oktober 2017, 16:15-17:15
Ort HG E 1.2
Abstract Stochastic partial differential equations arise naturally as limits of finite systems of weighted interacting particles. For a variety of purposes, it is useful to keep the particles in the limit obtaining an infinite exchangeable system of stochastic differential equations for the particle locations and weights. The corresponding de Finetti measure then gives the solution of the SPDE. These representations frequently simplify existence, uniqueness, and convergence results. Beginning with the classical McKean-Vlasov limit, the basic results on exchangeable systems will be discussed along with new applications to SPDEs with boundary conditions.
Particle representations for stochastic partial differential equations read_more
HG E 1.2
25. Oktober 2017
16:15-17:15 		Prof. Dr. Marlis Hochbruck
Karlsruhe Institute of Technology
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Zurich Colloquium in Applied and Computational Mathematics

Titel On the numerical solution of linear Maxwell's equations
Referent:in, Affiliation Prof. Dr. Marlis Hochbruck, Karlsruhe Institute of Technology
Datum, Zeit 25. Oktober 2017, 16:15-17:15
Ort HG E 1.2
Abstract In this talk we present an overview on the numerical solution of time-dependent linear Maxwell's equations with an emphasis on the timeintegration. For the space discretization we consider discontinuos Galerkin methods which can handle complex geometries by using unstructured, possibly locally refined meshes. For the time integration we discuss different options, starting with standard explicit and implicit methods. After a general introduction, our main interest is in problems where the spatial mesh contains only a small number of tiny mesh elements (i.e. elements with a very small diameter) while most of the mesh are coarse. Solving such problems with an explicit time integration scheme requires a constraint on the time step size related to the diameter of the smallest mesh element to ensure stability, the well-known CFL condition. This makes the simulation inefficient, in particular if the number of tiny mesh elements is small compared to the total number of elements. A natural way to overcome this restriction is using implicit time integrators but these come with the expense of having to solve a large linear system in each time step. A more suitable approach consists in treating only the tiny mesh elements implicitly while retaining an explicit time integration for the remaining coarse elements. This results in so-called locally implicit methods. We will show how such methods can be constructed and implemented efficiently, present results of a rigorous error analysis, and close with numerical examples.
On the numerical solution of linear Maxwell's equationsread_more
HG E 1.2
1. November 2017
16:15-17:15 		Prof. Dr. Shi Jin
University of Wisconsin-Madison, U.S.A
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Zurich Colloquium in Applied and Computational Mathematics

Titel Uncertainty quantification for multiscale kinetic equations with uncertain coefficients
Referent:in, Affiliation Prof. Dr. Shi Jin, University of Wisconsin-Madison, U.S.A
Datum, Zeit 1. November 2017, 16:15-17:15
Ort HG E 1.2
Abstract In this talk we will study the generalized polynomial chaos-stochastic Galerkin (gPC-SG) approach to kinetic equations with uncertain coefficients/inputs, and multiple time or space scales, and show that they can be made asymptotic-preserving, in the sense that the gPC-SG scheme preserves various asymptotic limits in the discrete space. This allows the implementation of the gPC methods for these problems without numerically resolving (spatially, temporally or by gPC modes) the small scales. Rigorous analysis, based on hypocoercivity of the collision operator, will be provided for general kinetic equations to prove uniform convergence toward the local or global equilibrium, and the spectral convergence of the gPC-SG method.
Uncertainty quantification for multiscale kinetic equations with uncertain coefficients read_more
HG E 1.2
8. November 2017
16:15-17:15 		Prof. Dr. Richard Craster
Imperial College London
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Zurich Colloquium in Applied and Computational Mathematics

Titel High frequency homogenization: Connecting the Microstructure to the Macroscale
Referent:in, Affiliation Prof. Dr. Richard Craster, Imperial College London
Datum, Zeit 8. November 2017, 16:15-17:15
Ort HG E 1.2
Abstract It is highly desirable to be able to create continuum equations that embed a known microstructure through effective or averaged quantities such as wavespeeds or shear moduli. The methodology for achieving this at low frequencies and for waves long relative to a microstructure is well-known and such static or quasi-static theories are well developed. However, at high frequencies the multiple scattering by the elements of the microstructure, which is now of a similar scale to the wavelength, has apparently prohibited any homogenization theory. Many interesting features of, say, periodic media: band gaps, localization etc occur at frequencies inaccessible to averaging theories. The materials exhibit effective anisotropy and this leads to topical effects such as cloaking/ invisibility, flat lensing, negative refraction and to inducing directional behaviour of the waves within a structure. Recently we have developed an asymptotic approach that overcomes this limitation, and continuum equations are developed, even though the microstructure and wavelength are now of the same order. The general theory will be described and applications to continuum, discrete and frame lattice structures will be outlined. The results and methodology are confirmed versus various illustrative exact/ numerical calculations showing that theory captures, for instance, all angle negative refraction, ultra refraction and localised defect modes.
High frequency homogenization: Connecting the Microstructure to the Macroscaleread_more
HG E 1.2
15. November 2017
16:15-17:15 		Dr. Felix Voigtlaender
TU Berlin
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Zurich Colloquium in Applied and Computational Mathematics

Titel Optimal approximation of piecewise smooth functions using deep ReLU neural networks
Referent:in, Affiliation Dr. Felix Voigtlaender, TU Berlin
Datum, Zeit 15. November 2017, 16:15-17:15
Ort HG E 1.2
Abstract Recently, machine learning techniques based on deep neural networks have significantly improved the state of the art in tasks like image classification and speech recognition. Nevertheless, a solid theoretical explanation of this success story is still missing. In this talk, we will present recent results concerning the approximation-theoretic propertiesof deep neural networks which help to explain some of the characteristics of such networks; in particular we will see that deeper networks can approximate certain classification functions much more efficiently than shallow networks. We emphasize though that these approximation theoretic properties do not explain why simple algorithms like stochastic gradient descent work so well in practice, or why deep neural networks tend to generalize so well; we purely focus on the expressive power of such networks. Precisely, as a model class for classifier functions we consider the class of (possibly discontinuous) piecewise smooth functions for which the different "smooth regions" are separated by smooth hypersurfaces. Given such a function, and a desired approximation accuracy, we construct a neural network which achieves the desired approximation accuracy, where the error is measured in L². We give precise bounds on the required size (in terms of the number of weights) and depth of the network, depending on the approximation accuracy, on the smoothness parameters of the given function, and on thedimension of its domain of definition.
Finally, we show that this size of the networks is optimal, and that networks of smaller depth wouldneed significantly more weights than the deep networks that we construct, in order to achieve the desired approximation accuracy.
Optimal approximation of piecewise smooth functions using deep ReLU neural networksread_more
HG E 1.2
22. November 2017
16:15-17:15 		Dr. Virginie Bonnaillie-Noel
Ecole Normale Superieure
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Zurich Colloquium in Applied and Computational Mathematics

Titel Minimal k-partition for the p-norm of the eigenvalues
Referent:in, Affiliation Dr. Virginie Bonnaillie-Noel, Ecole Normale Superieure
Datum, Zeit 22. November 2017, 16:15-17:15
Ort HG E 1.2
Abstract In this talk,we would like to analyze the connections between the nodal domains of the eigenfunctions of the Dirichlet-Laplacian and thepartitions of the domain by k open sets D_i which are minimal in the sense that the maximum over the D_i's of the groundstate energy of the Dirichlet realization of the Laplacian is minimal. Instead of considering the maximum among the first eigenvalues, we can also consider the p-norm of the vector composed by the first eigenvaluesof each subdomain.
Minimal k-partition for the p-norm of the eigenvaluesread_more
HG E 1.2
29. November 2017
16:15-17:15 		Prof. Dr. Marcus Grote
Departement Mathematik, Universität Basel
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Zurich Colloquium in Applied and Computational Mathematics

Titel High-Order Explicit Local Time-Stepping Methods For Wave Propagation
Referent:in, Affiliation Prof. Dr. Marcus Grote, Departement Mathematik, Universität Basel
Datum, Zeit 29. November 2017, 16:15-17:15
Ort HG E 1.2
Abstract In the presence of complex geometry, adaptivity and mesh refinement are certainly key for the efficient numerical simulation of wave phenomena. Locally refined meshes, however, impose severe stability constraints on any explicit time-marching scheme, where the maximal time-step allowed by the CFL condition is dictated by the smallest elements in the mesh. When mesh refinement is restricted to a small subregion, the use of implicit methods, or a very small time-step in the entire computational domain, are very high a price to pay. Explicit local time-stepping schemes (LTS) overcome the bottleneck due to a few small elements by using smaller time-steps precisely where the smallest elements in the mesh are located. When combined with a finite element discretization in space with an essentially diagonal mass matrix, the resulting time-marching schemes are fully explicit and thus inherently parallel.
High-Order Explicit Local Time-Stepping Methods For Wave Propagationread_more
HG E 1.2
6. Dezember 2017
16:15-17:15 		Prof. Dr. A. Nouy
Ecole Centrale de Nantes
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Zurich Colloquium in Applied and Computational Mathematics

Titel Learning high-dimensional functions with tree-based tensor formats.
Referent:in, Affiliation Prof. Dr. A. Nouy, Ecole Centrale de Nantes
Datum, Zeit 6. Dezember 2017, 16:15-17:15
Ort HG E 1.2
Abstract Tensor methods are among the most prominent tools for the approximation of high-dimensional functions. Such approximation problems naturally arise in statistical learning, stochastic analysis and uncertainty quantification. In many practical situations, the approximation of high-dimensional functions is made computationally tractable by using rank-structured approximations. In this talk, we give an introduction to tree-based (hierarchical) tensor formats and then present adaptive algorithms for the approximation in these formats using statistical methods.
Learning high-dimensional functions with tree-based tensor formats.read_more
HG E 1.2
11. Dezember 2017
16:15-17:15 		Dr. Sarah Eberle
Mathematisches Institut, Universität Tübingen
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Zurich Colloquium in Applied and Computational Mathematics

Titel FEM-BEM coupling for wave-type equations in 3d
Referent:in, Affiliation Dr. Sarah Eberle, Mathematisches Institut, Universität Tübingen
Datum, Zeit 11. Dezember 2017, 16:15-17:15
Ort HG E 22
Abstract In this talk we give an insight in the coupling for interior and exterior problems of wave-type equations as for example the elastic and poroelastic wave equation, where we introduce transparent boundary conditions. In more detail, we start with the background of the Calderon operator, which builds the basis of the stability analysis and the convergence proof for the full discretized problem. After that, we go over to the implementation of the numerical methods and present results of the numerical experiments
FEM-BEM coupling for wave-type equations in 3d read_more
HG E 22
13. Dezember 2017
16:15-17:15 		Prof. Dr. Olaf Steinbach
TU Graz, Österreich
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Zurich Colloquium in Applied and Computational Mathematics

Titel Coercive space-time finite element methods
Referent:in, Affiliation Prof. Dr. Olaf Steinbach, TU Graz, Österreich
Datum, Zeit 13. Dezember 2017, 16:15-17:15
Ort HG E 1.2
Abstract For the numerical solution of time-dependent partial differential equations, we consider space-time finite element methods which allow for an adaptive meshing simultaneously in space and time, and for a parallel solution of the global linear system. For the model problem of the heat equation we present a Galerkin--Petrov variational formulation where the test and ansatz spaces are of the same regularity. To prove the related stability condition, we introduce and discuss a Hilbert type transformation operator. This concept is then also used to derive suitable variational formulations for the wave equation. Numerical results are given, and we will discuss future topics, challenges, and applications.
Coercive space-time finite element methodsread_more
HG E 1.2
20. Dezember 2017
16:15-17:15 		Dr. Gabriel Peyré
Ecole Normale Superieure
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Zurich Colloquium in Applied and Computational Mathematics

Titel Computational Optimal Transport and its Applications
Referent:in, Affiliation Dr. Gabriel Peyré, Ecole Normale Superieure
Datum, Zeit 20. Dezember 2017, 16:15-17:15
Ort HG E 1.2
Abstract Optimal transport (OT) has become a fundamental mathematical tool at the interface between calculus of variations, partial differential equations and probability. It took however much more time for this notion to become mainstream in numerical applications. This situation is in large part due to the high computational cost of the underlying optimization problems. There is however a recent wave of activity on the use of OT-related methods in fields as diverse as computer vision, computer graphics, statistical inference, machine learning and image processing. In this talk, I will review an emerging class of numerical approaches for the approximate resolution of OT-based optimization problems. These methods make use of an entropic regularization of the functionals to be minimized, in order to unleash the power of optimization algorithms based on Bregman-divergences geometry (see [2] for a theoretical analysis and a litterature review). This results in fast, simple and highly parallelizable algorithms, in sharp contrast with traditional solvers based on the geometry of linear programming. For instance, they allow for the first time to compute barycenters (according to OT distances) of probability distributions discretized on computational 2-D and 3-D grids with millions of points [1]. This offers a new perspective for the application of OT in machine learning (to perform clustering or classification of bag-of-features data representations) and imaging sciences (to perform color transfer or shape and texture morphing [6]). These algorithms also enable the computation of gradient flows for the OT metric, and can thus for instance be applied to simulate crowd motions with congestion constraints [4]. We will also discus various extensions of classical OT, such as handling unbalanced transportation between arbitrary positive measures [3] (the so-called Hellinger-Kantorovich/Wasserstein-Fisher-Rao problem) and the computation of OT between different metric spaces (the so-called Gromov-Wasserstein problem) [7, 5].
References: [1] J-D. Benamou, G. Carlier, M. Cuturi, L. Nenna, G. Peyré. Iterative Bregman Projections for Regularized Transportation Problems. SIAM Journal on Scientific Computing, 37(2), pp. A1111-A1138, 2015.
[2] G. Carlier, V. Duval, G. Peyré, B. Schmitzer. Convergence of Entropic Schemes for Optimal Transport and Gradient Flows, to appear in SIAM Journal on Mathematical Analysis, 2017.
[3] L. Chizat, G. Peyré, B. Schmitzer, F-X. Vialard. Scaling Algorithms for Unbalanced Transport Problems. Preprint Arxiv:1607.05816, 2016.
[4] G. Peyré. Entropic Approximation of Wasserstein Gradient Flows. SIAM Journal on Imaging Sciences, 8(4), pp. 2323-2351, 2015.
[5] G. Peyré, M. Cuturi, J. Solomon. Gromov-Wasserstein Averaging of Kernel and Distance Matrices. In Proc. ICML’16, pp. 2664-2672, 2016.
[6] J. Solomon, F. de Goes, G. Peyré, M. Cuturi, A. Butscher, A. Nguyen, T. Du, L. Guibas. Convolutional Wasserstein Distances: Efficient Optimal Transportation on Geometric Domains. ACM Transactions on Graphics (Proc. SIGGRAPH 2015), 34(4), pp. 66:1-66:11, 2015.
[7] J. Solomon, G. Peyré, V. Kim, S. Sra. Entropic Metric Alignment for Correspondence Problems. ACM Transactions on Graphics (Proc. SIGGRAPH 2016), 35(4), pp. 72:1-72:13, 2016.
Computational Optimal Transport and its Applicationsread_more
HG E 1.2

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