Zurich colloquium in applied and computational mathematics

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Autumn Semester 2019

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Speaker

Title

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18 September 2019
16:15-17:15

Prof. Dr. Erik Burman
University College London

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Ill-posed problems and stabilized Finite Element Methods
Speaker, Affiliation	Prof. Dr. Erik Burman, University College London
Date, Time	18 September 2019, 16:15-17:15
Location	HG E 1.2
Abstract	In this talk we will consider some ill-posed elliptic equations and their discretization using Finite Element Methods. The standard approach to ill-posed problems is to regularize the continuous problem so that existence and uniqueness is guaranteed. The regularized problem can then be solved using standard Finite Element Methods. When using this strategy, in order to optimize accuracy, the regularization parameter must be chosen as a function both of the stability properties of the ill-posed problem, the mesh parameter and perturbations in data. Here we will propose a different approach [1], where the ill-posed pde is discretized in an optimization framework, prior to regularization. To ensure discrete well-posedness we add stabilizing terms to the formulation, drawing on experience from stabilized FEM and discontinuous Galerkin methods. The error in the resulting Finite Element reconstructions is then analyzed using Carleman estimates on the continuous problem. This results in approximations that are optimal with respect to the approximation order of the Finite Element space and the stability of the computed quantity. The mesh parameter here plays the role of the regularization parameter. Mesh resolution can be chosen independently of the stability properties of the physical problem, but must match perturbations in data, in a way made explicit in the estimates. Some examples of problems analyzed in this framework will be presented, selectedfrom recent work on the Helmholtz equation [4], the convection diffusion equation [5], Stokes'equations [2] and Darcy's equation [3]. Keywords: ill-posed problems, data assimilation, stabilized Finite Element Methods Mathematics Subject Classifications (2010): 65N30, 35R25, 65N20. [1] E. Burman. Stabilised Finite Element Methods for ill-posed problems with conditional stability. Building bridges: connections and challenges in modern approaches to numerical partial differential equations, 93127,Lect. Notes Comput. Sci. Eng., 114, Springer, [Cham], 2016., Dec. 2015. [2] E. Burman, P. Hansbo, Stabilized nonconforming Finite Element Methods for data assimilation in incompressible flows. Math. Comp. 87, no. 311, 2018. [3] E. Burman, M. G. Larson, L. Oksanen. Primal dual mixed Finite Element Methods for the elliptic Cauchy problem, arXiv:1712.10172, Siam J. Num. Anal., to appear, 2018. [4] E. Burman, M. Nechita, L. Oksanen. Unique continuation for the Helmholtz equation using stabilized Finite Element Methods. Journal de Mathematiques Pures et Appliquees,https://doi.org/10.1016/j.matpur.2018.10.003. [5] E. Burman, M. Nechita, L. Oksanen. A stabilized Finite Element Method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime. arXiv:1811.00431, 2018.

Ill-posed problems and stabilized Finite Element Methodsread_more

HG E 1.2

25 September 2019
16:15-17:15

Prof. Dr. Wolfgang Hackbusch
Max-Planck-Institut für Mathematik

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Computation of Best Exponential Sums Approximating 1/x in the Maximum Norm
Speaker, Affiliation	Prof. Dr. Wolfgang Hackbusch, Max-Planck-Institut für Mathematik
Date, Time	25 September 2019, 16:15-17:15
Location	HG E 1.2
Abstract	Exponential sums consist of the terms a_i*exp(-b_ix). Such approximations for the functions 1/x and 1/sqrt(x) are of particular interest. We give examples of their use in quantum chemistry and tensor calculus. The best approximation of 1/x with respect to the maximum norm is theoretically well understood, and rather sharp error estimates are known. The error decays exponentially in the number of terms. The optimal exponential sum is characterised by the equioscillation property. The Remez algorithm is the standard method in the case of polynomial approximation. The existing literature about the numerical computation of the best approximation by exponential sums shows that all authors faced severe numerical difficulties. This may lead to the wrong impression that the problem is illposed. In the lecture a stable method is described which is used to compute best approximations for various parameters up to 57 terms. Literature: W. Hackbusch: Computation of best L^{\infty} exponential sums for 1/x by Remez' algorithm. Comput. Vis. in Sci. (2019) 20:1-11

Computation of Best Exponential Sums Approximating 1/x in the Maximum Normread_more

HG E 1.2

2 October 2019
16:15-17:15

Dr. Carolina Urzua-Torres
University of Oxford

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	A new approach to Space-Time Boundary Integral Equations for the Wave Equation
Speaker, Affiliation	Dr. Carolina Urzua-Torres, University of Oxford
Date, Time	2 October 2019, 16:15-17:15
Location	HG E 1.2
Abstract	Space-time discretization methods are becoming increasingly popular, since they allow adaptivity in space and time simultaneously, and can use parallel iterative solution strategies for time-dependent problems. However, in order to exploit these advantages, one needs to have a complete numerical analysis of the corresponding Galerkin methods. Different strategies have been used to derive variational methods for the time domain boundary integral equations for the wave equation. The more established and succesful ones include weak formulations based on the Laplace transform, and also time-space energetic variational formulations. However, their corresponding numerical analyses are still incomplete and present difficulties that are hard to overcome, if possible at all. As an alternative, we pursue a new approach to formulate the boundary integral equations for the wave equation, which aims to provide the missing mathematical analysis for space-time boundary element methods. In this talk, I will give a short introduction to boundary element methods; briefly explain the current formulations for the wave equation; and discuss the new approach and our preliminary results.

A new approach to Space-Time Boundary Integral Equations for the Wave Equationread_more

HG E 1.2

7 October 2019
16:15-17:15

Prof. Dr. Gitta Kutyniok
TU Berlin

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Deep Learning meets Modeling: Taking the Best out of Both Worlds
Speaker, Affiliation	Prof. Dr. Gitta Kutyniok, TU Berlin
Date, Time	7 October 2019, 16:15-17:15
Location	HG G 19.2
Abstract	Inverse problems in imaging such as denoising, recovery of missing data, or the inverse scattering problem appear in numerous applications. However, due to their increasing complexity, model-based methods are often today not sufficient anymore. At the same time, we witness the tremendous success of data-based methodologies, in particular, deep neural networks for such problems. However, pure deep learning approaches often neglect known and valuable information from the modeling world and also currently still lack a profound theoretical understanding. In this talk, we will provide an introduction to this problem complex and then focus on the inverse problem of computed tomography, where one of the key issues is the limited angle problem. For this problem, we will demonstrate the success of hybrid approaches. We will develop a solver for this severely ill-posed inverse problem by combining the model-based method of sparse regularization by shearlets with the data-driven method of deep learning. Our approach is faithful in the sense that we only learn the part which cannot be handled by model-based methods, while applying the theoretically controllable sparse regularization technique to all other parts. We further show that our algorithm significantly outperforms previous methodologies, including methods entirely based on deep learning. Finally, we will discuss how similar ideas can also be used to solve related problems such as detection of wavefront sets.

Deep Learning meets Modeling: Taking the Best out of Both Worldsread_more

HG G 19.2

30 October 2019
16:15-17:15

Prof. Dr. Ian Hawke
University of Southhampton

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Numerical simulations of Gravitational Waves from Neutron Stars
Speaker, Affiliation	Prof. Dr. Ian Hawke, University of Southhampton
Date, Time	30 October 2019, 16:15-17:15
Location	HG E 1.2
Abstract	In late 2017 the merger of two neutron stars was detected both in gravitational and electromagnetic waves. This gave unprecedented information on the behaviour of objects more massive than the sun, squashed into the size of a city, with magnetic fields many orders of magnitude larger than Earth's. As more detections come in, we will need to significantly improve the models we use to describe these events, and the numerical methods used to simulate them. Here I will discuss the difficulties in coupling Einstein's theory of General Relativity to ever more complex matter models, with a focus on recent multi-phase descriptions of charged species needed to describe non-ideal elastic and magnetic behaviour. This necessarily requires non-conservative numerical methods for the matter, numerically enforced constraints for the magnetic fields, and adaptive techniques to cover the large range of scales involved.

Numerical simulations of Gravitational Waves from Neutron Stars read_more

HG E 1.2

4 December 2019
16:15-17:15

Gregor Gassner
Universität zu Köln

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Splitform Discontinuous Galerkin for the ideal MHD equations: energy, Lorentz force, entropy and divergence B
Speaker, Affiliation	Gregor Gassner, Universität zu Köln
Date, Time	4 December 2019, 16:15-17:15
Location	HG E 1.2
Abstract	In this talk, we show how to construct a discontinuous Galerkin type discretisation of the ideal MHD equations based on first principles. By carefully choosing the form of the PDEs (divergence, advective, splitform, etc) it is possible to design a compatible discretisation where e.g. kinetic energy is preserved, with the right Lorentz force behavior, where we recover a discrete entropy evolution and where zero divergence of the B field is satisfied discretely. We will demonstrate these properties for a 3D ideal MHD test case simulated with the open source framework FLUXO (github.com/project-fluxo).

Splitform Discontinuous Galerkin for the ideal MHD equations: energy, Lorentz force, entropy and divergence Bread_more

HG E 1.2

11 December 2019
16:15-17:15

Prof. Dr. Steffen Börm
Christian-Albrechts-Universität zu Kiel

Event Details

Zurich Colloquium in Applied and Computational Mathematics

Title	Fast Boundary Element Methods for Electrostatic Field Computations
Speaker, Affiliation	Prof. Dr. Steffen Börm, Christian-Albrechts-Universität zu Kiel
Date, Time	11 December 2019, 16:15-17:15
Location	HG E 1.2
Abstract	We consider the computation of electrostatic potentials by the boundary element method. In order to obtain O(h²) convergence of discrete solutions, we have to employ piecewise linear basis functions and piecewise quadratic parametrizations of the surface. Constructing the data-sparse approximations of the integral operators required for high accuracies poses several challenges: methods like ACA or GCA require the computation of individual matrix entries, and since the supports of basis functions are spread across multiple triangles, this computation is far more computationally expensive than for simple discontinuous basis functions. Alternative techniques like HCA allow us to significantly reduce the computational work. Another challenge is the parametrization of the curved triangles: the Gramian and the normal vector are no longer constant on each triangle, but have to be computed in each quadrature point, and this increases the necessary work even further. Combining efficient quadrature techniques with HCA matrix compression, algebraic coarsening and recompression, and Krylov solvers allows us to handle surface meshes with up to 18 million triangles on relatively affordable servers while preserving the theoretically predicted convergence rate of the underlying discretization scheme.

Fast Boundary Element Methods for Electrostatic Field Computationsread_more

HG E 1.2

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