Zurich Colloquium in Applied and Computational Mathematics

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Archive 2022

Date / Time

Speaker

Title

Location

16 March 2022
16:30-18:00

Prof. Dr. Michael Feischl
TU Wien

Event Details

Speaker invited by

Prof. Dr. Christoph Schwab

Abstract

We prove new optimality results for adaptive mesh refinement algorithms for non-symmetric, indefinite, and time-dependent problems by proposing a generalization of quasi-orthogonality which follows directly from the inf-sup stability of the underlying problem. This completely removes a central technical difficulty in modern proofs of optimal convergence of adaptive mesh refinement algorithms. The main technical tools are new stability bounds for the LU-factorization of matrices together with a recently established connection between quasi-orthogonality and matrix factorization.

Towards optimal adaptivity for time-dependent problems

Y27 H 35/36
Zoom Meeting

23 March 2022
16:30-18:00

Prof. Dr. Walter Boscheri
Department of Mathematics and Computer Science, University of Ferrara

Event Details

Speaker invited by	Rémi Abgrall

On the construction of conservative semi-Lagrangian IMEX advection schemes for multiscale time dependent PDE

Y27 H 35/36
Zoom Meeting

6 April 2022
16:30-18:00

Dr. Martin Licht
EPF Lausanne

Event Details

Speaker invited by

Prof. Ralf Hiptmair

Abstract

We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Clément interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert Lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over simplicial triangulations.

Local finite element approximation of Sobolev differential forms

Y27 H 35/36
Zoom Meeting

11 May 2022
16:15-17:15

Prof. Dr. Barbara Verfürth
Department of Mathematics, KIT

Event Details

Speaker invited by

Stefan Sauter

Abstract

Many applications, such as geophysical flow problems or scattering from Kerr-type media, require the combination of nonlinear material laws and multiscale features, which together pose a huge computational challenge. In this talk, we discuss how to construct a problem-adapted multiscale basis in a linearized and localized fashion for nonlinear problems such as the quasilinear diffusion equation or the nonlinear Helmholtz equation. For this, we will adapt two different perspectives: (a) determining a fixed multiscale space for the nonlinear problem or (b) adaptively and iteratively updating the multiscale space during an iteration scheme for the nonlinear problem. We prove optimal error estimates for the corresponding generalized finite element methods. In particular, neither higher regularity of the exact solution nor structural properties of the coefficients such as scale separation or periodicity need to be assumed. Numerical examples show very promising results illustrating the theoretical convergence rates.

Numerical homogenization for nonlinear multiscale problems

Y27 H 35/36
Zoom Meeting

19 September 2022
16:30-17:30

Prof. Dr. Björn Engquist
Oden Inst. Texas

Event Details

Speaker invited by

Prof. Dr. Ralf Hiptmair

Abstract

We will develop a new stochastic gradient descent algorithm. By adaptively controlling the variance in the noise term based on the objective function value we can prove global algebraic convergence rate. Earlier results only gave a logarithmic rate. The focus will mainly be on algorithms where the stochastic component is added for global convergence rather than when sampling is used for efficient approximation of the objective or loss function. We will also see that this methodology extends to a gradient free setting.

Globally convergent stochastic gradient descent

HG D 1.2

21 September 2022
16:30-17:30

Dr. Nils Vu
Max Planck Inst. for Gravitational Physics

Event Details

Speaker invited by

Prof. Dr. Christoph Schwab

Abstract

Numerical simulations of merging black holes and neutron stars are essential for the emerging era of gravitational-wave astronomy, but computationally very challenging. Discontinuous Galerkin (DG) methods and a task-based approach to parallelism help us scale these simulations to supercomputers. In this seminar I present our discontinuous Galerkin scheme for the elliptic Einstein constraint equations of general relativity, and applications to problems involving black holes. Our numerical scheme accommodates curved manifolds, nonlinear boundary conditions, and hp-nonconforming meshes. Our generalized internal-penalty numerical flux and our Schur-complement strategy of eliminating auxiliary degrees of freedom make the scheme compact without requiring equation-specific modifications. I also outline our strategy for solving the DG-discretized elliptic problems effectively on supercomputers.

Simulating black holes with discontinuous Galerkin methods

HG E 1.2
Zoom

3 October 2022
15:15-16:15

Prof. Dr. Hrushikesh Mhaskar
Claremont Graduate University, USA

Event Details

Speaker invited by

Prof. Dr. Siddhartha Mishra

Abstract

Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator between metric spaces X and Y. We study the problem of determining the degree of approximation of a such operators on a compact subset KX X using a finite amount of information. If F : KX ! KY, a well established strategy to approximate F(F) for some F 2 KX is to encode F (respectively, F(F)) in terms of a finite number d (respectively m) of real numbers. Together with appropriate reconstruction algorithms (decoders), the problem reduces to the approximation of m functions on a compact subset of a high dimensional Euclidean space Rd, equivalently, the unit sphere Sd embedded in Rd+1. The problem is challenging because d, m, as well as the complexity of the approximation on Sd are all large, and it is necessary to estimate the accuracy keeping track of the inter-dependence of all the approximations involved. In this paper, we establish constructive methods to do this efficiently; i.e., with the constants involved in the estimates on the approximation on Sd being O(d1=6). We study different smoothness classes for the operators, and also propose a method for approximation of F(F) using only information in a small neighborhood of F, resulting in an effective reduction in the number of parameters involved.To further mitigate the problem of large number of parameters, we propose prefabricated networks, resulting in a substantially smaller number of effective parameters. The problem is studied in both deterministic and probabilistic settings.

Local approximation of operators

HG G 19.1

5 October 2022
16:30-17:30

Dr. Vesa Kaarnioja
Freie Universität Berlin

Event Details

Speaker invited by

Prof. Dr. Ch. Schwab

Abstract

We describe a fast method for solving elliptic PDEs with uncertain coefficients using kernel-based interpolation over a rank-1 lattice point set. By representing the input random field of the system using a model proposed by Kaarnioja, Kuo, and Sloan (2020), in which a countable number of independent random variables enter the random field as periodic functions, it is shown that the kernel interpolant can be constructed for the PDE solution (or some quantity of interest thereof) as a function of the stochastic variables in a highly efficient manner using fast Fourier transform. The method works well even when the stochastic dimension of the problem is large, and we obtain rigorous error bounds which are independent of the stochastic dimension of the problem. We also outline some techniques that can be used to further improve the approximation error. This talk is based on joint work with Yoshihito Kazashi, Frances Kuo, Fabio Nobile, and Ian Sloan.

Fast kernel interpolation over lattice point sets with application to uncertainty quantification

HG E 1.2

12 October 2022
16:30-17:30

Prof. Dr. Alex Townsend
Cornell University, USA

Event Details

Speaker invited by

Prof. Dr. Rima Alaifari

Abstract

Can one learn a differential operator from pairs of solutions and righthand sides? If so, how many pairs are required? These two questions have received significant research attention in partial differential equation (PDE) learning. Given input-output pairs from an unknown elliptic or parabolic PDE, we will derive a theoretically rigorous scheme for learning the associated Green's function. By exploiting the hierarchical low-rank structure of Green’s functions and randomized linear algebra, we will have a provable learning rate. Along the way, we will develop essential new Green's function theory associated with parabolic PDEs and a more general theory for the randomized singular value decomposition.

Learning Green's functions associated with elliptic and parabolic PDEs

HG E 1.2

19 October 2022
16:30-17:30

Prof. Dr. Jean-François Remacle
Université catholique de Louvain

Event Details

Speaker invited by

Prof. Dr. Rémi Abgrall

Abstract

In this presentation, we develop an innovative approach - X-MESH - to overcome a major difficulty associated with numerical simulation in engineering: we aim to provide a revolutionary way to track physical interfaces in finite element simulations. The idea is to use so-called extreme mesh deformations. This new approach should allow low computational cost simulations as well as high robustness and accuracy. X-MESH is designed to avoid the pitfalls of current ALE methods by allowing topological changes on fixed mesh. The key idea of X-MESH is to allow elements to deform until they reach a zero measure. For example, a triangle can deform into an edge or even a point. This idea is rather extreme and completely revisits the interaction between the meshing community and the computational community, which for decades have been trying to interact through beautiful meshes.
In this talk, we will focus on both the mathematical issues related to the use of zero-measure elements and the X-MESH resolution scheme. Several applications will be targeted: the Stefan model of phase change, two-phase flows and contact between deformable solids.

The X-MESH method for capturing interfaces

HG E 1.2

26 October 2022
16:30-17:30

Prof. Dr. Kristin Kirchner
TU Delft

Event Details

Speaker invited by

Prof. Dr. Christoph Schwab

Abstract

Gaussian processes play an important role in statistics for making inference about spatial or spatiotemporal data. Traditionally, the dependence structures of these random processes (in space or space-time) are defined via their covariance kernels. Since the computational costs of these kernel-based approaches for applications such as predictions are, in general, cubic in the number of data points, a vibrant research area has evolved, where various methods for “big data” are proposed. In the last decade, the Stochastic Partial Differential Equation (SPDE) approach has proven to be very efficient for tackling the conflict between limited computing power and desired modeling capabilities. Motivated by a well-known relation between the Gaussian Matérn class and fractional-order SPDEs, the key idea of this approach is to define Gaussian processes as solutions to appropriate SPDEs and to use efficient numerical methods, such as the Finite Element Method (FEM) or wavelets, for approximating them. In this talk I will give an introduction to the (spatial) SPDE approach and discuss several recent developments, in particular with regard to the quality and computational costs of FEM approximations in statistical applications. Finally, I will give an outlook on spatiotemporal models which are based on SPDEs involving fractional powers of parabolic space-time differential operators. This talk is based on joint works with David Bolin, Sonja Cox, Lukas Herrmann, Mihály Kovács, Christoph Schwab and Joshua Willems.

The SPDE approach for Gaussian processes

HG E 1.2

9 November 2022
16:30-17:30

Dr. Yunan Yang
ETH Zürich

Event Details

Speaker invited by

Prof. Dr. Ralf Hipmtair

Abstract

Many models in machine learning and PDE-based inverse problems exhibit intrinsic spectral properties, which have been used to explain the generalization capacity and the ill-posedness of such problems. In this talk, we discuss weighted training for computational learning and inversion with noisy data. The highlight of the proposed framework is that we allow weighting in both the parameter space and the data space. The weighting scheme encodes both a priori knowledge of the object to be learned and a strategy to weight the contribution of training data in the loss function. We demonstrate that appropriate weighting from prior knowledge can improve the generalization capability of the learned model in both machine learning and PDE-based inverse problems.

Benefits of Weighted Training in Machine Learning and PDE-based Inverse Problems

HG E 1.2

8 December 2022
17:00-18:30

Prof. Dr. Wolfgang Hackbusch
Max-Planck-Institut für Mathematik in den Naturwissenschaften

Event Details

Speaker invited by

Prof. Dr. Stefan Sauter

Abstract

The best approximation of a matrix by a low-rank matrix can be obtained by the singular value decomposition. For large-sized matrices this approach is too costly. Instead we use a block decomposition. Approximating the small submatrices by low-rank matrices and agglomerating them into a new, coarser block decomposition, we obtain a recursive method. The required computational work is O(rnm) where r is the desired rank and nxm is the size of the matrix. We discuss error estimates for A-B and M-A where A is the result of the recursive trunction applied to M, while B is the best rank-r approximation. Numerical tests show that the approximate trunction is very close to the best one.

Recursive low-rank trunction of matrices

Y27 H 25

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