Zurich Colloquium in Applied and Computational Mathematics

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Archive 2021

Date / Time

Speaker

Title

Location

24 February 2021
16:15-17:15

Dr. José Luis Romero
University of Vienna

Event Details

Speaker invited by

Prof. Rima Alaifari

Abstract

The sampling problem concerns the reconstruction of every function within a given class from their values observed only at certain points (samples). A density theorem gives precise necessary or sufficient conditions for such reconstruction in terms of an adequate notion of density of the set of samples. The most classical density theory, due to Shannon and Beurling, concerns bandlimited functions (that is, functions whose Fourier transforms are supported on the unit interval) and provides a sharp geometric characterization of all configurations of points that lead to stable reconstruction. I will present recent variants of such results and their applications in other fields, including the asymptotic equidistribution of Coulomb gases at low temperatures.

Sampling, density, and equidistribution

Zoom Meeting

10 March 2021
16:15-17:15

Prof. Dr. Mikko Salo
University of Jyväskylä

Event Details

Speaker invited by

Prof. Rima Alaifari

Abstract

Many inverse and imaging problems, such as image deblurring or electrical/optical tomography, are known to be highly sensitive to noise. In these problems small errors in the measurements may lead to large errors in reconstructions. Such problems are called ill-posed or unstable, as opposed to being well-posed (a notion introduced by J. Hadamard in 1902). The inherent reason for instability is easy to understand in linear inverse problems like image deblurring. For more complicated nonlinear imaging problems the instability issue is more delicate. We will discuss a general framework for understanding ill-posedness in inverse problems based on smoothing/compression properties of the forward map together with estimates for entropy and capacity numbers in relevant function spaces. The methods apply to various inverse problems involving general geometries and low regularity coefficients. We will use Electrical Impedance Tomography as a guiding example in the presentation. This talk is based on joint work with Herbert Koch (Bonn) and Angkana Rüland (Heidelberg).

Why are inverse problems ill-posed?

Zoom Meeting

24 March 2021
16:15-17:15

Prof. Dr. Hans Feichtinger
University of Vienna

Event Details

Speaker invited by

Prof. Rima Alaifari

Abstract

Gabor Analysis goes back to the fundamental paper of D. Gabor from 1946, who expressed (resp. conjectured) that every function can be expanded into a series of time-frequency shifted version of the standard Gaussian, meaning by building blocks of the form

g_{k,n}(x) = exp(2 \pi i b n} g_0(x - ak),

with $g_0(t) = exp(-\pi t^2)$. By choosing $a=1=b$ he was hoping to expand "every signal in a unique way", thus having a natural interpretation of the energy at position $(k,n)$ in phase-space through the (expected) unique coefficients for such an expansion.
Only since 35 years mathematicians have taken care of this problem, which provides a number of interesting challenges, going far beyond the original problem. From the speaker's point of view there is not only the computational problem arising (since Gabor Analysis can also be realized in the context of finite, Abelian groups), but also a variety of functional analytic concepts, including the idea of Banach frames and Banach Gelfand Triples.
The methods developed in the last 25 years are also relevant for the teaching of classical Fourier Analysis (such a course has been held at ETH last semester by the speaker), but also provides a good platform for the formulation of "Conceptual Harmonic Analysis", where the main question concerns the relationship between the continuous formulation of a problem and the corresponding discrete version and its computational realization.

Gabor Analysis: Background, Concepts and Computational Issues

Zoom Meeting

31 March 2021
16:15-17:15

Dr. Anirbit Mukherjee
University of Pennsylvania, USA

Event Details

Speaker invited by

Prof. Siddhartha Mishra

Abstract

One of the most intriguing mathematical mysteries of our times is to be able to explain the phenomenon of deep-learning. Neural nets can be made to paint while imitating classical art styles or play chess better than any machine or human ever and they seem to be the closest we have ever come to achieving "artificial intelligence". But trying to reason about these successes quickly lands us into a plethora of extremely challenging mathematical questions - typically about discrete stochastic processes. Some of these questions remain unsolved for even the smallest neural nets! In this talk we will describe two of the most recent themes of our work in this direction. Firstly, we will explain how under mild distributional conditions we can construct iterative algorithms which can train a ReLU gate in the realizable setting in linear time while also keeping track of mini-batching. We will show how this algorithm does approximate training when there is a data-poisoning attack on the training labels. Such convergence proofs remain unknown for S.G.D, but we will show via experiments that our algorithm very closely mimics the behaviour of S.G.D. Lastly, we will review this very new concept of "local elasticity" of a learning process and demonstrate how it appears to reveal certain universal phase changes during neural training. Then we will introduce a mathematical model which reproduces some of these key properties in a semi-analytic way. We will end by delineating various open questions in this theme of macroscopic phenomenology with neural nets. This is joint work with Weijie Su (Wharton, Statistics), Sayar Karmakar (U Florida, Statistics) and Phani Deep (Amazon, India)

Some Recent Progresses in the Mathematics of Neural Training

Zoom Meeting

14 April 2021
16:15-17:15

Dr. Gerhard Unger
TU Graz

Event Details

Speaker invited by

Prof. Ralf Hiptmair

Abstract

The use of boundary integral equations enables a reduction of scattering resonance problems in acoustics and electromagnetics to the boundary of the scatterer. Boundary integral formulations of resonance problems always lead to nonlinear eigenvalue problems with respect to the frequency parameter even if the original resonance problem is a linear one, as e.g. in the case of non-dispersive scatterers. The reason for that is that the frequency parameter occurs nonlinearly in the fundamental solution of the involved partial differential operator. Boundary integral formulations of resonance problems can be considered as eigenvalue problems for holomorphic Fredholm operator-valued functions. For such kind of eigenvalue problems a comprehensive spectral theory exists. Moreover, abstract results on the convergence for the approximation of such kind of eigenvalue problems are available and applicable to standard Galerkin approximations of boundary integral formulations of scattering resonance problems. The resulting Galerkin approximations are eigenvalue problems for holomorphic matrix-valued functions. The contour integral method is a reliable method for the approximation of such kind of algebraic eigenvalue problems. The method is based on the contour integration of the inverse of the occurring matrix-valued function of the eigenvalue problem and utilizes that the eigenvalues are poles of it. By contour integration a reduction of the eigenvalue problem for a holomorphic matrix-valued function to an equivalent linear matrix eigenvalue problem is possible such that the eigenvalues of the linear eigenvalue problem coincide with the eigenvalues of the nonlinear eigenvalue problem inside the contour. For the practical application of this method an efficient approximation of the contour integral over the inverse of the underlying matrix-valued function of the eigenvalue problem is necessary. This can be achieved for example by the composite trapezoidal rule, which requires the solution of several linear systems involving boundary element matrices related to the eigenvalue problem.

Contour integral method for scattering resonance problems

Zoom Meeting

21 April 2021
16:15-17:15

Dr. Benjamin Scellier
Google Zurich

Event Details

Speaker invited by

Prof. Siddhartha Mishra

Abstract

We present a mathematical framework for machine learning, which allows us to train "physical systems with adjustable parameters" by gradient descent. Our framework applies to a very broad class of systems, namely those whose state or dynamics are described by variational equations. This includes physical systems whose equilibrium state is the minimum of an energy function, and physical systems whose trajectory minimizes an action functional (principle of least action). We present a simple procedure to compute the loss gradients in such systems. This procedure, called equilibrium propagation (EqProp), requires solely locally available information for each trainable parameter. In particular, our framework offers the possibility to build and train neural networks in substrates that directly exploit the laws of physics. As an example, we show how to use our framework to train a class of electrical circuits called nonlinear resistive networks. We also sketch a path to apply our framework to spiking neural networks (specifically spiking electrical circuits), by showing that nonlinear RLC circuits satisfy a principle of least action.

A deep learning theory for neural networks grounded in physics

Zoom Meeting

28 April 2021
16:15-17:15

Prof. Dr. Richardo Nochetto
University of Maryland, College Park, USA

Event Details

Speaker invited by	Prof. Christoph Schwab
Abstract	This is a survey talk about fractional diffusion. It describes its formulation via the integral Laplacian, the regularity of solutions on bounded domains and the approximation by finite element methods. It will emphasize recent research about Besov regularity on Lipschitz domains, BPX preconditioning and local energy error estimates.

What is fractional diffusion?

Zoom Meeting

2 June 2021
16:15-17:15

Prof. Dr. Carsten Carstensen
Institut für Mathematik, Humboldt-Universität

Event Details

Speaker invited by	Prof. Dr. Stefan Sauter

Eigenvalue Computation for Symmetric PDEs

Zoom Meeting

13 October 2021
16:15-17:15

Prof. Dr. Kirill Cherednichenko
University of Bath

Event Details

Speaker invited by	Prof. Habib Ammari
Abstract	I shall discuss a novel approach to the homogenisation of critical-contrast periodic PDEs, which yields an explicit construction of their norm-resolvent asymptotics. A practically relevant outcome of this result is that it interprets composite media with micro-resonators as a class of (temporally and spatially) dispersive media. This is joint work with Yulia Ershova and Alexander Kiselev.

Metamaterials, or media with time and spatial dispersion: a new approach to the analysis of composites with contrast

Zoom Meeting

27 October 2021
16:15-17:15

Wasilij Barsukow
MPI Munich, CNRS

Event Details

Speaker invited by

Prof. Rémi Abgrall

Abstract

In the limit of low Mach number, the compressible Euler equations become incompressible. In the compressible case, numerical stabilization (upwinding) for explicit methods for a long time has been inspired by Riemann problems, i.e. high-Mach compressible phenomena. It has been noticed that this kind of stabilization introduces strong numerical errors for low Mach number flow. Modifications of Riemann solvers have been proposed which allow usage of coarse grids in the low Mach number regime, but the modifications are ad hoc and generally affect stability. In the talk I will show new approaches to achieving all-Mach number methods which are stable, and can be derived from first principles. The methods are truly multi-dimensional, reflecting the fact that incompressible flow is only nontrivial in multiple spatial dimensions.

New strategies for all-Mach number finite volume methods

HG E 1.2
Zoom Meeting

3 November 2021
16:15-17:15

Prof. Dr. Qiang Du
Columbia University, New York

Event Details

Speaker invited by

Prof. Ch. Schwab

Abstract

Nonlocality has become increasingly noticeable in nature. The modeling and simulation of its presence and impact motivate new development of mathematical theory. In this lecture, we focus on nonlocal models with a finite horizon of interactions, and illustrate their roles in the understanding of various phenomena involving anomalies, singularities and other effects due to nonlocal interactions. We also present some recent analytical studies concerning nonlocal operators and nonlocal function spaces. The theoretical advances are making nonlocal modeling and simulations more reliable, effective and robust for applications ranging from classical mechanics to traffic flows of autonomous and connected vehicles.

Recent progress on nonlocal modeling, analysis and applications

Zoom Meeting

10 November 2021
16:15-17:15

Dr. Chupeng Ma
Universität Heidelberg

Event Details

Speaker invited by

Prof. Ralf Hiptmair

Abstract

In this talk, I will present generalized FEMs with optimal local approximation spaces for solving Helmholtz equation with heterogeneous coefficient and high wavenumber. The optimal local approximation spaces are constructed by eigenvectors of local eigenvalue problems involving a partition of unity function defined on generalized harmonic spaces. Nearly exponential convergent and wavenumber explicit local approximation errors are derived both at the continuous and fully discrete level. The method can be viewed as an extension of Trefftz methods to Helmholtz equation with heterogeneous coefficients. This is a joint work with Robert Scheichl.

Generalized FEMs based on locally optimal spectral approximations for Helmholtz equation with heterogeneous coefficient

HG E 1.2
Zoom Meeting

24 November 2021
16:15-17:15

Dr. Barbara Verfürth
Fakultät für Mathematik, KIT

Event Details

Speaker invited by

Prof.Dr. 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 (CANCELLED)

HG E 1.2

1 December 2021
16:15-17:15

Prof. Dr. Wolfgang Hackbusch
Max Planck Leipzig

Event Details

Speaker invited by

Prof. Stefan Sauter

Abstract

Since tensor spaces may have a huge dimension, it is often not possible to store tensors by all their entries. Instead one uses certain representations (also called 'formats'), which describe a subset of tensors. For some formats used in practice the set of representable tensors is not closed. This leads to an instability comparable with the cancellation effect in the case of numerical differentiation. Under general conditions we prove for the finite-dimensional case that there is some minimal strength of the instability. For the special case of the 2-term format a quantitative result can be proved. In the infinite-dimensional case with a tensor norm not weaker than the injective crossnorm, the same instability behaviour can be proved. Even the constants in the estimates are under control. As a result, it is sufficient to study the instability behaviour for finite-dimensional model spaces.

On nonclosed tensor formats

Zoom Meeting

8 December 2021
16:15-14:15

Dr. Michael Dumbser
University of Trento, Italy

Event Details

Speaker invited by

Prof. Rémi Abgrall

Abstract

In the first part of this talk we present the unified first order hyperbolic formulation of Newtonian continuum mechanics proposed by Godunov, Peshkov and Romenski (GPR). The governing PDE system can be derived from a variational principle and belongs to the class of symmetric hyperbolic and thermodynamically compatible systems (SHTC), which have been studied for the first time by Godunov in 1961 and later in a series of papers by Godunov & Romenski. An important feature of the model is that the propagation speeds of all physical processes, including dissipative processes, are finite. The GPR model is a geometric approach to continuum mechanics that is able to describe the behavior of nonlinear elasto-plastic solids at large deformations, as well as viscous Newtonian and non-Newtonian fluids within one and the same governing PDE system. This is achieved via appropriate relaxation source terms in the evolution equations for the distortion field and the thermal impulse. It can be shown that the GPR model reduces to the compressible Navier-Stokes equations in the stiff relaxation limit, i.e. when the relaxation times tend to zero. The unified system is also able to describe material failure, such as crack generation and fatigue. In the absence of source terms, the homogeneous part of the GPR model is endowed with involutions, namely the distortion field A and the thermal impulse J need to remain curl-free. In the second part of the talk we therefore present a new structure-preserving scheme that is able to preserve the curl-free property of both fields exactly also on the discrete level. This is achieved via the definition of appropriate and compatible discrete gradient and curl operators on a judiciously chosen staggered grid. Furthermore, the pressure terms are discretized implicitly, in order to capture the low Mach number limit of the equations properly, while all other terms are discretized explicitly. In this manner, the resulting pressure system is symmetric and positive definite and can be solved with efficient iterative solvers like the conjugate gradient method. Last but not least, the new staggered semi-implicit scheme is asymptotic-preserving and thus also able to reproduce the stiff relaxation limit of the governing PDE system properly, recovering an appropriate discretization of the compressible Navier-Stokes equations and of the incompressible equations in the low Mach number limit. In the final part of the talk we present a new thermodynamically compatible finite volume scheme that is exactly compatible with the overdetermined structure of the model at the semi-discrete level, making use of a discrete form of the continuous formalism introduced by Godunov in 1961. A very particular feature of our new thermodynamically compatible finite volume scheme is the fact that it directly discretizes the entropy inequality, rather than the total energy conservation law. Energy conservation is instead achieved as a mere consequence of the scheme, thanks to the thermodynamically compatible discretization of all the other equations.

Numerical schemes for a unified first order hyperbolic system of continuum mechanics

HG E 1.2

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