Zurich Colloquium in Applied and Computational Mathematics

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Archive 2018

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Speaker

Title

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21 February 2018
16:15-17:15

Prof. Dr. Jun Zou
Chinese University of Hong Kong

Event Details

Speaker invited by	Ralf Hiptmair
Abstract	In this talk we discuss several preconditioners for the large-scale discrete systems arising from finite element discretizations of various Maxwell systems, including the time-harmonic Maxwell system and Maxwell scattering systems.

Precondioners and analyses for Maxwell systems

Y27 H25

28 February 2018
16:15-17:15

Prof. Dr. Christophe Chalons
Université Versailles Saint Quentin

Event Details

Speaker invited by	Remi Abgrall

Title T.B.A.

Y27 H25

7 March 2018
16:15-17:15

Prof. Dr. Olivier Faugeras
INRIA

Event Details

Speaker invited by

Habib Ammari

Abstract

We have developed a new method for establishing the thermodynamic limit of a network of fully connected rate neurons with correlated, Gaussian distributed, synaptic weights, and random inputs. The method is based on the formulation of a large deviation principle (LDP) for the probability distribution of the neuronal activity of a sequence of networks of increasing sizes. The motivation for using random connections comes from the fact that connections in neural networks are complex, poorly known and heterogeneous. The motivation for introducing correlation is the emphasis in computational modelling of neuroscience that neural networks are modular, and the correlations in the connection distribution reproduce this modularity, unlike in previous work. The limiting probability law is Gaussian and its mean and covariance functions are computed using a very quickly converging fixed point algorithm. Our outstanding new result is that in the thermodynamic limit the network does not become asynchronous, there is no propagation of chaos: neurons remain correlated and the amount of correlation can be described precisely from the correlation between the synaptic weights.

Neural networks do not become asynchronous in the large size limit: there is no propagation of chaos

Y27 H25

14 March 2018
16:15-17:15

Prof. Dr. Peter Kritzer
Johann Radon Institute for Comp. and Appl. Mathematics

Event Details

Speaker invited by

Arnulf Jentzen

Abstract

The (fast) component-by-component construction of lattice point sets and polynomial lattice point sets is a powerful method to obtain quadrature rules for approximating integrals over the d-dimensional unit cube. In this talk, we present modifications of the component-by-component algorithm and of the more recent successive coordinate search algorithm, which yield savings of the construction cost for lattice rules and polynomial lattice rules in weighted function spaces. The idea is to reduce the size of the search space for coordinates which are associated with small weights and are therefore of less importance to the overall error compared to coordinates associated with large weights. We analyze tractability conditions of the resulting quasi-Monte Carlo rules, and show some numerical results. The talk is based on joint work with J. Dick (UNSW Sydney), A. Ebert (KU Leuven), G. Leobacher (KFU Graz), and F. Pillichshammer (JKU Linz).

Modified component-by-component constructions of (polynomial) lattice points

Y27 H 25

21 March 2018
16:15-17:15

Prof. Dr. Yalchin Efendiev
Texas A&M University

Event Details

Speaker invited by

Christoph Schwab

Abstract

Subsurface formations often display high degrees of variability over multiple length scales. This permeability's spatial variations and complex connectivity impact flow and transport. For this reason, these effects must be included in simulations of flows. Upscaling techniques are introduced to coarsen these geological models for flow calculations. The main idea of upscaling techniques is to formulate macroscopic equations on a coarse grid and ways to compute macroscopic parameters. It is important that these coarsened flow models replicate the fine scale characterizations. Recently, multiscale methods are introduced to perform coarse-grid simulations. In this talk, I will give an overview of some multiscale methods and discuss their relation to flow-based upscaling. I will mostly focus on single-phase flow and show how to derive upscaled models and compute effective properties using multiscale methods and flow-based upscaling. I will describe subgrid errors in these methods and show a relation between multiscale methods and flow-based upscaling methods. Then, I will describe a general multiscale framework and show how one can achieve an accurate coarse-grid models using multi-continuum and non-local upscaling. I will describe some applications of multiscale methods to Bayesian approaches and inverse problems.

Generalized multiscale methods for porous media flows and their applications

Y27 H 25

11 April 2018
16:15-17:15

Dr. Irene Waldspurger
CEREMADE

Event Details

Speaker invited by

Habib Ammari

Abstract

The problem of finding best approximating pairs consists, given two closed sets in a metric space, in finding two points, one in each set, such that the distance between the points is minimal. We will discuss the case where the sets are convex polyhedrons in R^n. In this situation, several algorithms are known. The simplest one is alternating projections, and its convergence speed is relatively well understood. However, in practice, another algorithm, Douglas-Rachford, often seems to perform on par or better than alternating projections. We will discuss the convergence speed of this second algorithm, globally as well as locally. This is a joint work with Stefanie Jegelka.

Convergence Rate of the Douglas-Rachford Method for Finding Best Approximating Pairs

Y27 H 25

25 April 2018
16:15-17:15

Prof. Dr. Sergey Repin
V.A. Steklov Institute of Mathematics

Event Details

Speaker invited by

Stefan Sauter

Abstract

We discuss mathematical questions that play a fundamental role in quantitative analysis of incompressible viscous fluids and other incompressible media. Reliable verification of the quality of approximate solutions requires explicit and computable estimates of the distance to the corresponding generalized solution. In the context of this problem, one of the most essential questions is how to estimate the distance (measured in terms of the gradient norm) to the set of divergence free fields. It is closely related to the so-called inf-sup (LBB) condition or stability lemma for the Stokes problem and requires estimates of the LBB constant. We discuss methods of getting computable bounds of the constant and respective estimates of the distance to exact solutions of the Stokes, generalized Oseen, and Navier-Stokes problems.

Estimates of the distance to the set of divergence free fields and applications to analysis of incompressible viscous flow problems

Y27 H 25

2 May 2018
16:15-17:15

Prof. Dr. Francesco Andriulli
Politecnico di Torino

Event Details

Speaker invited by

Ralf Hiptmair

Abstract

When solving electromagnetic scattering problems with the boundary element method, the Electric Field and the Magnetic Field Integral Operators are the building blocks for a large number of formulations in literature. When modelling increasingly low frequency scenarios, however, the electric operator is known to give rise, upon discretization, to increasingly ill-conditioned problems whose iterative solution converges very slowly (a problem traditionally handled by leveraging on Helmholtz-Hodge decompositions). The magnetic operator, instead, gives rise to uniformly well-conditioned matrices (on simply connected geometries) independently of the frequency. At low-frequency, when the magnetic operator associated problems are solved via an iterative procedure, they converge rapidly but, lamentably, this rapid convergence is towards a severely incomplete solution since numerical cancellations occur in finite precision. We will discuss several aspects of this issue in detail delineating effective strategies to handle it and to obtain electromagnetic full-wave solvers providing stable solutions from high to arbitrarily low frequency.

Handling Magnetic Field Integral Operators at Extremely Low Frequency

Y27 H 25

23 May 2018
16:15-17:15

Prof. Dr. Markus Bachmayr
Institut für Numerische Simulation, Universität Bonn

Event Details

Speaker invited by

Christoph Schwab

Abstract

Folding grid value vectors into high-order tensors, combined with low-rank representation in the tensor train format, has been shown to lead to highly efficient approximations for various classes of functions. These include solutions of elliptic PDEs on nonsmooth domains or with oscillatory data. This tensor-structured approach is attractive because it leads to highly compressed, adaptive approximations based on simple discretizations. Straightforward choices of the underlying basis, such as piecewise multilinear finite elements on uniform tensor product grids, lead to the well-known basis ill-conditioning of discretized operators. We demonstrate that for low-rank representations, the use of tensor structure additionally leads to representation ill-conditioning, a new effect specific to computations in tensor networks. We construct an explicit tensor-structured representation of a BPX preconditioner with ranks independent of the number of discretization levels, which combined with a carefully chosen representation of its product with the stiffness matrix turns out to remove both basis and representation ill-conditioning. Numerical tests, including problems with highly oscillatory coefficients, show that one arrives at reliable and efficient solvers which remain numerically stable for mesh sizes near machine precision.

Stability of Low-Rank Tensor Representations and Structured Multilevel Preconditioning for Elliptic PDEs

Y27 H 25

19 September 2018
16:15-17:15

Dr. Philipp Petersen
Oxford University

Event Details

Speaker invited by

Christoph Schwab

Abstract

Novel machine learning techniques based on deep learning, i.e., the data-driven manipulation of neural networks, have reported remarkable results in many areas such as image classification, game intelligence, or speech recognition. Driven by these successes, many scholars have started using them in areas which do not focus on traditional machine learning tasks. For instance, more and more researchers are employing neural networks to develop tools for the discretisation and solution of partial differential equations. Two reasons can be identified to be the driving forces behind the increased interest in neural networks in the area of the numerical analysis of PDEs. On the one hand, powerful approximation theoretical results have been established which demonstrate that neural networks can represent functions from the most relevant function classes with a minimal number of parameters. On the other hand, highly efficient machine learning techniques for the training of these networks are now available and can be used as a black box.
In this talk, we will give an overview of some approaches towards the numerical treatment of PDEs with neural networks and study the two aspects above. We will recall some classical and some novel approximation theoretical results and tie these results to PDE discretisation. Afterwards, providing a counterpoint, we analyse the structure of network spaces and deduce considerable problems for the black box solver. In particular, we will identify a number of structural properties of the set of neural networks that render optimisation over this set especially challenging and sometimes impossible.

Deep Neural Networks and Partial Differential Equations: Approximation Theory and Structural Properties

HG E 1.2

10 October 2018
16:15-17:15

Prof. Dr. Steffen Börm
Institut für Informatik, Universität Kiel

Event Details

Speaker invited by

Stefan Sauter

Abstract

Fast summation methods are well-established for non-local operators arising in the context for electrostatics, molecular dynamics, or linear elastostatics, where they can reduce the complexity from O(n²) to O(n)or O(n log n). In the case of high-frequency Helmholtz equations, the situation is significantly more challenging: although the kernel function is still analytic, standard approximations, e.g., by polynomials, converge only slowly. This problem can be solved by splitting the kernel function into a smooth factor and a plane wave and approximating only the smooth factor, e.g., by interpolation. We obtain a fast summation scheme, but the storage requirements are fairly high: on one hand, multiple directions have to be handled simultaneously to reach a suitable accuracy. On the other hand, the computational domain has to be split into fairly small subdomains. Fortunately, we can combine the interpolation-based approximation with algebraic techniques to reduce the storage requirements, and this allows us to handle large problems efficiently. This talk gives an introduction to the directional interpolation approach, illustrates its properties in numerical examples, describes the algebraic re-compression, and demonstrates that it can significantly improve the overall performance of the method.

Hybrid compression of boundary element matrices for high-frequency Helmholtz problems

HG E 1.2

17 October 2018
16:15-17:15

Prof. Dr. Lourenco Beirao da Veiga
Università di Milano-Bicocca

Event Details

Speaker invited by

Remi Abgrall

Abstract

The Virtual Element Method (VEM), is a very recent technology introduced in [Beirao da Veiga, Brezzi, Cangiani, Manzini, Marini, Russo, 2013, M3AS] for the discretization of partial differential equations, that has shared a good success in recent years. The VEM can be interpreted as a generalization of the Finite Element Method that allows to use general polygonal and polyhedral meshes, still keeping the same coding complexity and allowing for arbitrary degree of accuracy. The Virtual Element Method makes use of local functions that are not necessarily polynomials and are defined in an implicit way. Nevertheless, by a wise choice of the degrees of freedom and introducing a novel construction of the associated stiffness matrixes, the VEM avoids the explicit integration of such shape functions. In addition to the possibility to handle general polytopal meshes, the flexibility of the above construction yields other interesting properties with respect to more standard Galerkin methods. For instance, the VEM easily allows to build discrete spaces of arbitrary C^k regularity, or to satisfy exactly the divergence-free constraint for incompressible fluids. The present talk is an introduction to the VEM, aiming at showing the main ideas of the method. We consider for simplicity a simple elliptic model problem (that is pure diffusion with variable coefficients) but set ourselves in the more involved 3D setting. In the first part we introduce the adopted Virtual Element space and the associated degrees of freedom, first by addressing the faces of the polyhedrons (i.e. polygons) and then building the space in the full volumes. We then describe the construction of the discrete bilinear form and the ensuing discretization of the problem. Furthermore, we show a set of theoretical and numerical results. In the very final part, we will give a glance at more involved problems, such as magnetostatics (mixed problem with more complex spaces interacting) and large deformation elasticity (nonlinear problem).

An introduction to virtual elements in 3D

HG E 1.2

24 October 2018
16:15-17:15

Prof. Dr. Claudia Schillings
Universitaet Mannheim

Event Details

Speaker invited by

Christoph Schwab

Abstract

Inverse problems arise in various fields of sciences and engineering. Methods to efficiently incorporate data into models are needed to reduce the overall uncertainty and to ensure the reliability of the simulations under real world conditions. The Bayesian approach to inverse problems provides a rigorous framework for the incorporation and quantification of uncertainties in measurements, parameters and models. The concentration of the posterior is a highly desirable situation in practice, since it relates to informative or large data. However, sampling methods for Bayesian inference show numerical instabilities in the case of concentrated posterior distributions. In this talk, we will discuss convergence results of Laplace’s approximation and analyze the use of the approximation within sampling methods. This is joint work with Bjoern Sprungk (U Goettingen) and Philipp Wacker (FAU Erlangen).

On the Convergence of Laplace's Approximation and Its Implications for Bayesian Computation

HG E 1.2

7 November 2018
16:15-17:15

Dr. Richard Küng
Caltech, Pasadena, USA

Event Details

Speaker invited by

Rima Alaifari

Abstract

We prove that low-rank matrices can be recovered efficiently from a small number of measurements that are sampled from orbits of a certain matrix group. As a special case, our theory makes statements about the phase retrieval problem. Here, the task is to recover a vector given only the amplitudes of its inner product with a small number of vectors from an orbit. Variants of the group in question have appeared under different names in many areas of mathematics. In coding theory and quantum information, it is the complex Clifford group; in time-frequency analysis the oscillator group; and in mathematical physics the metaplectic group. It affords one particularly small and highly structured orbit that includes and generalizes the discrete Fourier basis: While the Fourier vectors have coefficients of constant modulus and phases that depend linearly on their index, the vectors in said orbit have phases with a quadratic dependence. Our proof methods could be adapted to cover orbits of other groups.

Low rank matrix recovery from group orbits

HG E 1.2

5 December 2018
16:15-17:15

Prof. Dr. Elena Cordero
Universita degli Studi di Torino

Event Details

Speaker invited by

Rima Alaifari

Abstract

One of the most popular time-frequency representations is certainly the Wigner distribution. Its quadratic nature, however, causes the appearance of unwanted interferences or artefacts. The desire to suppress these artefacts is the reason why engineers, mathematicians and physicists have been looking for related time-frequency distributions, many of them are members of the Cohen class. Among them, the Born-Jordan distribution has recently attracted the attention of many authors, since the so-called "ghost frequencies" are damped quite well, and the noise is, in general, reduced. The very insight relies on the kernel of such a distribution, which contains the \emph{sinus cardinalis} $\mathrm{sinc}$, the Fourier transform\, of the first B-Spline $B_{1}$. Replacing the function $B_{1}$ with the spline or order $n$, denoted by $B_{n}$, yields the function $(\mathrm{sinc})^{n}$ on the Fourier side, whose decay at infinity increases with $n$. The related Cohen's Kernel is given by $\Theta^{n}(z_{1},z_{2})=\mathrm{sinc}^{n}(z_{1}\cdot _{2})$, $n\in\bN$. We study properties of the time-frequency distribution, called \emph{generalized Born-Jordan distribution of order $n$}, arising from these new kernels. Such representations display a great capacity of damping interferences and the reduction increases with $n$. This talk will show the different facets of this phenomenon, from visual comparisons to rigorous mathematical explanations. This is a joint work with Monika Dörfler, Maurice de Gosson (University of Vienna) and Fabio Nicola (Politecnico di Torino).

Generalized Born-Jordan Distributions and Applications to the Reduction of Interferences

HG E 1.2

12 December 2018
16:15-17:15

Prof. Dr. Nilima Nigam
Simon Fraser University

Event Details

Speaker invited by

Habib Ammari

Abstract

Approximations via conforming and non-conforming finite elements can be used to construct validated and computable bounds on eigenvalues for the Dirichlet Laplacian in certain domains. If these are to be used as part of a proof, care must be taken to ensure each step of the computation is validated and verifiable. In this talk we present a long-standing conjecture in spectral geometry, and its resolution using validated finite element computations. Schiffer's conjecture states that if a bounded domain omega in Rn has any nontrivial Neumann eigenfunction which is a constant on the boundary, then omega must be a ball. This conjecture is open. A modification of Schiffer's conjecture is: for regular polygons of at least 5 sides, we can demonstrate the existence of a Neumann eigenfunction which is does not change sign on the boundary. In this talk, we provide a recent proof using finite element calculations forthe regular pentagon. The strategy involves iteratively bounding eigenvalues for a sequence of polygonal subdomains of the triangle with mixed Dirichlet and Neumann boundary conditions. We use a learning algorithm to find and optimize this sequence of subdomains, and use non-conforming linear FEM to compute validated lower bounds for the lowest eigenvalue in each of these domains. The linear algebra is performed within interval arithmetic. This is joint work with Bartlomiej Siudeja and Ben Green, at U. Oregon.

A modification of Schiffer's Conjecture, and a proof via Finite Elements

HG E 1.2

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