Zurich colloquium in applied and computational mathematics

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Spring Semester 2017

Note: The highlighted event marks the next occurring event.

Date / Time Speaker Title Location
6 March 2017
Prof. Dr. Bjorn Engquist
ICES, University of Texas
Seismic imaging and the Monge-Ampère equation Y27  H 28
8 March 2017
Dr. Carola-Bibiane Schönlieb
University of Cambridge
Bilevel learning of variational models  Y27  H 25
Abstract: When assigned with the task of reconstructing an image from imperfect data the first challenge one faces is the derivation of a truthful image and data model. In the context of regularised reconstructions, some of this task amounts to selecting an appropriate regularisation term for the image, as well as an appropriate distance function for the data fit. This can be determined by the a-priori knowledge about the image, the data and their relation to each other. The source of this knowledge is either our understanding of the type of images we want to reconstruct and of the physics behind the acquisition of the data or we can thrive to learn parametric models from the data itself. The common question arises: how can we optimise our model choice? In this talk we discuss a bilevel optimization method for learning optimal variational regularisation models. Parametrising the regularisation and data fidelity terms, we will learn optimal total variation type regularisation models for image and video de-noising, and optimal data fidelity functions for pure and mixed noise corruptions. By considering quotients of desirable and undesirable image structures, we will also show how optimization strategies can be used to favour particular structures over others. I will also give an outlook on how such an approach can be used to learn optimal sampling patterns for magnetic resonance tomography. This presentation contains joint work with M. Benning, L. Calatroni, C. Chung, J. C. De Los Reyes, M. Ehrhardt, G. Gilboa, J. Grah, G. Maierhofer, T. Valkonen, and V. Vlacic
15 March 2017
Prof. Dr. Jean-Michel Coron
Université Pierre et Marie Curie
Finite-time stabilization  Y27  H25
Abstract: We present various results on the finite-time stabilization of control systems. This includes control systems in finite dimension (with an application to a quadcopter sliding on a plane) as well as control systems modeled by means of partial differential (1-D linear hyperbolic systems and 1-D linear parabolic equations).
22 March 2017
Dr. Joscha Gedicke
Fakultät für Mathematik, Universität Wien
An adaptive C^0 interior penalty method for a biharmonic obstacle problem  Y27  H 25
Abstract: C^0 interior penalty methods are discontinuous Galerkin methods for fourth order elliptic boundary value problems that are easier to implement than C^1 continuous finite elements. They were recently extended to the obstacle problem of clamped Kirchhoff plates. In this talk we present the residual based a posteriori error analysis for the fourth order obstacle problem following the approach of Braess for second order obstacle problems. We show that the resulting a posteriori error estimator is similar to the one for the fourth order boundary value problem. Moreover, we apply the resulting adaptive finite element algorithm to a specific optimal control problem, that can be reformulated as fourth order obstacle problem. Numerical examples in 2d and 3d illustrate the proven reliability and efficiency of the a posteriori error estimator.
29 March 2017
Prof. Dr. Mathias Fink
Wave Control and Holography with Time Transformations  Y27  H 25
Abstract: Because time and space play a similar role in wave propagation, wave control can be achieved or by manipulating spatial boundaries or by manipulating time boundaries. Here we emphasize the role of time boundaries manipulation. We show that sudden changes of the medium properties generate instant wave sources that emerge instantaneously from the entire wavefield and can be used to control wavefield and to revisit the holographic principles and the way to create time-reversed waves. Experimental demonstrations of this approach with water waves will be presented and the extension of this concept to acoustic and electromagnetic waves will be discussed. More sophisticated time manipulations can also be studied in order to extend the concept of photonic crystals and wave localization in the time domain.
5 April 2017
Prof. Dr. Eitan Tadmor
University of Maryland, ETH-ITS
Hierarchical construction of images and the problem of Bourgain-Brezis  Y27  H 25
Abstract: Edges are noticeable features in images which can be extracted from noisy data using different variational models. The analysis of such variational models leads to the question of representing general images as the divergence of uniformly bounded vector fields. We construct uniformly bounded solutions of div(U)= F for general F’s in the critical regularity space L^d(T^d). The study of this equation and related problems was motivated by recent results of Bourgain & Brezis. The intriguing aspect here is that although the problems are linear, construction of their solution is not. These constructions are special cases of a rather general framework for solving linear equations in critical regularity spaces. The solutions are realized in terms of nonlinear hierarchical representations U = \sum_j u_j which we introduced earlier in the context of image processing. The u_j’s are constructed recursively as proper minimizers, yielding a multi-scale decomposition of “images” U.
12 April 2017
Dr. Máté Gerencsér
Institute of Science and Technology, Austria
Characteristics of SPDEs  Y27  H 25
Abstract: We discuss Feynman-Kac formulae for linear stochastic PDEs. Due to the adapted randomness of the equations to be represented, the associated backward flow does not make (Itô) sense, and hence the temporal inversion has to be replaced by a spatial inversion. Some applications of such formulae to numerics and the theory of SPDEs will be outlined. Based on joint work with I. Gyöngy.
26 April 2017
Prof. Dr. Xue-Mei Li
University of Warwick
Weighted heat kernels and 'Brownian bridges'  Y27  H 25
Abstract: Gaussian upper and lower bounds for heat kernels are the basic tools for large deviation estimates. There are two well known characterisations on the derivatives of heat semi-grouops: the lower bound of the Ricci curvature by gradient bounds on the heat semi-group; and the validity of the Logarithmic Sobolev inequality for the distributions of the Brownian motion by bounds on the Ricci curvature. What can we say about their second order derivatives? What can we say about the kernels of the self-adjoint operator, which is the sum of the Laplace-Beltrami operator plus a gradient vector field and a potential function? This talk will not be technical. We will discuss why the stochastic damped parallel translation and the doubly damped stochastic parallel translation equation are the natural companions for the heat equations, we will also discuss the associated estimates, the second order Feynman-Kac formulas, and the role of the Brownian bridges and the semi-classical Brownian bridges.
3 May 2017
Prof. Dr. Dmitri Kuzmin
Technische Universität Dortmund
Flux-Corrected Transport Schemes for High-Order Bernstein Finite Elements  Y27  H 25
Abstract: This talk presents the first extensions of the flux-corrected transport (FCT) methodology to discontinuous and continuous high-order finite element discretizations of scalar conservation laws. Using Bernstein polynomials as local basis functions, we constrain the variation of the numerical solution by imposing local discrete maximum principles on the coefficients of the Bezier net. The design of accuracy-preserving FCT schemes for high-order Bernstein-Bezier finite elements requires a major upgrade of algorithms tailored for linear and multilinear Lagrange elements. The proposed ingredients include (i) a new discrete upwinding strategy leading to low order approximations with compact stencils, (ii) a variational stabilization operator based on the difference between two gradient approximations, and (iii) new localized limiters for antidiffusive element contributions. The optional use of a smoothness indicator based on a second derivative test makes it possible to avoid unnecessary limiting at smooth extrema and achieve optimal convergence rates for problems with smooth solutions. The accuracy of the proposed schemes is assessed in numerical studies for the linear transport equation in 1D and 2D. This is a joint work with R. Anderson, V. Dobrev, Tz. Kolev, C. Lohmann, M. Quezada de Luna, S. Mabuza, R. Rieben, J.N. Shadid, and V. Tomov.
10 May 2017
Prof. Dr. Ivan Oseledets
INM RAS and SkolTech, Moscow
Deep learning and tensors for the approximation of multivariate functions: recent results and open problems.  Y27  H 25
Abstract: In this talk I overview recent results in the algorithms and theory for the approximation of multivariate functions using low-rank tensor decompositions and deep neural networks (DNN), outline connections between two areas and also discuss open problems that need to be addressed. Tensor decompositions can be applied in DNN in several ways: first, they can be used to compress layers of DNN; second, DNN can be viewed as a generalized tensor network. A separate part will be denoted to the generalization ability of DNN, which is not fully described by the standard methods, and I will show our recent experimental study of the existence of "bad" local minima for neural networks.
17 May 2017
Prof. Dr. Thanasis Fokas
University of Cambridge
Revisiting the greats: Fourier, Laplace and Riemann  Y27  H 25
Abstract: The unified transform (also referred to as the Fokas method) will be reviewed. In particular, it will be shown that this transform yields unexpected results for such classical problems as the heat equation on the half line which was first investigated by Fourier,as well as for the Laplace equation in the interior of a polygon. Interesting connections of this approach with the Riemann hypothesis has led to the proof of the Lindelof hypothesis for a close variant of the Riemann zeta function.
24 May 2017
Prof. Dr. Mikhail Shashkov
Los Alamos National Laboratory
Modern numerical methods for high-speed, compressible, multi-physics, multi-material flows  KOL  G 201
Abstract: Computational experiment is among the most significant developments in the practice of the scientific inquiry in the 21th century. Within last four decades, computational experiment has become an important contributor to all scientific research programs. It is particular important for the solution of the research problems that are insoluble by traditional theoretical and experimental approaches, hazardous to study in the laboratory, or time consuming or expensive to solve by traditional means. Computational experiment includes several important ingredients: creating mathematical model, discretization, solvers, coding, verification and validation, visualization, analysis of the results, etc. In this talk we will describe some aspects of the modern numerical methods for high-speed, compressible, multi-physics, multi-material flows. We will address meshing issues, mimetic discretizations of equations of the Lagrangian gas dynamics and diffusion equation on general polygonal meshes, mesh adaptation strategies, methods for dealing with shocks, interface reconstruction needed for multi-material flows, closure models for multi-material cells, time discretizations, etc.
31 May 2017
Dr. Maxim Rakhuba
SkTech Institute, Moscow, Russia
Tensor solvers for high-dimensional eigenvalue problems  Y27  H 25
Abstract: This talk focuses on the solution of high-dimensional eigenvalue problems, which arise, e.g. in quantum chemistry and in the modelling of quantum spin systems. The key assumption is that eigenvectors can be approximated in low-rank tensor formats. This a priori knowledge allows us to reduce the number of parameters and to improve the convergence of iterative methods. We present new iterative solvers that explicitly account for the low-rank structure of the eigenvectors and that are capable of computing hundreds of eigenstates in high dimensions. We showcase our method by solving vibrational Scrödinger equation as well as equations arising in density functional theory.

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