The goal of this conference is to provide an overview of the major recent developments in scientific computing, numerical analysis, and neighboring areas.
Speakers are supposed to motivate their work and embed it into a general context, talks are supposed to be also accessible to non-experts.
The format of the conference is supposed to be Oberwolfach-style, i.e. with sufficient time for questions and discussions.
Speakers
Sören Bartels
Universität Freiburg
Mikhail Belkin
Ohio State University
Stéphane Boucheron
Université Paris Diderot - Paris 7
Annalisa Buffa
Istituto di Matematica Applicata e Tecnologie Informatiche, Pavia
Mixed finite elements, as developed by Raviart, Thomas, Nédélec, Brezzi and others can be given a unified presentation in the language of differential forms. As pointed out by Bossavit, the resulting spaces are related to constructs from differential topology dating back to de Rham and Whitney. This connection, developed in particular by Hiptmair, Arnold, Falk and Winther has led to the subject of Finite Element Exterior Calculus. I will try to give an account of this synthesis and present my related notion of Finite Element Systems.
Symmetry, and the study of invariant and equivariant objects, is a deep and unifying principle underlying a variety of mathematical fields. In particular, geometric mechanics is characterized by the application of symmetry and differential geometric techniques to Lagrangian and Hamiltonian mechanics, and geometric integration is concerned with the construction of numerical methods with geometric invariant and equivariant properties.
Computational geometric mechanics blends these fields, and uses a self-consistent discretization of geometry and mechanics to systematically construct geometric structure-preserving numerical schemes. In this talk, we will introduce a systematic method of constructing geometric integrators using ideas from geometric mechanics, and discuss generalizations that allow one to systematically model complex hierarchical systems, flows on Lie groups, and field theories.
One main focus of applied harmonic analysis is to develop representation systems based on particular partitions of Fourier domain aiming to provide sparse approximations for certain function classes. A well-known example are wavelet systems, which provide optimally sparse approximations of isotropic features. Since multivariate problems are however typically governed by anisotropic features in the sense of singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shock fronts in solutions of transport dominated equations, directional representation systems such as curvelets and shearlets have recently been introduced, with the novel concept of parabolic molecules providing a unified framework for sparse approximation properties of those systems.
Compressed sensing is a new research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that high-dimensional signals, which allow a sparse approximation by a suitable basis or, more generally, a frame, can be recovered from what was previously considered highly incomplete linear measurements by using, for instance, convex optimization or greedy type algorithms.
In this talk we will first provide an introduction to these two research areas and highlight recent developments. We will then discuss novel methodologies, which require a careful combination of those two fields, to solve problems in imaging sciences such as recovery of missing data and separation of morphologically distinct features, and present both analytic and numerical results. A discussion of future research directions will complete the talk.
Multidimensional problems are notoriously difficult due to the curse of dimensionality. However, high-dimensional problems are usually the most interesting ones and moreover, if the problem is of a considerable practical interest, there is a method that solves it. The most vivid example is the Schrodinger equation in quantum chemistry, where efficient solution methods have been proposed.
However, such methods are usually problem-specific, require a lot of efforts to implement and difficult to be applied in other areas. In the recent year, active development of mathematical foundations for the algorithms for the solution of high-dimensional problem has begun. Novel tensor formats (Hierarchical Tucker, Tensor Train) as well as surprinsing connections with other research areas (MPS, PEPS, tensor networks, graphical models) form a new research area with new fascinating theoretical and algorithmic problems and new applications in chemistry, biology and data-mining.
This talk will be a review of the known results and as well as recent advances in several areas, including low-rank methods for solving integro-differential equations, new computation of the convolution, application to data-mining and global optimization.
The entropy method has proved to be a handy device to derive concentration inequalities for functions of independent random variables. So far, it has delivered the tightest general inequalities concerning suprema of bounded empirical processes, conditional Rademacher averages or self-bounded functionals like VC-entropies. It has also been pointed out that off-the-shelf exponential Efron-Stein inequalities delivered by the entropy method may not be as tight as they should, they may fail to capture the so-called super-concentration phenomenon. We show that in the simplest setting of this phenomenon, suprema of Gaussian vectors, complementing the entropy method with appropriate representations provides a pedestrian derivation of tight "super"-concentration inequalities.
In this talk I will discuss some mathematical aspects of machine learning. I will start by describing the basic problems and challenges of learning from high-dimensional data and proceed by concentrating on the role of understanding "the shape of the data" through its differential geometry as given by the Laplace-Beltrami operator and the corresponding heat kernel. I will describe various connections, algorithms and theoretical results.
In micro-fluidic applications where the scales are small and viscous effects dominant, the Stokes equations are often applicable. The suspension dynamics of fluids with immersed rigid particles and fibers are very complex also in this Stokesian regime, and surface tension effects are strongly pronounced at interfaces of immiscible fluids. Simulation methods can be developed based on boundary integral equations, which leads to discretizations of the boundaries of the domain only, and hence fewer unknowns compared to a discretization of the PDE.
Two main challenges associated with boundary integral discretizations are to construct accurate quadrature methods for singular and nearly singular integrands, as well as to accelerate the solution of the linear systems, that will have dense system matrices. If these issues are properly addressed, boundary integral based simulations can be both highly accurate and very efficient. I will present a spectrally accurate FFT based Ewald method developed for the purpose of accelerating simulations and will discuss its application to simulations of periodic suspensions of rigid particles and rigid fibers in 3D. I will also discuss a method for highly accurate simulations of interacting drops in 2D.
Applications from molecular dynamics, material science, biology, or atmosphere/ocean sciences present new challenges for applied and numerical mathematics. These applications typically involve systems whose dynamics span a very wide range of spatio-temporal scales, and are subject to random perturbations of thermal or other origin. This second aspect especially complicates the modeling and computation of these systems and requires one to revisit standard tools from numerical analysis from a probabilistic perspective. For example, I will show how tools from Freidlin-Wentzell theory of large deviations and potential theoretic approaches to metastability can be used to develop numerical algorithms to accelerate the computations of reactive events, how averaging theorems for singularly perturbed Markov processes can help develop schemes bridging micro- to macro-scales of description or compute free energies, etc. As illustrations, I will use a selection of examples from molecular dynamics, material sciences, and fluid dynamics and show how the confrontation with actual problems not only profits from the theory but also enriches it.
We introduce a novel class of algorithms for the estimation of probability measures in high-dimensional spaces, given a finite number of samples. We are particularly interested in the case when the probability measure is concentrated near a low-dimensional set. These algorithms are based on geometric multiscale decompositions of probability measures, and we prove that with high probability, given a sufficiently large but finite number of samples, the algorithm returns a probability measure which is close, in Wasserstein-Kantorovich distance, to the target probability measure. We discuss applications to modeling high-dimensional noisy data sets, and anomaly detection in time-varying data.
During this talk, I will discuss the challenges for the numerical analysis of partial differential equations, with a special attention to the impact of this activity in the real world, and in particual in the product design process. This perspective open new questions in numerical analysis and computational geometry, some of them having a very theoretical flavour.
Isogeometric analysis (IGA), introduced by Hughes et al. in 2005, is a novel numerical approach that try to tackle these challenges by redesigning numerical techniques for PDEs. Within the isogeometric approach, the interaction of geometric modelers and PDE solvers is drastically simplified, and, moreover, the isogeometric numerical methods enjoys interesting properties that would be hard to obtain with classical finite elements. I will discuss our recent advances in this field.
Sampling problems are ubiquitous: Bayesian inference, computational statistical physics, uncertainty quantification, reliability analysis are examples of scientific fields which require advanced Monte Carlo methods to sample measures, in high or even infinite dimension. In this talk, I will present two recent works demonstrating the interest of partial differential equations approaches to study advanced stochastic algorithms: accelerated dynamics techniques (used in molecular dynamics to sample paths) and optimal scaling for Metropolis Hastings algorithms (used in computational statistics to sample high-dimensional measures). The mathematical analysis is based respectively on spectral analysis (semi-classical analysis for Witten boundary Laplacians) and on functional inequalities (entropy estimates for nonlinear partial differential equations).
References:D. Aristoff and T. Lelièvre, Mathematical analysis of Temperature Accelerated Dynamics, arxiv:1305.6569 B. Jourdain, T. Lelièvre and B. Miasojedow, Optimal scaling for the transient phase of Metropolis Hastings algorithms: the longtime behavior, arxiv:1212.5517 to appear in Bernoulli B. Jourdain, T. Lelièvre and B. Miasojedow, Optimal scaling for the transient phase of Metropolis Hastings algorithms: the mean-field limit, arxiv:1210.7639 C. Le Bris, T. Lelièvre, M. Luskin and D. Perez, A mathematical formalization of the parallel replica dynamics, Monte Carlo Methods and Applications, 18(2), 119-146, 2012 T. Lelièvre and F. Nier, Low temperature asymptotics for Quasi-Stationary Distributions in a bounded domain, arxiv:1309.3898
For many questions of scientific interest, all-atom molecular dynamics simulations are still out of reach, for example when accurate and hence computationally expensive force-fields are used, or in materials engineering where large numbers of atoms are required. A variety of coarse-graining techniques exist to reduce computational costs. In this talk, I will discuss the role that numerical analysis can (should?) play in this field.
For the majority of my presentation, I will focus on the particular example of atomistic-to-continuum coupling, which is a popular coarse-graining scheme for material defects. I will demonstrate how classical numerical analysis concepts can be applied to (1) understand the various errors committed in the coarse-graining process and (2) how to apply this analysis to optimise practical computational schemes.
The classical approach to the numerical analysis of approximation schemes for partial differential equations consists in the concept that stability and consistency imply convergence. For nonlinear equations often uniqueness and regularity properties are not available and weaker methods have to be employed to justify numerical schemes. The use of modern analytical techniques in the numerical analysis of nonlinear partial differential equations is discussed for certain classes of constrained or nondifferentiable minimization problems and nonsmooth evolution equations. This allows to prove the asymptotic accuracy of the proposed numerical methods. The application to particular instances such as the large deformation of thin elastic objects, the denoising of images, and rate-independent, inelastic material behavior provide empirical evidence of the qualitative accuracy and practical efficiency of the devised algorithms.
Participants
Ramy Badr
University of Leipzig
Sören Bartels
Universität Freiburg
Mikhail Belkin
Ohio State University, Columbus
Peter Benner
Max Planck Institute for Dynamics of Complex Technical Systems
Stéphane Boucheron
Université Paris Diderot - Paris 7
Annalisa Buffa
Istituto di Matematica Applicata e Tecnologie Informatiche, Pavia
Snorre H. Christiansen
University of Oslo
Wolfgang Dahmen
RWTH Aachen
Peter Druschel
Max Planck Institute for Software Systems
Roland Griesmaier
Universität Leipzig
Wolfgang Hackbusch
Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig
Venera Khoromskaya
MPI MIS, Leipzig
Boris Khoromskij
MPI MiS, Leipzig
Oswald Knoth
Leibniz-Institut für Troposphärenforschung e.V. (TROPOS), Leipzig
Christian Kuehn
Vienna University of Technology
Gitta Kutyniok
Technische Universität, Berlin
Tony Lelièvre
Ecole des Ponts ParisTech
Dominik Lellek
Philipps-Universität Marburg
Melvin Leok
University of California, San Diego
Stephan Luckhaus
Universität Leipzig
Mauro Maggioni
Duke University, Durham
Georg Martius
MPI MIS Leipzig
Ingo Nitschke
Technische Universität Dresden
Christoph Ortner
University of Warwick
Ivan Oseledets
Skolkovo Institute of Science and Technology
Dmitry Savostyanov
University of Southampton
Matti Schneider
TU Chemnitz
Bernhard Schölkopf
Universität Tübingen
Eric Sonnendrücker
Max-Planck-Institut für Plasmaphysik, Garching
Anna-Karin Tornberg
Royal Institute of Technology, Stockholm
Eric Vanden-Eijnden
Courant Institute, NYU, New York
Axel Voigt
TU Dresden
Harry Yserentant
TU Berlin
Scientific Organizers
Wolfgang Dahmen
RWTH Aachen
Jürgen Jost
Max-Planck-Institut für Mathematik in den Naturwissenschaften
Homepage
Felix Otto
Max-Planck-Institut für Mathematik in den Naturwissenschaften
Homepage
Administrative Contact
Jörg Lehnert
Max-Planck-Institut für Mathematik in den Naturwissenschaften
Contact by email
Valeria Hünniger
Max Planck Institute for Mathematics in the Sciences
Contact by email
