ICOSAHOM 2020
International Conference on Spectral and High Order Methods
12th  16th July 2021  Vienna, Austria
Conference Agenda
Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).
Please note that all times are shown in the time zone of the conference. The current conference time is: 4th Dec 2022, 06:31:15pm CET

Session Overview 
Session  
plenary 6: Virginie Ehrlacher
 
Presentations  
Tensor methods and greedy algorithms for highdimensional problems: applications in materials science. Ecole des Ponts ParisTech & INRIA, France Highdimensional problems are ubiquitous in a large variety of applications: materials science, finance, uncertainty quantification, data science, stochastic game theory etc. However, standard numerical methods cannot be used in practice for the resolution of Partial Differential Equations the solutions of which depend on a large number of variables because of the socalled curse of dimensionality. The most direct manifestation of this curse lies in the fact that the complexity of the representation of a function depending on d variables with a fixed number of degrees of freedom per variable grows exponentially with d. In the last decades, several dedicated numerical strategies have been developped by applied mathematicians in order to circumvent this curse for the resolution of highdimensional Partial Differential Equations. Among these, tensor methods have been a very active field of research in the past few years and are nowadays one of the most successfull family of approaches for the resolution of such problems. The bottom line of these methods is to use the wellknown principle of separation of variables to define appropriate subsets of functions, called tensor formats, depending on a large number of variables and which can be represented with low complexity. There exists a wide variety of tensor formats, the most widely used of those being for instance the socalled Canonical Polyadic, Tucker, Tensor Train or Hierarchical Tucker formats. A first objective of this talk is to give a comprehensive introduction to these tensor formats, and explain how these methods can be used for the resolution of highdimensional Partial Differential Equations. A particular emphasis will be put on theoretical results concerning the analysis of numerical methods which consists in combining these tensor formats with socalled greedy algorithms from nonlinear approximation theory. A second objective of this talk is to illustrate the efficiency of such approaches for the resolution various highdimensional problems stemming from materials science applications. Indeed, interacting particle systems are ubiquitous in materials science applications in order to understand the macroscopic properties of materials from its microscopic or mesoscopic features. Several mathematical models exist to account for the evolution of such systems at different scales. Among those, let us mention for instance kinetic models, FokkerPlanck equations for molecular dynamics or quantum models for electronic structure calculations. All these models are defined on highdimensional spaces, the high dimension stemming either from the large number of particles in the system of interest or the high number of features characterizing the state of each particle. A specific focus will be put in this talk on kinetic equations, which are mesoscopic models (used for instance for the study of plasmas, neutronics or electron transport) which describe the state of large particle systems at the statistical level by a timedependent probability distribution function, which encodes the probability of finding a particle at a certain position in space and with a certain speed. This distribution function is thus defined on a highdimensional phase space and its evolution is typically modeled via a Boltzmann Partial Differential Equation. Numerical results will be presented which illustrates the successful use of tensor methods and greedy algorithms for the resolution of such models, in particular the resolution of the socalled VlasovPoisson system in some 3d3d test cases. 
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