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Stochastic polynomial approximation of PDEs with random coefficients

le 9 février 2011

14H - Groupe de travail "Applications des Mathématiques"

ENS Rennes Bâtiment Sauvy, Salle 5 (rdc)

Séminaire de Fabio Nobile (Politecnico di Milano) au groupe de travail "Applications des mathématiques"

Lien vers la page Web de l'orateur Résumé : When building a mathematical model to describe the behavior of a physical system, one has often to face a certain level of uncertainty in the proper characterization of the model parameters and input data. An example is given by the study of groundwater flow, where the subsurface permeability is largely unknown and often reconstructed from few available measurements via geostatistical techniques. In this talk we focus on models based on Partial Differential Equations with random coefficients or forcing terms, where randomness is used to model our insufficient knowledge or intrinsic variability of the physical system. We first parametrize the random input data by a finite number or random variables. Then we approximate the functional dependence of the solution of the PDE on the random variables by global multivariate polynomials, exploiting the fact that such functional dependence is often highly smooth (even analytic). We consider both Galerkin projection on polynomial spaces and Collocation type approximation on sparse grids of Gauss points. We focus, in particular, on the optimal choice of the polynomial space / sparse grid depending on the features of the differential problem at hand and the type of stochastic model for the coefficients. Our recipe for building the polynomial spaces based on a priori estimates is shown to be very effective on few numerical examples.

Recherche - Valorisation
Erwan Faou et Yannick Privat

Mise à jour le 8 février 2011