Séminaire de Mécanique d'Orsay
Mardi 21 juin à 14h au LIMSI
Prospects for breaking the curse of dimensionality of stochastic PDEs
Christophe Audouze
LSS, Sup'Elec
 

This talk is concerned with the numerical resolution of stochastic partial differential equations (SPDEs) where uncertain input data are described by means of random variables. For instance, the data randomness concerns the geometry, some physical parameters of the model, boundary conditions or initial values. This framework typically allows to quantify the uncertainty propagation in the model, namely to predict the dependency of the SPDE solution with respect to the random input data. In a computational point of view, the numerical resolution of SPDEs can be very costly and even prohibitive when the number of random variables becomes large. Various numerical techniques have been proposed for solving SPDEs: (i) non-intrusive methods such as collocation schemes, least squares fit or sparse grids; (ii) intrusive methods such as chaos-based stochastic Galerkin projection schemes or more recently proper generalized decomposition (PGD) schemes. After a review of the existing methods, we present a new numerical strategy based on the high-dimensional model representation (HDMR) scheme of Rabitz and co-workers. The key idea consists in decoupling the stochastic weak formulation into independent parametrized subproblems of very low dimension in the parameter space, each subproblem being solved with stochastic Galerkin projection schemes. The different steps of the proposed approach are detailed showing how the computational cost is reduced. Numerical illustrations are then provided for several stochastic models: (i) steady-state diffusion equation; (ii) time-dependent diffusion equation; (iii) Burgers's equation with a random viscosity. We finally conclude by giving some perspectives of further challenging works to be investigated.