Numerical Solution of Partial Differential Equations with Stochastic Inputs

The numerical solution of initial boundary value problems of partial differential equations with random input data by generalized polynomial chaos (gpc) and multilevel Monte-Carlo (MLMC) methods is considered.

In numerical methods based on gpc expansions, random coefficients are parametrized in terms of countably many random variables via a Karhunen-Loeve (KL) or a multiresolution (MR) expansion, and random solutions are represented in terms of polynomial chaos expansions of the inputs' coordinates. Thus, the PDE problems are reformulated as parametric families of deterministic initial boundary value problems on infinite dimensional parameter spaces. Their solutions are represented as gpc expansions in the (possibly countably many) input parameters. Convergence rates for best N-term approximations of the parametric solutions and Galerkin and Collocation algorithms which realize these best N-term approximation rates are presented. The complexity of these algorithms is compared to those of MLMC space-time discretizations, in terms of the regularity of the input data, in particular for PDEs with propagation of singularities.

Joint work with Siddartha Mishra, Roman Andreev, Andrea Barth, Claude Gittelson, Jonas Sukys, David Berhardsgruetter of SAM, ETH and with Albert Cohen (Paris), R. DeVore (Texas A&M) and Viet-Ha Hoang (NTU Singapore)