Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients

Abstract

A scalar, elliptic boundary-value problem in divergence form with stochastic diffusion coefficient a(x, ) in a bounded domain D R d is reformulated as a deterministic, infinite-dimensional, parametric problem by separation of deterministic (x D) and stochastic ( ) variables in a(x, ) via Karhnen-Love or Legendre expansions of the diffusion coefficient. Deterministic, approximate solvers are obtained by projection of this problem into a product probability space of finite dimension M and sparse discretizations of the resulting M-dimensional parametric problem. Both Galerkin and collocation approximations are considered. Under regularity assumptions on the fluctuation of a(x, ) in the deterministic variable x, the convergence rate of the deterministic solution algorithm is analysed in terms of the number N of deterministic problems to be solved as both the chaos dimension M and the multiresolution level of the sparse discretization resp. the polynomial degree of the chaos expansion increase simultaneously.