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, ω) in a bounded domain D ⊂ d is reformulated as a deterministic, infinite-dimensional, parametric problem by separation of deterministic (x ∈ D) and stochastic (ω ∈ Ω) variables in a(x, ω) via Karhúnen-Loève 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, ω) 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.