Quantifying Uncertainty
Todor, R.A. and Schwab, C.. "Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients" IMA J. of Numer. Anal.. 27
(2).
APR 2007.
pp. 232--261.
A scalar, elliptic boundary-value problem in divergence form with stochastic diffusion coefficient a(x, omega) in a bounded domain D subset of < Ropf >(d) is reformulated as a deterministic, infinite-dimensional, parametric problem by separation of deterministic (x is an element of D) and stochastic (omega is an element of Omega) variables in a(x, omega) via Karhunen-Loeve 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, omega) 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.