N-term Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs

Abstract

We consider diffusion in a random medium modeled as diffusion equation with lognormal Gaussian diffusion coefficient. Sufficient conditions on the log permeability are provided in order for a weak solution to exist in certain Bochner–Lebesgue spaces with respect to a Gaussian measure. The stochastic problem is reformulated as an equivalent deterministic parametric problem on RN. It is shown that the weak solution can be represented as Wiener–Itˆo Polynomial Chaos series of Hermite Polynomials of a countable number of i.i.d standard Gaussian random variables taking values in R1. We establish sufficient conditions on the random inputs for weighted sequence norms of the Wiener–Itˆo decomposition coefficients of the random solution to be p-summable for some 0 < p < 1. For random inputs with additional spatial regularity, stronger norms of the weighted coefficient sequence in the random solutions’ Wiener–Itˆo decomposition are shown to be p-summable for the same value of 0 < p < 1. We prove rates of nonlinear, best N-term Wiener Polynomial Chaos approximations of the random field, as well as of Finite Element discretizations of these approximations from a dense, nested family V0 ⊂ V1 ⊂ V2 ⊂ ·· · V of finite element spaces of continuous, piecewise linear Finite Elements.