A Non-adapted Sparse Approximation of PDEs with Stochastic Inputs

Abstract

We propose a method for the approximation of solutions of PDEs with stochastic coefficients based on the direct non-adapted, i.e., non-adapted, sampling of solutions. This sampling can be done by using any legacy code for the deterministic problem as a black box. The method converges in probability (with probabilistic error bounds) as a consequence of sparsity and a concentration of measure phenomenon on the empirical correlation between samples. We show that the method is well suited for truly high-dimensional problems (with slow decay in the spectrum).