A non-adapted sparse approximation of PDEs with stochastic inputs

TL;DR

This paper introduces a non-adapted sampling method for approximating solutions to PDEs with stochastic inputs, leveraging existing deterministic solvers and exploiting sparsity for high-dimensional problems.

Contribution

It presents a novel non-adapted sampling approach that converges with probabilistic error bounds, suitable for high-dimensional stochastic PDEs.

Findings

Method converges in probability with error bounds.

Effective for high-dimensional problems with slow spectral decay.

Compatible with legacy deterministic PDE solvers.

Abstract

We propose a method for the approximation of solutions of PDEs with stochastic coefficients based on the direct, 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).