An adaptive stochastic Galerkin method based on multilevel expansions of random fields: Convergence and optimality

TL;DR

This paper introduces an adaptive stochastic Galerkin method that efficiently solves elliptic PDEs with random coefficients by combining operator compression and wavelet approximation, achieving near-optimal complexity.

Contribution

It presents a novel multilevel expansion-based stochastic Galerkin method that attains optimal complexity even with limited regularity of the random field.

Findings

Achieves optimal computational complexity up to a logarithmic factor.

Convergence rates are maintained despite limited regularity of the random field.

Numerical experiments confirm theoretical convergence and complexity estimates.

Abstract

The subject of this work is a new stochastic Galerkin method for second-order elliptic partial differential equations with random diffusion coefficients. It combines operator compression in the stochastic variables with tree-based spline wavelet approximation in the spatial variables. Relying on a multilevel expansion of the given random diffusion coefficient, the method is shown to achieve optimal computational complexity up to a logarithmic factor. In contrast to existing results, this holds in particular when the achievable convergence rate is limited by the regularity of the random field, rather than by the spatial approximation order. The convergence and complexity estimates are illustrated by numerical experiments.