Uncertainty Quantification for PDEs with Anisotropic Random Diffusion

TL;DR

This paper studies elliptic PDEs with anisotropic random diffusion, analyzing how the solution depends on random parameters and enabling advanced numerical methods for uncertainty quantification.

Contribution

It establishes regularity results based on the Karhunen-Loeve expansion decay, facilitating the use of sophisticated quadrature methods for PDE uncertainty quantification.

Findings

Regularity of solutions depends on the decay of the random vector field's expansion.

Enables use of quasi-Monte Carlo and sparse grid quadrature methods.

Numerical examples demonstrate the theory in three dimensions.

Abstract

In this article, we consider elliptic diffusion problems with an anisotropic random diffusion coefficient. We model the notable direction in terms of a random vector field and derive regularity results for the solution's dependence on the random parameter. It turns out that the decay of the vector field's Karhunen-Loeve expansion entirely determines this regularity. The obtained results allow for sophisticated quadrature methods, such as the quasi-Monte Carlo method or the anisotropic sparse grid quadrature, in order to approximate quantities of interest, like the solution's mean or the variance. Numerical examples in three spatial dimensions are provided to supplement the presented theory.