# Multilevel Methods for Uncertainty Quantification of Elliptic PDEs with Random Anisotropic Diffusion

## TL;DR

This paper develops multilevel methods for efficiently quantifying uncertainty in elliptic PDEs with random anisotropic diffusion, showing how the decay of the Karhunen-Loève expansion influences solution regularity and computational complexity.

## Contribution

It establishes a link between the decay of the Karhunen-Loève expansion and solution regularity, enabling multilevel methods to efficiently approximate statistical quantities.

## Key findings

- Decay of the Karhunen-Loève expansion determines solution regularity.
- Multilevel collocation and quadrature methods achieve expected convergence rates.
- Numerical examples validate the theoretical results in three dimensions.

## Abstract

We consider elliptic diffusion problems with a random anisotropic diffusion coefficient, where, in a notable direction given by a random vector field, the diffusion strength differs from the diffusion strength perpendicular to this notable direction. The Karhunen-Lo\`eve expansion then yields a parametrisation of the random vector field and, therefore, also of the solution of the elliptic diffusion problem. We show that, given regularity of the elliptic diffusion problem, the decay of the Karhunen-Lo\`eve expansion entirely determines the regularity of the solution's dependence on the random parameter, also when considering this higher spatial regularity. This result then implies that multilevel collocation and multilevel quadrature methods may be used to lessen the computation complexity when approximating quantities of interest, like the solution's mean or its second moment, while still yielding the expected rates of convergence. Numerical examples in three spatial dimensions are provided to validate the presented theory.