Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

TL;DR

This paper extends multilevel Monte Carlo methods for elliptic PDEs with complex, non-uniform random coefficients, providing new error analysis and improved estimators for challenging non-smooth and discontinuous problems.

Contribution

It advances the theoretical analysis of multilevel Monte Carlo for elliptic PDEs with irregular coefficients and introduces level-dependent truncations to enhance estimator performance.

Findings

Proved convergence for a broader class of elliptic PDEs with irregular coefficients.

Extended finite element error analysis to non-smooth domains and discontinuous coefficients.

Demonstrated improved numerical performance with level-dependent Karhunen-Loève truncations.

Abstract

We consider the application of multilevel Monte Carlo methods to elliptic PDEs with random coefficients. We focus on models of the random coefficient that lack uniform ellipticity and boundedness with respect to the random parameter, and that only have limited spatial regularity. We extend the finite element error analysis for this type of equation, carried out recently by Charrier, Scheichl and Teckentrup, to more difficult problems, posed on non--smooth domains and with discontinuities in the coefficient. For this wider class of model problem, we prove convergence of the multilevel Monte Carlo algorithm for estimating any bounded, linear functional and any continuously Fr\'echet differentiable non--linear functional of the solution. We further improve the performance of the multilevel estimator by introducing level dependent truncations of the Karhunen--Lo\`eve expansion of the random coefficient. Numerical results complete the paper.