Multilevel Monte Carlo methods for highly heterogeneous media

TL;DR

This paper explores the use of multilevel Monte Carlo methods to efficiently solve elliptic PDEs with random, highly heterogeneous coefficients, advancing numerical analysis for uncertainty quantification in subsurface flow modeling.

Contribution

It extends existing analysis of multilevel Monte Carlo methods to tensor-valued coefficients and point evaluations, including log-normal random coefficients.

Findings

Enhanced understanding of multilevel Monte Carlo convergence for complex coefficients

Application to log-normal random coefficients in subsurface flow

Broader analysis covering tensor-valued coefficients and point evaluations

Abstract

We discuss the application of multilevel Monte Carlo methods to elliptic partial differential equations with random coefficients. Such problems arise, for example, in uncertainty quantification in subsurface flow modeling. We give a brief review of recent advances in the numerical analysis of the multilevel algorithm under minimal assumptions on the random coefficient, and extend the analysis to cover also tensor--valued coefficients, as well as point evaluations. Our analysis includes as an example log--normal random coefficients, which are frequently used in applications.