Multilevel Monte Carlo on a high-dimensional parameter space for transmission problems with geometric uncertainties

TL;DR

This paper demonstrates that multilevel Monte Carlo methods are effective for high-dimensional, non-smooth uncertainty quantification problems, such as transmission problems with uncertain interfaces, by analyzing solution regularity and optimizing sampling strategies.

Contribution

It provides a space regularity analysis for non-smooth solutions and establishes convergence results for Monte Carlo methods in high-dimensional uncertain transmission problems.

Findings

MLMC efficiently computes moments despite non-smoothness

Convergence results hold in high-dimensional settings

Optimal sampling strategies improve computational efficiency

Abstract

In the framework of uncertainty quantification, we consider a quantity of interest which depends non-smoothly on the high-dimensional parameter representing the uncertainty. We show that, in this situation, the multilevel Monte Carlo algorithm is a valid option to compute moments of the quantity of interest (here we focus on the expectation), as it allows to bypass the precise location of discontinuities in the parameter space. We illustrate how such lack of smoothness occurs for the point evaluation of the solution to a (Helmholtz) transmission problem with uncertain interface, if the point can be crossed by the interface for some realizations. For this case, we provide a space regularity analysis for the solution, in order to state converge results in the L1-norm for the finite element discretization. The latter are then used to determine the optimal distribution of samples among the Monte Carlo levels. Particular emphasis is given on the robustness of our estimates with respect to the dimension of the parameter space.

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