Adaptive Multilevel Monte Carlo Methods for Stochastic Variational Inequalities

TL;DR

This paper introduces an adaptive multilevel Monte Carlo finite element method for stochastic variational inequalities, combining spatial adaptivity with MLMC to improve efficiency and providing convergence analysis supported by numerical experiments.

Contribution

It develops a novel adaptive MLMC finite element approach for stochastic variational inequalities, integrating deterministic adaptive mesh refinement with convergence guarantees.

Findings

The method achieves improved computational efficiency.

Numerical experiments confirm theoretical convergence.

Adaptive refinement enhances accuracy for stochastic PDEs.

Abstract

While multilevel Monte Carlo (MLMC) methods for the numerical approximation of partial differential equations with random coefficients enjoy great popularity, combinations with spatial adaptivity seem to be rare. We present an adaptive MLMC finite element approach based on deterministic adaptive mesh refinement for the arising "pathwise" problems and outline a convergence theory in terms of desired accuracy and required computational cost. Our theoretical and heuristic reasoning together with the efficiency of our new approach are confirmed by numerical experiments.