Adaptive Multilevel Monte Carlo Methods for Stochastic Variational Inequalities
This work addresses the gap between MLMC methods and spatial adaptivity for stochastic PDEs, offering a theoretically grounded and efficient numerical method.
The authors develop an adaptive multilevel Monte Carlo method for stochastic variational inequalities, combining deterministic adaptive mesh refinement with MLMC. Numerical experiments confirm the efficiency and convergence of the approach in terms of accuracy and computational cost.
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.