Uncertainty Quantification by Multilevel Monte Carlo and Local Time-Stepping for Wave Propagation

Because of their robustness, efficiency, and non intrusiveness, Monte Carlo methods are probably the most popular approach in uncertainty quantification for computing expected values of quantities of interest. Multilevel Monte Carlo (MLMC) methods significantly reduce the computational cost by distributing the sampling across a hierarchy of discretizations and allocating most samples to the coarser grids. For time dependent problems, spatial coarsening typically entails an increased time step. Geometric constraints, however, may impede uniform coarsening thereby forcing some elements to remain small across all levels. If explicit time-stepping is used, the time step will then be dictated by the smallest element on each level for numerical stability. Hence, the increasingly stringent CFL condition on the time step on coarser levels significantly reduces the advantages of the multilevel approach. To overcome that bottleneck we propose to combine the multilevel approach of MLMC with local time-stepping. By adapting the time step to the locally refined elements on each level, the efficiency of MLMC methods is restored even in the presence of complex geometry without sacrificing the explicitness and inherent parallelism. In a careful cost comparison, we quantify the reduction in computational cost for local refinement either inside a small fixed region or towards a reentrant corner.