Uncertainty quantification by multilevel Monte Carlo and local time-stepping

Abstract

Because of their robustness, efficiency, and non intrusiveness, Monte Carlo methods are probablythe most popular approach in uncertainty quantification for computing expected values of quantitiesof interest. Multilevel Monte Carlo (MLMC) methods significantly reduce the computational costby distributing the sampling across a hierarchy of discretizations and allocating most samples tothe coarser grids. For time dependent problems, spatial coarsening typically entails an increasedtime step. Geometric constraints, however, may impede uniform coarsening thereby forcing someelements to remain small across all levels. If explicit time-stepping is used, the time step will thenbe dictated by the smallest element on each level for numerical stability. Hence, the increasinglystringent CFL condition on the time step on coarser levels significantly reduces the advantages of themultilevel approach. To overcome that bottleneck we propose to combine the multilevel approach ofMLMC with local time-stepping. By adapting the time step to the locally refined elements on eachlevel, the efficiency of MLMC methods is restored even in the presence of complex geometry withoutsacrificing the explicitness and inherent parallelism. In a careful cost comparison, we quantify thereduction in computational cost for local refinement either inside a small fixed region or towards areentrant corner