  Because of their robustness, efficiency and non-intrusiveness, Monte Carlo
methods are probably the most popular approach in uncertainty quantification to
computing expected values of quantities of interest (QoIs). 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 (LTS). 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.
