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. By
adapting the time-step to the locally refined elements on each level, local
time-stepping (LTS) methods permit to restore the efficiency of MLMC methods
even in the presence of complex geometry without sacrificing the explicitness
and inherent parallelism