2021/06/21 by Marcus J. Grote, Grote, Marcus J., Simon Michel +3
Decision Sciences · Engineering · #Probabilistic and Robust Engineering Design #Scientific Measurement and Uncertainty Evaluation #Structural Health Monitoring Techniques
paper · pdf · doi:10.48550/arxiv.2106.11117
Because of their robustness, efficiency and non-intrusiveness, Monte Carlo\nmethods are probably the most popular approach in uncertainty quantification to\ncomputing expected values of quantities of interest (QoIs). Multilevel Monte\nCarlo (MLMC) methods significantly reduce the computational cost by\ndistributing the sampling across a hierarchy of discretizations and allocating\nmost samples to the coarser grids. For time dependent problems, spatial\ncoarsening typically entails an increased time-step. Geometric constraints,\nhowever, may impede uniform coarsening thereby forcing some elements to remain\nsmall across all levels. If explicit time-stepping is used, the time-step will\nthen be dictated by the smallest element on each level for numerical stability.\nHence, the increasingly stringent CFL condition on the time-step on coarser\nlevels significantly reduces the advantages of the multilevel approach. To\novercome that bottleneck we propose to combine the multilevel approach of MLMC\nwith local time-stepping (LTS). By adapting the time-step to the locally\nrefined elements on each level, the efficiency of MLMC methods is restored even\nin the presence of complex geometry without sacrificing the explicitness and\ninherent parallelism. In a careful cost comparison, we quantify the reduction\nin computational cost for local refinement either inside a small fixed region\nor towards a reentrant corner.\n