Intrinsic fault tolerance of multilevel Monte Carlo methods

Monte Carlo (MC) and multilevel Monte Carlo (MLMC) methods applied to solvers for Partial Differential Equations with random input data are proved to exhibit intrinsic failure resilience. Sufficient conditions are provided for non-recoverable loss of a random fraction of MC samples not to fatally damage the asymptotic accuracy vs. work of a MC simulation. Specifically, the convergence behavior of MLMC methods on massively parallel hardware with runtime faults is analyzed mathematically and investigated computationally. Our mathematical model assumes node failures which occur uncorrelated of MC sampling and with general sample failure statistics on the different levels and which also assume absence of checkpointing, i.e., we assume irrecoverable sample failures with complete loss of data. Modifications of the MLMC with enhanced resilience are proposed. The theoretical results are obtained under general statistical models of CPU failure at runtime. Particular attention is paid to node failures with so-called Weibull failure models. We discuss the resilience of massively parallel stochastic Finite Volume computational fluid dynamics simulations.