Intrinsic Fault Tolerance of Multi Level Monte Carlo Methods

Abstract

Monte Carlo (MC) and Multilevel Monte Carlo (MLMC) methods applied to solvers for Partial Differential Equations with random input data are shown to exhibit intrinsic failure resilience. Sufficient conditions are provided for non-recoverable loss of a random fraction of samples not to fatally damage the asymptotic accuracy vs. work of an MC simulation. Specifically, the convergence behavior of MLMC methods on massively parallel hardware is analyzed mathematically and computationally, under general assumptions on the node failures and on the sample failure statistics on the different MC levels, in the 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. Specifically, node failures with the so-called Weibull failure models on massively parallel stochastic Finite Volume computational fluid dynamics simulations are discussed.