An overview of p-refined Multilevel quasi-Monte Carlo Applied to the Geotechnical Slope Stability Problem

Abstract

[EN] Problems in civil engineering are often characterized by significant uncertainty in their material parameters. Sampling methods are a straightforward manner to account for this uncertainty, which is typically modeled as a random field. A popular sampling method consists of the classic Multilevel Monte Carlo method (h-MLMC). Its most distinctive feature consists of a hierarchy of h-refined meshes, where most of the samples are taken on coarse and computationally inexpensive meshes, and few are taken on finer but computationally expensive meshes. We present an improvement upon the classic Multilevel Monte Carlo, called the prefined Multilevel quasi-Monte Carlo method (p-MLQMC). Its key features consist of a mesh hierarchy constructed from a p-refinement scheme combined with a deterministic set of samples points (quasi-Monte Carlo points). In this work we show how the uncertainty needs to be accounted for and present results comparing the total computational cost of the h-ML(Q)MC and p-MLQMC method. Specifically, we present two novel approaches in order to account for the uncertainty in case of p-MLQMC. We benchmarking the different multilevel methods on a slope stability problem, and find that p-MLQMC outperforms h-MLMC up to several orders of magnitude.