Benchmarking the P-MLQMC method on a geotechnical engineering problem

Abstract

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 novel method developed by the authors called p-refined Multilevel Quasi-Monte Carlo (p-MLQMC), achieves a significant computational cost reduction with respect to classic Multilevel Monte Carlo (h-MLMC). p-MLQMC uses a hierarchy of p-refined Finite Element meshes, combined with a deterministic Quasi-Monte Carlo (QMC) sampling rule. A non-negligible part of modeling the stochastics in non-deterministic engineering problems consists in adequately incorporating the uncertainty in the Finite Element model. This is typically done by evaluating the random field at certain carefully chosen evaluation points, and assigning the resulting scalars to the different finite elements. For the h-MLMC method, these evaluation points consist of the centroids of the elements. For the p-MLQMC method, we distinguish two different approaches to select the evaluation points, the Non-Nested Approach (NNA) and the Local Nested Approach(LNA). We investigate how these approaches affect the variance reduction over the levels and the total computational cost. We benchmark the p-ML(Q)MC-LNA, p-ML(Q)MC-NNA and h-MLMC method on a slope stability problem where the uncertainty is located in the soil's cohesion. We show that the p-MLQMC-LNA method outperforms all other considered methods in terms of computational cost.

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