An overview of p-refined Multilevel quasi-Monte Carlo Applied to the Geotechnical Slope Stability Problem
Authors: Blondeel Philippe , Robbe Pieterjan , François Stijn , Lombaert Geert , Vandewalle Stefan 
Published: May 2022

Engineering problems are often characterized by significant uncertainty in their material parameters. Multilevel sampling methods are a straightforward manner to account for this uncertainty. The most well known multilevel method is the Multilevel Monte Carlo method (MLMC). First developed by Giles, see [1], this method relies on a hierarchy of successive refined Finite Element meshes of the considered engineering problem, in order to achieve a computational speedup. Most of the samples are taken on coarse and computationally cheap meshes, while a decreasing number of samples are taken on finer and computationally expensive meshes. Classically, the mesh hierarchy is constructed by selecting a coarse mesh discretization of the problem, and recursively applying an h-refinement approach to it, see [2]. This will be referred to as h-MLMC. However, in the h-MLMC mesh hierarchy, the number of degrees of freedom increases almost geometrical with increasing level, leading to a large computational cost. An efficient manner to reduce this computational cost, is by means of the novel sampling method called p-refined Multilevel Quasi-Monte Carlo (p-MLQMC), see [3]. The p-MLQMC method uses a hierarchy of p-refined Finite Element meshes, combined with a deterministic Quasi-Monte Carlo sampling rule. This combination significantly reduced the computational cost with respect to h-MLMC. However, the p-MLQMC method presents the practitioner with a challenge. This challenge consists in adequately incorporating the uncertainty, represented as a random field, in the Finite Element model. In previous work, see [4], we have tackled this challenge by investigating how the evaluation points, used to calculate point evaluations of the random field by means of the Karhunen-Loève (KL) expansion, need to be selected in order to achieve the lowest computational cost. We found that using sets of nested evaluation points across the mesh hierarchy, i.e., the Local Nested Approach (LNA), yields a speedup up to a factor 5 with respect to sets consisting of non-nested evaluation points, i.e., the Non-Nested Approach (NNA). Furthermore, we have shown that p-MLQMC-LNA yields a speedup up to a factor 70 with respected to h-MLMC. Currently, our research focus lies on implementing the use of higher order Quasi-Monte Carlo rules, and hierarchical shape functions in p-MLQMC. Both paths show promising results for further computational savings in the p-MLQMC method. All the aforementioned implementations are benchmarked on a slope stability problem, with spatially varying uncertainty in the ground. The chosen quantity of interest (QoI) consists of the vertical displacement of the top of the slope.[1] Michael B. Giles. Multilevel Monte Carlo path simulation. Oper. Res., 56(3):607–617, 2008. [2] K. A. Cliffe, M. B. Giles, R. Scheichl, and A. L. Teckentrup. Multilevel Monte Carlo methods and applications to elliptic pdes with random coefficients. Comput. Vis. Sci., 14(1):3, Aug 2011. [3] Philippe Blondeel, Pieterjan Robbe, Cédric Van hoorickx, Stijn François, Geert Lombaert, and Stefan Vandewalle. p-refined multilevel quasi-monte carlo for galerkin finite element methods with applications in civil engineering. Algorithms, 13(5), 2020. [4] Philippe Blondeel, Pieterjan Robbe, Stijn François, Geert Lombaert, and Stefan Vandewalle. On the selection of random field evaluation points in the p-mlqmc method. arXiv, 2020.