p-Refined Multilevel Quasi-Monte Carlo for Galerkin Finite Element Methods with Applications in Civil Engineering

• Published in: Algorithms Vol. 13; no. 5; p. 110
• Main Authors: Blondeel, Philippe, Robbe, Pieterjan, Van hoorickx, Cédric, François, Stijn, Lombaert, Geert, Vandewalle, Stefan
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.05.2020
• Subjects: Algorithms; Civil engineering; Discretization; Expected values; Fields (mathematics); Finite element method; Galerkin method; Geotechnical engineering; h- and p-refinement; Mean square errors; Methods; Monte Carlo simulation; Multilevel; Multilevel Monte Carlo; Multilevel Quasi-Monte Carlo; Parameter uncertainty; Partial differential equations; Random variables; Slope stability; Stability analysis; Stochastic processes; structural engineering; uncertainty quantification; Variance
• ISSN: 1999-4893; 1999-4893
• DOI: 10.3390/a13050110

Civil engineering applications are often characterized by a large uncertainty on the material parameters. Discretization of the underlying equations is typically done by means of the Galerkin Finite Element method. The uncertain material parameter can be expressed as a random field represented by, for example, a Karhunen–Loève expansion. Computation of the stochastic responses, i.e., the expected value and variance of a chosen quantity of interest, remains very costly, even when state-of-the-art Multilevel Monte Carlo (MLMC) is used. A significant cost reduction can be achieved by using a recently developed multilevel method: p-refined Multilevel Quasi-Monte Carlo (p-MLQMC). This method is based on the idea of variance reduction by employing a hierarchical discretization of the problem based on a p-refinement scheme. It is combined with a rank-1 Quasi-Monte Carlo (QMC) lattice rule, which yields faster convergence compared to the use of random Monte Carlo points. In this work, we developed algorithms for the p-MLQMC method for two dimensional problems. The p-MLQMC method is first benchmarked on an academic beam problem. Finally, we use our algorithm for the assessment of the stability of slopes, a problem that arises in geotechnical engineering, and typically suffers from large parameter uncertainty. For both considered problems, we observe a very significant reduction in the amount of computational work with respect to MLMC.