h- and p-refined Multilevel Monte Carlo Methods for Uncertainty Quantification in Structural Engineering

TL;DR

This paper explores advanced multilevel Monte Carlo methods combining h- and p-refinement hierarchies to efficiently quantify uncertainty in structural engineering problems, demonstrating significant computational speedups and cost reductions.

Contribution

It introduces novel combinations of h- and p-refinement with MLMC and MLQMC methods and evaluates their efficiency on complex structural problems with uncertain material properties.

Findings

MLMC and MLQMC significantly faster than standard MC

MLQMC cost scales as 1/epsilon under certain conditions

p-refinement reduces computational cost more effectively than h-refinement when modeling uncertainty as a random field

Abstract

Practical structural engineering problems are often characterized by significant uncertainties. Historically, one of the prevalent methods to account for this uncertainty has been the standard Monte Carlo (MC) method. Recently, improved sampling methods have been proposed, based on the idea of variance reduction by employing a hierarchy of mesh refinements. We combine an h- and p-refinement hierarchy with the Multilevel Monte Carlo (MLMC) and Multilevel Quasi-Monte Carlo (MLQMC) method. We investigate the applicability of these novel combination methods on three structural engineering problems, for which the uncertainty resides in the Young's modulus: the static response of a cantilever beam with elastic material behavior, its static response with elastoplastic behavior, and its dynamic response with elastic behavior. The uncertainty is either modeled by means of one random variable sampled from a univariate Gamma distribution or with multiple random variables sampled from a gamma random field. This random field results from a truncated Karhunen-Lo\`eve (KL) expansion. In this paper, we compare the computational costs of these Monte Carlo methods. We demonstrate that MQLMC and MLMC have a significant speedup with respect to MC, regardless of the mesh refinement hierarchy used. We empirically demonstrate that the MLQMC cost is optimally proportional to 1/epsilon under certain conditions, where epsilon is the tolerance on the root-mean-square error (RMSE). In addition, we show that, when the uncertainty is modeled as a random field, the multilevel methods combined with p-refinement have a significant lower computation cost than their counterparts based on h-refinement. We also illustrate the effect the uncertainty models have on the uncertainty bounds in the solutions.