Quasi-Monte Carlo sparse grid Galerkin finite element methods for linear elasticity equations with uncertainties

TL;DR

This paper develops a numerical framework combining quasi-Monte Carlo, sparse grids, and finite element methods to efficiently estimate expected values of solutions to linear elasticity equations with random parameters, addressing high-dimensional uncertainty quantification.

Contribution

It introduces a combined approach of QMC, sparse grids, and FEM for high-dimensional stochastic elasticity problems, with rigorous error analysis and regularity results.

Findings

Error estimates for truncation, FEM, and QMC methods are established.

The method achieves high accuracy in approximating expected values.

Numerical experiments validate the theoretical convergence rates.

Abstract

We explore a linear inhomogeneous elasticity equation with random Lam\'e parameters. The latter are parameterized by a countably infinite number of terms in separated expansions. The main aim of this work is to estimate expected values (considered as an infinite dimensional integral on the parametric space corresponding to the random coefficients) of linear functionals acting on the solution of the elasticity equation. To achieve this, the expansions of the random parameters are truncated, a high-order quasi-Monte Carlo (QMC) is combined with a sparse grid approach to approximate the high dimensional integral, and a Galerkin finite element method (FEM) is introduced to approximate the solution of the elasticity equation over the physical domain. The error estimates from (1) truncating the infinite expansion, (2) the Galerkin FEM, and (3) the QMC sparse grid quadrature rule are all studied. For this purpose, we show certain required regularity properties of the continuous solution with respect to both the parametric and physical variables. To achieve our theoretical regularity and convergence results, some reasonable assumptions on the expansions of the random coefficients are imposed. Finally, some numerical results are delivered.