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

M. Clarke, J. Dick, Q. T. Le Gia, K. Mustapha, T. Tran

TL;DR

This work tackles uncertainty quantification for linear elasticity with random Lamé parameters expanded in infinite series, aiming to estimate the expected value of a linear functional of the displacement. It combines truncation of the infinite parameter expansions, conforming Galerkin FEM for spatial discretization, and high-order quasi-Monte Carlo (QMC) or sparse-grid quadrature to approximate the high-dimensional parametric integral $\Xi_{\mathbf{u}} := \iint_U \mathcal{L}(\mathbf{u}(\cdot,\mathbf{y},\mathbf{z}))\,d\mathbf{y}\,d\mathbf{z}$. The paper establishes regularity and error estimates for the truncated problem, FE discretization, and QMC quadrature, including sparse-grid variants, under precise assumptions on the coefficient expansions and tails. It also provides extensive numerical experiments in 2D validating the predicted convergence rates and demonstrating practical scalability, thereby delivering a robust framework for efficient uncertainty quantification in elasticity with infinite-dimensional random inputs.

Abstract

We explore a linear inhomogeneous elasticity equation with random Lamé 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.

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

TL;DR

This work tackles uncertainty quantification for linear elasticity with random Lamé parameters expanded in infinite series, aiming to estimate the expected value of a linear functional of the displacement. It combines truncation of the infinite parameter expansions, conforming Galerkin FEM for spatial discretization, and high-order quasi-Monte Carlo (QMC) or sparse-grid quadrature to approximate the high-dimensional parametric integral . The paper establishes regularity and error estimates for the truncated problem, FE discretization, and QMC quadrature, including sparse-grid variants, under precise assumptions on the coefficient expansions and tails. It also provides extensive numerical experiments in 2D validating the predicted convergence rates and demonstrating practical scalability, thereby delivering a robust framework for efficient uncertainty quantification in elasticity with infinite-dimensional random inputs.

Abstract

We explore a linear inhomogeneous elasticity equation with random Lamé 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.
Paper Structure (7 sections, 9 theorems, 96 equations, 3 figures, 5 tables)

This paper contains 7 sections, 9 theorems, 96 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Weak formulation and regularity
  4. A truncated problem
  5. Finite element approximation
  6. QMC method and sparse grids
  7. Numerical experiments

Key Result

Theorem 2.1

Let ${\bf u}$ be the solution of problem eq:L1 and ${\bf u}_{{\bf s}_h}$ be the Galerkin finite element solution of degree $\le r$ (with $r\ge 1$) defined as in FE solution with $y_j=z_k=0$ for $j>s_1\ge 1$ and $k>s_2\ge 1.$ For $i=1,\,2$, let $N_i = b^{m_i}$ with $m_i$ being positive integers and $ where $N = N_1 N_2$ is the total number of QMC quadrature points. The constant $C$ depends on $b,p,

Figures (3)

  • Figure 1: Pointwise nodal errors in the displacement, $|u_1-u_{1_h}|$ on right and $|u_2-u_{2_h}|$ on left.
  • Figure 2: Numerical errors (err$_{N_1}$) vs. $N_1^{-2}$ for Example 2
  • Figure 3: Numerical errors (err$_{N_2}$) vs. $N_2^{-2}$ for Example 3

Theorems & Definitions (16)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • ...and 6 more