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.

翻译：本文研究了具有随机拉梅参数的线性非齐次弹性方程组。随机参数通过可数无穷项分离展开进行参数化。本研究的主要目标是估计作用于弹性方程解上的线性泛函的期望值（视为随机系数参数空间上的无穷维积分）。为此，我们截断随机参数的展开，采用高阶拟蒙特卡洛（QMC）方法结合稀疏网格技术逼近高维积分，并通过伽辽金有限元方法（FEM）近似物理域上弹性方程的解。本文分别研究了以下三个方面的误差估计：(1) 无穷展开截断误差，(2) 伽辽金有限元近似误差，(3) QMC稀疏网格求积规则误差。为此，我们证明了连续解关于参数变量和物理变量的某些必要正则性性质。为获得理论正则性和收敛性结果，我们对随机系数展开施加了合理假设。最后给出了数值计算结果。