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：本文研究了一个具有随机Lamé参数的线性非齐次弹性方程。这些参数通过可数无穷项分离展开式进行参数化。本工作的主要目标是估计作用于弹性方程解的线性泛函的期望值（该期望值被视为对应于随机系数的参数空间上的无穷维积分）。为实现此目标，我们对随机参数的展开式进行了截断，结合高阶拟蒙特卡洛方法与稀疏网格方法以逼近高维积分，并引入伽辽金有限元方法来逼近物理域上的弹性方程解。我们系统研究了（1）无穷展开截断、（2）伽辽金有限元法以及（3）拟蒙特卡洛稀疏网格求积法则所产生的误差估计。为此，我们证明了连续解关于参数变量和物理变量所需满足的特定正则性。为实现理论正则性与收敛性结果，我们对随机系数的展开式施加了若干合理假设。最后，本文给出了若干数值计算结果。