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：本文研究具有随机拉梅参数的线性非齐次弹性方程。拉梅参数通过可数无限项分离展开进行参数化。本文的主要目标是估计弹性方程解的线性泛函的期望值（将其视为随机系数参数空间上的无限维积分）。为此，我们截断随机参数的无穷展开，将高阶拟蒙特卡洛方法（QMC）与稀疏网格方法相结合以近似高维积分，并引入伽辽金有限元方法（FEM）来近似物理域上弹性方程的解。本文分别研究了（1）无穷展开截断、（2）伽辽金有限元方法以及（3）QMC稀疏网格求积规则引起的误差估计。为达到这一目的，我们证明了连续解在参数变量和物理变量上的必要正则性质。为获得理论正则性和收敛性结果，我们对随机系数的展开施加了一些合理假设。最后，给出了若干数值结果。