Consider a linear elliptic PDE defined over a stochastic stochastic geometry a function of N random variables. In many application, quantify the uncertainty propagated to a quantity of interest (QoI) is an important problem. The random domain is split into large and small variations contributions. The large variations are approximated by applying a sparse grid stochastic collocation method. The small variations are approximated with a stochastic collocation-perturbation method and added as a correction term to the large variation sparse grid component. Convergence rates for the variance of the QoI are derived and compared to those obtained in numerical experiments. Our approach significantly reduces the dimensionality of the stochastic problem making it suitable for large dimensional problems. The computational cost of the correction term increases at most quadratically with respect to the number of dimensions of the small variations. Moreover, for the case that the small and large variations are independent the cost increases linearly.

考虑在随机几何上定义的线性椭圆 PDE，它是N 个随机变量的函数。在许多应用中，量化传播到感兴趣量 (QoI) 的不确定性是一个重要问题。随机域被分为大变化贡献和小变化贡献。通过应用稀疏网格随机配置方法来近似大的变化。小变化用随机搭配扰动方法来近似，并作为校正项添加到大变化稀疏网格分量中。导出 QoI 方差的收敛率，并将其与数值实验中获得的收敛率进行比较。我们的方法显着降低了随机问题的维数，使其适用于大维问题。校正项的计算成本相对于小变化的维数至多以二次方增加。此外，对于小变化和大变化独立的情况，成本线性增加。