This paper proposes a novel collocation-type numerical stochastic homogenization method for prototypical stochastic homogenization problems with random coefficient fields of small correlation lengths. The presented method is based on a recently introduced localization technique that enforces a super-exponential decay of the basis functions relative to the underlying coarse mesh, resulting in considerable computational savings during the sampling phase. More generally, the collocation-type structure offers a particularly simple and computationally efficient construction in the stochastic setting with minimized communication between the patches where the basis functions of the method are computed. An error analysis that bridges numerical homogenization and the quantitative theory of stochastic homogenization is performed. In a series of numerical experiments, we study the effect of the correlation length and the discretization parameters on the approximation quality of the method.

翻译：本文针对具有小相关长度随机系数场的典型随机均匀化问题，提出了一种新型配点型数值随机均匀化方法。该方法基于近期引入的局部化技术，通过强制基函数相对于底层粗网格呈超指数衰减，在采样阶段实现了显著的计算量节省。更一般地，配点型结构在随机设定下提供了一种特别简单且计算高效的构造方式，同时将计算基函数所需的各局部区域间的通信降至最低。我们进行了融合数值均匀化与随机均匀化定量理论的误差分析，并通过一系列数值实验研究了相关长度与离散化参数对方法逼近质量的影响。