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，提出了一种新颖的配置型数值随机均匀化方法。该方法基于最近引入的局部化技术，该技术强制基函数相对于底层粗网格呈现超指数衰减，从而在采样阶段实现显著的计算节省。更一般地，配置型结构在随机设置中提供了一种特别简单且计算高效的构造，同时最小化了计算该方法基函数的各单元块之间的通信。我们进行了连接数值均匀化与随机均匀化定量理论的误差分析。在一系列数值实验中，我们研究了相关长度和离散化参数对该方法近似质量的影响。