In this paper, we propose an offline-online strategy based on the Localized Orthogonal Decomposition (LOD) method for elliptic multiscale problems with randomly perturbed diffusion coefficient. We consider a periodic deterministic coefficient with local defects that occur with probability $p$. The offline phase pre-computes entries to global LOD stiffness matrices on a single reference element (exploiting the periodicity) for a selection of defect configurations. Given a sample of the perturbed diffusion the corresponding LOD stiffness matrix is then computed by taking linear combinations of the pre-computed entries, in the online phase. Our computable error estimates show that this yields a good coarse-scale approximation of the solution for small $p$, which is illustrated by extensive numerical experiments. This makes the proposed technique attractive already for moderate sample sizes in a Monte Carlo simulation.
翻译:在本文中,我们提议了一种离线战略,其依据是:对随机扰动扩散系数的椭圆形多尺度问题采用局部性矫形分解法(LOD)方法。我们考虑的是周期性确定系数,其局部缺陷发生概率为1美元。离线阶段预先计算了一个单一参考元素(利用周期)的全球LOD硬度矩阵的条目,用于选择缺陷配置。根据受扰扩散的样本,随后在网上阶段对预算的条目进行线性组合,计算出相应的LOD硬度矩阵。我们的计算误差估计表明,通过大量数字实验,这一计算得出了小额美元解决方案的粗略近似值。这使得拟议的技术在蒙特卡洛模拟中已经对中度样本大小具有吸引力。