In this paper, we consider an elliptic eigenvalue problem with multiscale, randomly perturbed coefficients. For an efficient and accurate approximation of the solutions for many different realizations of the coefficient, we propose a computational multiscale method in the spirit of the Localized Orthogonal Decomposition (LOD) method together with an offline-online strategy similar to [M{\aa}lqvist, Verf\"urth, ESIAM Math. Model. Numer. Anal., 56(1):237-260, 2022]. The offline phase computes and stores local contributions to the LOD stiffness matrix for selected defect configurations. Given any perturbed coefficient, the online phase combines the pre-computed quantities in an efficient manner. We further propose a modification in the online phase, for which numerical results indicate enhanced performances for moderate and high defect probabilities. We show rigorous a priori error estimates for eigenfunctions as well as eigenvalues.

翻译：本文研究一类具有多尺度随机扰动系数的椭圆特征值问题。为高效精确地逼近系数多种不同实现对应的解，我们提出一种计算多尺度方法，其思想基于局部正交分解（LOD）方法，并结合与[M{\aa}lqvist, Verf\"urth, ESIAM Math. Model. Numer. Anal., 56(1):237-260, 2022]类似的离线-在线策略。离线阶段计算并存储选定缺陷构型下对LOD刚度矩阵的局部贡献量。给定任意扰动系数后，在线阶段以高效方式组合这些预计算量。我们进一步提出在线阶段的改进方案，数值结果表明该改进在中等及高缺陷概率情况下具有更优性能。我们给出了特征函数与特征值的严格先验误差估计。