一种基于缺陷型理论的控制变量方法用于随机同化中的方差减少
A control variate approach based on a defect-type theory for variance reduction in stochastic homogenization
AI总结：
本文提出一种基于缺陷型理论的控制变量方法，用于减少随机同化中方差，通过构建近似模型提高计算效率，并通过数值实验验证其有效性。
摘要：
我们考虑了一种用于随机同化中方差减少的控制变量方法。虽然精确的同化系数是确定性的，但其实际近似值是随机的。我们引入了控制变量技术以减少计算近似值的方差。我们的方法基于一种受缺陷型理论启发的替代模型，其中完美的周期性材料受到罕见缺陷的扰动。该模型最初在[A. Anantharaman和C. Le Bris, CRAS 2010]中提出，用于弱随机模型。在本工作中，我们处理完全随机的情况，并展示[ A. Anantharaman和C. Le Bris, CRAS 2010, MMS 2011]中提出的扰动方法可以转化为高效的控制变量。我们通过简单情况理论上证明了方法的有效性。随后提供了示例数值结果，并与其他方差减少策略进行比较。我们还展示了如何使用[C. Le Bris和F. Thomines, Chinese Ann. Math. 2012]中提出的降阶方法，以限制构建替代模型的成本。
英文摘要：
We consider a variance reduction approach for the stochastic homogenization of divergence form linear elliptic problems. Although the exact homogenized coefficients are deterministic, their practical approximations are random. We introduce a control variate technique to reduce the variance of the computed approximations of the homogenized coefficients. Our approach is based on a surrogate model inspired by a defect-type theory, where a perfect periodic material is perturbed by rare defects. This model has been introduced in [A. Anantharaman and C. Le Bris, CRAS 2010] in the context of weakly random models. In this work, we address the fully random case, and show that the perturbative approaches proposed in [A. Anantharaman and C. Le Bris, CRAS 2010, MMS 2011] can be turned into an efficient control variable. We theoretically demonstrate the efficiency of our approach in simple cases. We next provide illustrating numerical results and compare our approach with other variance reduction strategies. We also show how to use the Reduced Basis approach proposed in [C. Le Bris and F. Thomines, Chinese Ann. Math. 2012] so that the cost of building the surrogate model remains limited.
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