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Optimal Artificial Boundary Condition for Random Elliptic Media
Foundations of Computational Mathematics ( IF 3 ) Pub Date : 2021-03-02 , DOI: 10.1007/s10208-021-09492-1
Jianfeng Lu , Felix Otto

We are given a uniformly elliptic coefficient field that we regard as a realization of a stationary and finite-range ensemble of coefficient fields. Given a right-hand side supported in a ball of size \(\ell \gg 1\) and of vanishing average, we are interested in an algorithm to compute the solution near the origin, just using the knowledge of the given realization of the coefficient field in some large box of size \(L\gg \ell \). More precisely, we are interested in the most seamless artificial boundary condition on the boundary of the computational domain of size L. Motivated by the recently introduced multipole expansion in random media, we propose an algorithm. We rigorously establish an error estimate on the level of the gradient in terms of \(L\gg \ell \gg 1\), using recent results in quantitative stochastic homogenization. More precisely, our error estimate has an a priori and an a posteriori aspect: with a priori overwhelming probability, the prefactor can be bounded by a constant that is computable without much further effort, on the basis of the given realization in the box of size L. We also rigorously establish that the order of the error estimate in both L and \(\ell \) is optimal, where in this paper we focus on the case of \(d=2\). This amounts to a lower bound on the variance of the quantity of interest when conditioned on the coefficients inside the computational domain, and relies on the deterministic insight that a sensitivity analysis with respect to a defect commutes with stochastic homogenization. Finally, we carry out numerical experiments that show that this optimal convergence rate already sets in at only moderately large L, and that more naive boundary conditions perform worse both in terms of rate and prefactor.



中文翻译:

随机椭圆介质的最优人工边界条件

给定一个均匀的椭圆系数场,我们将其视为系数场的固定和有限范围集合的实现。给定一个右手边,球的大小为\(\ ell \ gg 1 \),且平均值消失,我们对仅使用给定实现的知识就可以计算出接近原点的解的算法感兴趣。某个大小为\(L \ gg \ ell \)的大盒子中的系数字段。更确切地说,我们对大小L的计算域边界上最无缝的人工边界条件感兴趣。基于最近在随机介质中引入的多极扩展,我们提出了一种算法。我们严格按照以下公式建立关于梯度水平的误差估计:\(L \ gg \ ell \ gg 1 \),使用最近的结果进行定量随机均质化。更准确地说,我们的误差估计具有先验和后验方面:在具有先验和后验的可能性的情况下,根据大小框中的给定实现,预因子可以由无需进一步努力即可计算的常数限制。大号。我们还严格地建立了L\(\ ell \)中误差估计的顺序都是最优的,在本文中,我们重点讨论\(d = 2 \)的情况。。当以计算域内的系数为条件时,这相当于感兴趣量方差的下限,并且依赖于确定性见解,即对缺陷的敏感性分析会随着随机均质化而转变。最后,我们进行了数值实验,结果表明,该最优收敛速度仅在中等偏大的L时就已设定,并且更幼稚的边界条件在速度和预因子方面都表现得较差。

更新日期:2021-03-03
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