随机椭圆介质最优人工边界条件
Optimal artificial boundary condition for random elliptic media
AI总结：
本文研究随机椭圆介质中计算解梯度的最优人工边界条件，通过定量随机正交化理论建立误差估计，并通过数值实验验证最优收敛率。
摘要：
我们给定一个均匀椭圆系数场，视为一个平稳且有限范围（例如范围为1）的系数场实现。给定一个（确定性）右端项，其在大小为ℓ≫1的球内支撑，且平均值趋于零，我们感兴趣于一种算法，仅利用系数场在大小为L≫ℓ的较大盒内的已知信息，计算近原点处的（解的）梯度。受最近引入的随机介质多重展开启发，我们提出一种算法。我们严格建立误差估计（在梯度水平上），以L≫ℓ≫1为参数，利用最近的定量随机正交化结果。更具体地说，我们的误差估计具有先验和后验方面：在先验概率下，随机的前因子可以被一个常数界所限制，该常数基于给定的盒内实现计算。我们还严格证明误差估计在L和ℓ的阶数是最佳的，本文聚焦于d=2的情况。这相当于在计算域内系数条件下感兴趣的量方差的下界，并依赖于确定性洞察：缺陷敏感性分析相对于缺陷与（随机）正交化交换。最后，我们进行了数值实验，显示这种最优收敛率在仅中等大小的L时就已出现，而更简单的边界条件在收敛率和前因子方面表现更差。
英文摘要：
We are given a uniformly elliptic coefficient field that we regard as a realization of a stationary and finite-range (say, range unity) ensemble of coefficient fields. Given a (deterministic) 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 (gradient of 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 (random) 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 wrt 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.
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