We are interested in numerical algorithms for computing the electrical field generated by a charge distribution localized on scale $l$ in an infinite heterogeneous correlated random medium, in a situation where the medium is only known in a box of diameter $L\gg l$ around the support of the charge. We show that the algorithm of Lu, Otto and Wang, suggesting optimal Dirichlet boundary conditions motivated by the multipole expansion of Bella, Giunti and Otto, still performs well in correlated media. With overwhelming probability, we obtain a convergence rate in terms of $l$, $L$ and the size of the correlations for which optimality is supported with numerical simulations. These estimates are provided for ensembles which satisfy a multi-scale logarithmic Sobolev inequality, where our main tool is an extension of the semi-group estimates established by the first author. As part of our strategy, we construct sub-linear second-order correctors in this correlated setting which is of independent interest.

翻译：我们关注数值算法，用于计算由尺度$l$上局域化的电荷分布生成的电场的无限非均匀相关随机介质中的电场，其中介质仅在电荷支持周围直径为$L\gg l$的盒子内已知。我们证明Lu、Otto和Wang提出的算法——该算法受Bella、Giunti和Otto的多极展开启发，建议最优Dirichlet边界条件——在相关介质中仍表现良好。在压倒性概率下，我们获得关于$l$、$L$和相关尺度的收敛速率，并通过数值模拟支持最优性。这些估计适用于满足多尺度对数Sobolev不等式的系综，我们的主要工具是第一位作者建立的半群估计的推广。作为策略的一部分，我们在这一相关设定中构造了次线性二阶校正子，这本身具有独立意义。