演化Multiply-Connected Domains中积分方程的并行离散化
Parallel Skeletonization for Integral Equations in Evolving Multiply-Connected Domains
摘要：
本文给出了一种将分层矩阵分解应用于含可能秩亏积分算子的边界积分方程数值解的一般方法。秩亏算子出现在边界积分方法中的椭圆偏微分方程与多个边界组件，如在多个囊泡的情况下，在粘性流体流动。我们的广义tenonization因式分解保留了“代理点方法”提供的局部属性，并允许并行实现，其中不同的处理器同时在边界的不同部分上工作。此外，当边界经历局部几何扰动（例如内部孔的移动）时，可以相对于修改的离散化节点的数量有效地重新计算因子分解。我们提出了一个应用程序，利用并行实现的形状优化制度中的更新。
英文摘要：
This paper presents a general method for applying hierarchical matrix skeletonization factorizations to the numerical solution of boundary integral equations with possibly rank-deficient integral operators. Rank-deficient operators arise in boundary integral approaches to elliptic partial differential equations with multiple boundary components, such as in the case of multiple vesicles in a viscous fluid flow. Our generalized skeletonization factorization retains the locality property afforded by the "proxy point method," and allows for a parallelized implementation where different processors work on different parts of the boundary simultaneously. Further, when the boundary undergoes local geometric perturbations (such as movement of an interior hole), the factorization can be recomputed efficiently with respect to the number of modified discretization nodes. We present an application that leverages a parallel implementation of skeletonization with updates in a shape optimization regime.
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