A Highly Scalable Boundary Integral Equation and Walk-On-Spheres (BIE-WOS) Method for the Laplace Equation with Dirichlet Data
A Highly Scalable Boundary Integral Equation and Walk-On-Spheres (BIE-WOS) Method for the Laplace Equation with Dirichlet Data
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DOI：
10.4208/cicp.oa-2020-0099
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发表时间：
2021-06
影响因子：
3.7
通讯作者：
Changhao Yan
中科院分区：
文献类型：
--
作者：
Changhao Yan
.基于边界积分方程与球面行走（BIE-WOS）方法相结合的混合方法，研究了拉普拉斯方程的一种高可扩展性的无通信并行区域边界分解算法，该算法提供了拉普拉斯方程Dirichlet-to-Neumann（DtN）映射的数值逼近. BIE-WOS是一种在区域边界上的局部方法，不需要结构化网格，只需要用面片覆盖区域边界，并为每个面片建立局部BIE网格。为了更好地控制误差，引入了一种新的第二类积分方程的BIE-WOS方法。分析了基于Feynman-Kac公式的路径积分WOS方法的误差对BIE-WOS方法整体精度的影响，特别是在BIE右侧的计算中。对于特殊的非光滑曲面，本文证明了BIE-WOS方法的第二类积分方程可以简化，从而使局部BIE解可以用封闭形式给出.并行BIE-WOS方法的一个关键优点是在计算边界的各个补丁上的DtN映射期间没有通信，从而使用大量的集群节点进行完全独立的计算。此外，BIE-WOS具有固有的艾级计算容错能力。在6400个CPU核的大规模集群上，并行BIE-WOS方法的近似线性可扩展性被验证，艾德用于计算多个域Dirichlet数据的外拉普拉斯问题的DtN映射.
. In this paper, we study a highly scalable communication-free parallel domain boundary decomposition algorithm for the Laplace equation based on a hybrid method combining boundary integral equations and walk-on-spheres (BIE-WOS) method, which provides a numerical approximation of the Dirichlet-to-Neumann (DtN) mapping for the Laplace equation. The BIE-WOS is a local method on the boundary of the domain and does not require a structured mesh, and only needs a covering of the domain boundary by patches and a local mesh for each patch for a lo-cal BIE. A new version of the BIE-WOS method with second kind integral equations is introduced for better error controls. The effect of errors from the Feynman-Kac formula based path integral WOS method on the overall accuracy of the BIE-WOS method is analyzed for the BIEs, especially in the calculation of the right hand sides of the BIEs. For the special case of flat patches, it is shown that the second kind integral equation of BIE-WOS method can be simplified where the local BIE solutions can be given in closed forms. A key advantage of the parallel BIE-WOS method is the absence of communications during the computation of the DtN mapping on individual patches of the boundary, resulting in a complete independent computation using a large number of cluster nodes. In addition, the BIE-WOS has an intrinsic capability of fault tolerance for exascale computations. The nearly linear scalability of the parallel BIE-WOS method on a large-scale cluster with 6400 CPU cores is verified for computing the DtN mapping of exterior Laplace problems with Dirichlet data for several domains.