A Highly Scalable Boundary Integral Equation and Walk-on-Spheres (BIE-WOS) Method for the Laplace Equation with Dirichlet Data
Year: 2021
Author: Changhao Yan, Wei Cai, Xuan Zeng
Communications in Computational Physics, Vol. 29 (2021), Iss. 5 : pp. 1446–1468
Abstract
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 local 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2020-0099
Communications in Computational Physics, Vol. 29 (2021), Iss. 5 : pp. 1446–1468
Published online: 2021-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 23
Keywords: Dirichlet-to-Neumann (DtN) mapping meshless methods parallel BIE-WOS method Monte Carlo method walk-on-spheres (WOS) boundary integral equations Laplace equation Dirichlet problem.
Author Details
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A Parallel Iterative Probabilistic Method for Mixed Problems of Laplace Equations with the Feynman--Kac Formula of Killed Brownian Motions
Ding, Cuiyang
Yan, Changhao
Zeng, Xuan
Cai, Wei
SIAM Journal on Scientific Computing, Vol. 44 (2022), Iss. 5 P.A3413
https://doi.org/10.1137/22M1478458 [Citations: 1]