Distributed ℋ2-Matrices for Boundary Element Methods

IF 2.7 1区 数学 Q2 COMPUTER SCIENCE, SOFTWARE ENGINEERING
S. Börm
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引用次数: 0

Abstract

Standard discretization techniques for boundary integral equations, e.g., the Galerkin boundary element method, lead to large densely populated matrices that require fast and efficient compression techniques like the fast multipole method or hierarchical matrices. If the underlying mesh is very large, running the corresponding algorithms on a distributed computer is attractive, e.g., since distributed computers frequently are cost-effective and offer a high accumulated memory bandwidth. Compared to the closely related particle methods, for which distributed algorithms are well-established, the Galerkin discretization poses a challenge, since the supports of the basis functions influence the block structure of the matrix and therefore the flow of data in the corresponding algorithms. This article introduces distributed ℋ2-matrices, a class of hierarchical matrices that is closely related to fast multipole methods and particularly well-suited for distributed computing. While earlier efforts required the global tree structure of the ℋ2-matrix to be stored in every node of the distributed system, the new approach needs only local multilevel information that can be obtained via a simple distributed algorithm, allowing us to scale to significantly larger systems. Experiments show that this approach can handle very large meshes with more than 130 million triangles efficiently.
边界元法的分布h - 2矩阵
边界积分方程的标准离散化技术,如伽辽金边界元方法,导致大量密集的矩阵,需要快速有效的压缩技术,如快速多极方法或分层矩阵。如果底层网格非常大,则在分布式计算机上运行相应的算法是有吸引力的,例如,因为分布式计算机通常具有成本效益,并且提供较高的累积内存带宽。与密切相关的粒子方法相比,Galerkin离散化提出了一个挑战,因为基函数的支持会影响矩阵的块结构,从而影响相应算法中的数据流。本文介绍了分布式h 2矩阵,它是与快速多极方法密切相关的一类层次矩阵,特别适合于分布式计算。早期的研究需要在分布式系统的每个节点中存储全局的h 2矩阵树结构,而新的方法只需要局部的多层信息,这些信息可以通过一个简单的分布式算法获得,从而使我们能够扩展到更大的系统。实验表明,该方法可以有效地处理超过1.3亿个三角形的超大网格。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
ACM Transactions on Mathematical Software
ACM Transactions on Mathematical Software 工程技术-计算机:软件工程
CiteScore
5.00
自引率
3.70%
发文量
50
审稿时长
>12 weeks
期刊介绍: As a scientific journal, ACM Transactions on Mathematical Software (TOMS) documents the theoretical underpinnings of numeric, symbolic, algebraic, and geometric computing applications. It focuses on analysis and construction of algorithms and programs, and the interaction of programs and architecture. Algorithms documented in TOMS are available as the Collected Algorithms of the ACM at calgo.acm.org.
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