论文标题
分布式$ \ MATHCAL {H}^2 $ - 边界元素方法的matrices
Distributed $\mathcal{H}^2$-matrices for boundary element methods
论文作者
论文摘要
边界积分方程的标准离散技术，例如Galerkin边界元素方法，导致大量填充的矩阵需要快速有效的压缩技术，例如快速多极方法或层次矩阵。如果基础网格非常大，则在分布式计算机上运行相应的算法很有吸引力，例如，由于分布式计算机经常具有成本效益，并提供高累积的内存带宽。 与密切相关的粒子方法相比，分布式算法是良好的，盖尔金离散构成了挑战，因为基本函数的支持会影响矩阵的块结构，因此相应的算法中的数据流。本文介绍了分布式$ \ MATHCAL {H}^2 $ -MATRICES，这是一类与快速多极方法密切相关的层次矩阵，特别适合分布式计算。尽管较早的努力要求$ \ Mathcal {H}^2 $ -Matrix的全球树结构存储在分布式系统的每个节点中，但新方法仅需要可以通过简单的分布式算法获得的本地多级信息，从而使我们能够扩展到显着更大的系统。 实验表明，这种方法可以有效地处理超过1.30亿美元的三角形的大型网眼。
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 $\mathcal{H}^2$-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 $\mathcal{H}^2$-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.
