当前位置: X-MOL 学术SIAM J. Matrix Anal. Appl. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Efficient Construction of an HSS Preconditioner for Symmetric Positive Definite $\mathcal{H}^2$ Matrices
SIAM Journal on Matrix Analysis and Applications ( IF 1.5 ) Pub Date : 2021-04-27 , DOI: 10.1137/20m1365776
Xin Xing , Hua Huang , Edmond Chow

SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 2, Page 683-707, January 2021.
In an iterative approach for solving linear systems with dense, ill-conditioned, symmetric positive definite (SPD) kernel matrices, both fast matrix-vector products and fast preconditioning operations are required. Fast (linear-scaling) matrix-vector products are available by expressing the kernel matrix in an $\mathcal{H}^2$ representation or an equivalent fast multipole method representation. This paper is concerned with preconditioning such matrices using the hierarchically semiseparable (HSS) matrix representation. Previously, an algorithm was presented to construct an HSS approximation to an SPD kernel matrix that is guaranteed to be SPD. However, this algorithm has quadratic cost and was only designed for recursive binary partitionings of the points defining the kernel matrix. This paper presents a general algorithm for constructing an SPD HSS approximation. Importantly, the algorithm uses the $\mathcal{H}^2$ representation of the SPD matrix to reduce its computational complexity from quadratic to quasilinear. Numerical experiments illustrate how this SPD HSS approximation performs as a preconditioner for solving linear systems arising from a range of kernel functions.


中文翻译:

对称正定$\mathcal{H}^2$矩阵的HSS预处理器的有效构造

SIAM 矩阵分析与应用杂志,第 42 卷,第 2 期,第 683-707 页,2021 年 1 月。
在求解具有密集、病态、对称正定 (SPD) 核矩阵的线性系统的迭代方法中,需要快速矩阵向量乘积和快速预处理操作。通过在 $\mathcal{H}^2$ 表示或等效的快速多极方法表示中表达内核矩阵,可以获得快速(线性缩放)矩阵向量乘积。本文涉及使用分层半可分 (HSS) 矩阵表示对此类矩阵进行预处理。之前,提出了一种算法来构建 SPD 内核矩阵的 HSS 近似,该矩阵保证是 SPD。但是,该算法具有二次成本,并且仅设计用于定义内核矩阵的点的递归二进制分区。本文提出了一种构建 SPD HSS 近似的通用算法。重要的是,该算法使用 SPD 矩阵的 $\mathcal{H}^2$ 表示将其计算复杂度从二次降低到拟线性。数值实验说明了这种 SPD HSS 近似如何作为求解由一系列核函数产生的线性系统的预处理器。
更新日期:2021-06-22
down
wechat
bug