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Efficient Construction of an HSS Preconditioner for Symmetric Positive Definite $\mathcal{H}^2$ Matrices

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.

Ketersediaan

Informasi Detail

Judul Seri

SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS; Vol.42 No.2 April-June 2021

No. Panggil

-

Penerbit

: .,

Deskripsi Fisik

p. 683 - 707

Bahasa

English

ISBN/ISSN

-

Klasifikasi

-

Tipe Isi

-

Tipe Media

-

Tipe Pembawa

-

Edisi

-

Subjek

KERNEL MATRIX
SYMMETRIC POSITIVE DEFINITE PRECONDITIONER
HSS MATRIX REPRESENTATION
$\MATHCAL{H}^2$ MATRIX REPRESENTATION

Info Detail Spesifik

https://doi.org/10.1137/20M1365776

Pernyataan Tanggungjawab

Xin Xing, Hua Huang, Edmond Chow

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