Efficient construction of an HSS preconditioner for symmetric positive definite $\mathcal{H}^2$ matrices

TL;DR

This paper introduces a new, efficient algorithm for constructing an SPD HSS matrix approximation from an $ ext{H}^2$ representation, significantly reducing computational complexity and improving preconditioning for kernel-based linear systems.

Contribution

The paper presents a general, quasilinear algorithm for constructing SPD HSS approximations from $ ext{H}^2$ matrices, enhancing preconditioning efficiency for symmetric positive definite kernel matrices.

Findings

The new algorithm reduces construction complexity from quadratic to quasilinear.

Numerical experiments demonstrate effective preconditioning across various kernel functions.

The method improves the scalability of solving large SPD linear systems.

Abstract

In an iterative approach for solving linear systems with 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. Preconditioning such matrices, however, requires a structured matrix approximation that is more regular than the $\mathcal{H}^2$ representation, such as the hierarchically semiseparable (HSS) matrix representation, which provides fast solve operations. 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.