Inexact versions of several block-splitting preconditioners for indefinite least squares problems

TL;DR

This paper develops inexact block-splitting preconditioners for indefinite least squares problems, analyzes their convergence and eigenvalue properties, and demonstrates their effectiveness in accelerating GMRES through numerical experiments.

Contribution

It introduces and analyzes inexact block-splitting preconditioners for indefinite least squares problems, providing convergence conditions and iteration bounds.

Findings

Eigenvalues of preconditioned matrices lie within a circle centered at (1,0) with radius 1.

Preconditioners effectively accelerate GMRES convergence.

Numerical experiments confirm the efficacy of the proposed preconditioners.

Abstract

This paper introduces inexact versions of several block-splitting preconditioners for solving the three-by-three block linear systems arising from a special class of indefinite least squares problems. We first establish the convergence conditions for the corresponding stationary iterative methods. Then, it follows that under these conditions, all eigenvalues of the preconditioned matrices are contained within a circle centered at $(1,0)$ with radius $1$. This property implies that these preconditioners are effective in accelerating the convergence of the GMRES method. Furthermore, we analyze the eigenpairs of the preconditioned matrices in detail and derive a theoretical upper bound on the number of GMRES iterations for solving the preconditioned systems. Ultimately, numerical experiments reveal the efficacy of the proposed preconditioners.