Inner-iteration preconditioning with a symmetric splitting matrix for rank-deficient least squares problems

TL;DR

This paper develops a theoretical framework for inner-iteration preconditioning with symmetric splitting matrices, enabling Krylov methods like CG and MINRES to effectively solve rank-deficient least squares problems.

Contribution

It establishes conditions for definite preconditioning matrices and extends convergence analysis of Krylov methods to rank-deficient systems with inner-iteration preconditioning.

Findings

Inner-iteration preconditioning matrices can be definite under certain conditions.

Preconditioned CG and MINRES methods solve symmetric and positive semidefinite systems, including singular cases.

Theoretical bounds and convergence justifications for these methods are provided.

Abstract

Stationary iterative methods with a symmetric splitting matrix are performed as inner-iteration preconditioning for Krylov subspace methods. We give conditions such that the inner-iteration preconditioning matrix is definite, and show that conjugate gradient (CG) method preconditioned by the inner iterations determines a solution of symmetric and positive semidefinite linear systems, and the minimal residual (MINRES) method preconditioned by the inner iterations determines a solution of symmetric linear systems including the singular case. These results are applied to the CG and MINRES-type methods such as the CGLS, LSMR, and CGNE methods preconditioned by inner iterations, and thus justify using these methods for solving least squares and minimum-norm solution problems whose coefficient matrices are not necessarily of full rank. Thus, we complement the convergence theories of these methods presented in [K. Morikuni and K. Hayami, SIAM J. Matrix Appl. Anal., 34 (2013), pp.1-22], [K. Morikuni and K. Hayami, SIAM J. Matrix Appl. Anal., 36 (2015), pp. 225-250], and give bounds for these methods.