MinAres: An Iterative Solver for Symmetric Linear Systems

TL;DR

MINARES is a new iterative solver for symmetric linear systems that minimizes the residual of the residuals, offering advantages over existing methods especially for inconsistent systems, with efficient computation and strong theoretical properties.

Contribution

Introduces MINARES, an iterative solver based on the symmetric Lanczos process that minimizes r_k instead of r_k, and analyzes its properties and relation to other methods.

Findings

MINARES is effective for both consistent and inconsistent symmetric systems.

MINARES requires only one matrix-vector product per iteration.

MINARES and CAR generate monotonic convergence in certain norms.

Abstract

We introduce an iterative solver named MINARES for symmetric linear systems $Ax \approx b$, where $A$ is possibly singular. MINARES is based on the symmetric Lanczos process, like MINRES and MINRES-QLP, but it minimizes $\|Ar_k\|$ in each Krylov subspace rather than $\|r_k\|$, where $r_k$ is the current residual vector. When $A$ is symmetric, MINARES minimizes the same quantity $\|Ar_k\|$ as LSMR, but in more relevant Krylov subspaces, and it requires only one matrix-vector product $Av$ per iteration, whereas LSMR would need two. Our numerical experiments with MINRES-QLP and LSMR show that MINARES is a pertinent alternative on consistent symmetric systems and the most suitable Krylov method for inconsistent symmetric systems. We derive properties of MINARES from an equivalent solver named CAR that is to MINARES as CR is to MINRES, is not based on the Lanczos process, and minimizes $\|Ar_k\|$ in the same Krylov subspace as MINARES. We establish that MINARES and CAR generate monotonic $\|x_k - x_{\star}\|$, $\|x_k - x_{\star}\|_A$ and $\|r_k\|$ when $A$ is positive definite.