A class of low-rank short recurrences for nonsymmetric linear matrix equations

TL;DR

This paper introduces a new class of short recurrences for efficiently solving nonsymmetric linear matrix equations, combining projection, rank truncation, and randomization to improve convergence and reduce memory use.

Contribution

It presents a novel iterative method that integrates local subspace projection, rank truncation, and randomization for solving complex nonsymmetric matrix equations.

Findings

Demonstrates effectiveness on benchmark problems.

Shows potential in solving diffusion equations with random inputs.

Abstract

We propose a new class of short matrix recurrences for the solution of nonsymmetric linear equations of the type $\mathbf{A}_1\mathbf{X}\mathbf{B}_1+\ldots+\mathbf{A}_p\mathbf{X}\mathbf{B}_p=CD^T$. These iterative methods combine local subspace projection to speed up convergence with rank truncation strategies and randomization procedures to limit memory consumption. Computational experiments on a benchmark problem as well as a challenging discretized mixed formulation of a diffusion equation with random inputs illustrate the potential of the proposed methodology.