Low-rank updates and a divide-and-conquer method for linear matrix equations

TL;DR

This paper introduces a tensorized Krylov subspace algorithm for efficiently updating solutions of linear matrix equations with low-rank coefficient changes, and proposes a divide-and-conquer method for structured matrices, improving computational efficiency.

Contribution

The paper presents a novel tensorized Krylov subspace algorithm for low-rank updates and a divide-and-conquer approach for structured matrix equations, enhancing speed and memory efficiency.

Findings

Algorithm accelerates solution updates for low-rank coefficient changes.

Divide-and-conquer method outperforms existing approaches in time and memory.

Numerical experiments validate the efficiency and effectiveness of the proposed methods.

Abstract

Linear matrix equations, such as the Sylvester and Lyapunov equations, play an important role in various applications, including the stability analysis and dimensionality reduction of linear dynamical control systems and the solution of partial differential equations. In this work, we present and analyze a new algorithm, based on tensorized Krylov subspaces, for quickly updating the solution of such a matrix equation when its coefficients undergo low-rank changes. We demonstrate how our algorithm can be utilized to accelerate the Newton method for solving continuous-time algebraic Riccati equations. Our algorithm also forms the basis of a new divide-and-conquer approach for linear matrix equations with coefficients that feature hierarchical low-rank structure, such as HODLR, HSS, and banded matrices. Numerical experiments demonstrate the advantages of divide-and-conquer over existing approaches, in terms of computational time and memory consumption.