A Low-Rank BUG Method for Sylvester-Type Equations

TL;DR

This paper presents a low-rank algorithm based on the BUG integrator for efficiently solving Sylvester-type equations by exploiting low-rank and sparsity structures, significantly reducing computational complexity.

Contribution

The paper introduces a novel low-rank BUG-based method that improves efficiency in solving Sylvester equations by leveraging structure and reducing complexity.

Findings

Reduces computational complexity to O(kr(n^2+m^2+mn+r^2))

Exploits low-rank and sparsity in solutions

Compatible with dense solvers like Bartels-Stewart

Abstract

We introduce a low-rank algorithm inspired by the Basis-Update and Galerkin (BUG) integrator to efficiently approximate solutions to Sylvester-type equations. The algorithm can exploit both the low-rank structure of the solution as well as any sparsity present to reduce computational complexity. Even when a standard dense solver, such as the Bartels-Stewart algorithm, is used for the reduced Sylvester equations generated by our approach, the overall computational complexity for constructing and solving the associated linear systems reduces to O(kr(n^2+m^2 +mn + r^2)), for X in R^{m \times n}, where k is the number of iterations and r the rank of the approximation.