# Krylov methods for low-rank commuting generalized Sylvester equations

## TL;DR

This paper introduces a new Krylov subspace projection method tailored for generalized Sylvester equations with low-rank commutators, enabling efficient low-rank approximations for large-scale problems in control and PDE discretization.

## Contribution

It develops an extended Krylov subspace method specifically adapted for low-rank structured Sylvester equations, with a constructive approach for initial vectors based on low-rank commutators.

## Key findings

- Effective for large-scale control and PDE problems
- Outperforms existing methods in efficiency and accuracy
- Enables low-rank solutions for complex matrix equations

## Abstract

We consider generalizations of the Sylvester matrix equation, consisting of the sum of a Sylvester operator and a linear operator $\Pi$ with a particular structure. More precisely, the commutator of the matrix coefficients of the operator $\Pi$ and the Sylvester operator coefficients are assumed to be matrices with low rank. We show (under certain additional conditions) low-rank approximability of this problem, i.e., the solution to this matrix equation can be approximated with a low-rank matrix. Projection methods have successfully been used to solve other matrix equations with low-rank approximability. We propose a new projection method for this class of matrix equations. The choice of subspace is a crucial ingredient for any projection method for matrix equations. Our method is based on an adaption and extension of the extended Krylov subspace method for Sylvester equations. A constructive choice of the starting vector/block is derived from the low-rank commutators. We illustrate the effectiveness of our method by solving large-scale matrix equations arising from applications in control theory and the discretization of PDEs. The advantages of our approach in comparison to other methods are also illustrated.