Paper Info
Title
Krylov methods for low-rank commuting generalized Sylvester 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 commutators 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, that is, 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 the 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.
Year
DOI
Venue
2018
10.1002/nla.2176
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
Keywords
Field
DocType
generalized Sylvester equation,iterative solvers,Krylov subspace,low-rank commutation,matrix equation,projection methods
Krylov subspace,Discretization,Applied mathematics,Sylvester equation,Matrix (mathematics),Mathematical analysis,Projection method,Linear map,Operator (computer programming),Sylvester matrix,Mathematics
Journal
Volume
Issue
ISSN
25.0
SP6.0
1070-5325
Citations 
PageRank 
References 
2
0.64
27
Authors
4
Name
Order
Citations
PageRank
Jarlebring Elias18411.48
Giampaolo Mele220.64
Davide Palitta341.35
Emil Ringh420.64