Paper Info
Title
A Riemannian Optimization Approach for Computing Low-Rank Solutions of Lyapunov Equations
Abstract
We propose a new framework based on optimization on manifolds to approximate the solution of a Lyapunov matrix equation by a low-rank matrix. The method minimizes the error on the Riemannian manifold of symmetric positive semidefinite matrices of fixed rank. We detail how objects from differential geometry, like the Riemannian gradient and Hessian, can be efficiently computed for this manifold. As a minimization algorithm we use the Riemannian trust-region method of [P.-A. Absil, C. Baker, and K. Gallivan, Found. Comput. Math., 7 (2007), pp. 303-330] based on a second-order model of the objective function on the manifold. Together with an efficient preconditioner, this method can find low-rank solutions with very little memory. We illustrate our results with numerical examples.
Year
DOI
Venue
2010
10.1137/090764566
SIAM J. Matrix Analysis Applications
Keywords
Field
DocType
riemannian manifold,low-rank solution,lyapunov matrix equation,93b40,65n22,low-rank matrix,c. baker,90c26,riemannian optimization approach,low-rank approximations,lyapunov equations,computing low-rank solutions,58c05,riemannian gradient,k. gallivan,riemannian trust-region method,large-scale equations msc : 65f10,65k05,differential geometry,: lyapunov matrix equations,numerical optimization on manifolds,symmetric positive semidefinite matrix,low rank approximation,matrix equation,objective function,positive semi definite,lyapunov equation,second order
Information geometry,Ricci curvature,Mathematical analysis,Riemannian manifold,Statistical manifold,Riemannian geometry,Fundamental theorem of Riemannian geometry,Exponential map (Riemannian geometry),Pseudo-Riemannian manifold,Mathematics
Journal
Volume
Issue
ISSN
31
5
0895-4798
Citations 
PageRank 
References 
23
1.33
19
Authors
2
Name
Order
Citations
PageRank
Bart Vandereycken119910.21
Stefan Vandewalle250162.63