Titelaufnahme

Titel
Low rank methods for a class of generalized Lyapunov equations and related issues / Peter Benner, Tobias Breiten
VerfasserBenner, Peter ; Breiten, Tobias
KörperschaftMax-Planck-Institut für Dynamik Komplexer Technischer Systeme
ErschienenMagdeburg : Max Planck Institute for Dynamics of Complex Technical Systems, February 21, 2012
Umfang1 Online-Ressource (25 Seiten = 0,44 MB) : Diagramme
SpracheEnglisch
SerieMax Planck Institute Magdeburg Preprints ; 12-03
URNurn:nbn:de:gbv:3:2-63947 
Zugriffsbeschränkung
 Das Dokument ist frei verfügbar
Dateien
Low rank methods for a class of generalized Lyapunov equations and related issues [0.44 mb]
Links
Nachweis
Klassifikation
Zusammenfassung

Abstract: In this paper, we study possible low rank solution methods for generalized Lyapunov equations arising in bilinear and stochastic control. We show that under certain assumptions one can expect a strong singular value decay in the solution matrix allowing for low rank approximations. Since the theoretical tools strongly make use of a connection to the standard linear Lyapunov equation, we can even extend the result to the d-dimensional case described by a tensorized linear system of equations. We further provide some reasonable extensions of some of the most frequently used linear low rank solution techniques such as the alternating directions implicit (ADI) iteration and the Krlyov-plus-inverse-Krylov (KPIK) method. By means of some standard numerical examples used in the area of bilinear model order reduction, we will show the efficiency of the new methods.

Keywords
Abstract: In this paper we study possible low rank solution methods for generalized Lyapunov equations arising in bilinear and stochastic control. We show that under certain assumptions one can expect a strong singular value decay in the solution matrix allowing for low rank approximations. Since the theoretical tools strongly make use of a connection to the standard linear Lyapunov equation we can even extend the result to the d-dimensional case described by a tensorized linear system of equations. We further provide some reasonable extensions of some of the most frequently used linear low rank solution techniques such as the alternating directions implicit (ADI) iteration and the Krlyov-plus-inverse-Krylov (KPIK) method. By means of some standard numerical examples used in the area of bilinear model order reduction we will show the efficiency of the new methods.