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Residual-based iterations for the generalized Lyapunov equation
Institute for Mathematics and Scientific Computing, Karl-Franzens-Universität, Graz, 8010, Austria.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.ORCID iD: 0000-0001-6279-6145
2019 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, E-ISSN 1572-9125, Vol. 59, no 4, p. 823-852Article in journal (Refereed) Published
Abstract [en]

This paper treats iterative solution methods for the generalized Lyapunov equation. Specifically, a residual-based generalized rational-Krylov-type subspace is proposed. Furthermore, the existing theoretical justification for the alternating linear scheme (ALS) is extended from the stable Lyapunov equation to the stable generalized Lyapunov equation. Further insights are gained by connecting the energy-norm minimization in ALS to the theory of H2-optimality of an associated bilinear control system. Moreover it is shown that the ALS-based iteration can be understood as iteratively constructing rank-1 model reduction subspaces for bilinear control systems associated with the residual. Similar to the ALS-based iteration, the fixed-point iteration can also be seen as a residual-based method minimizing an upper bound of the associated energy norm.

Place, publisher, year, edition, pages
Springer Netherlands, 2019. Vol. 59, no 4, p. 823-852
Keywords [en]
Alternating linear scheme, Bilinear control systems, Generalized Lyapunov equation, H2-optimal model reduction, Matrix equations, Projection methods, Rational Krylov
National Category
Other Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-263286DOI: 10.1007/s10543-019-00760-9ISI: 000499644400001Scopus ID: 2-s2.0-85067791778OAI: oai:DiVA.org:kth-263286DiVA, id: diva2:1367845
Note

QC 20191105

Available from: 2019-11-05 Created: 2019-11-05 Last updated: 2020-01-02Bibliographically approved

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Ringh, Emil

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