Cross-Gramian-based dominant subspaces
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A standard approach for model reduction of linear input-output systems is balanced truncation, which is based on the controllability and observability properties of the underlying system. The related dominant subspaces projection model reduction method similarly utilizes these system properties, yet instead of balancing, the associated subspaces are directly conjoined. In this work, we extend the dominant subspace approach by computation via the cross Gramian for linear systems, and describe an a-priori error indicator for this method. Furthermore, efficient computation is discussed alongside numerical examples illustrating these findings.
KeywordsControllability Observability Cross Gramian Model reduction Dominant subspaces HAPOD DSPMR
Mathematics Subject Classification (2010)93A15 93B11 93B20
This work is dedicated to the late Thilo Penzl, who wrote the preprint version of  20 years (at this time of writing) ago, in 1999, and, moreover, 2019 marks the year of his 20th death anniversary. Thilo Penzl died December 17, 1999, but his work and ideas inspire researchers in model reduction and matrix equations to date.
The authors thank the two anonymous reviewers for their helpful feedback and comments.
Open access funding provided by Max Planck Society. This study is supported by the German Federal Ministry for Economic Affairs and Energy (BMWi), in the joint project: “MathEnergy – Mathematical Key Technologies for Evolving Energy Grids,” sub-project: Model Order Reduction (Grant No. 0324019B).
Code availability section
The source code of the presented numerical examples can be obtained from: http://runmycode.org/companion/view/3270 and is authored by: Christian Himpe.
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
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