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Cross-Gramian-based dominant subspaces

  • Peter Benner
  • Christian HimpeEmail author
Open Access
Article
Part of the following topical collections:
  1. Model reduction of parametrized Systems

Abstract

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.

Keywords

Controllability Observability Cross Gramian Model reduction Dominant subspaces HAPOD DSPMR 

Mathematics Subject Classification (2010)

93A15 93B11 93B20 

Notes

Acknowledgments

This work is dedicated to the late Thilo Penzl, who wrote the preprint version of [32] 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.

Funding information

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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© The Author(s) 2019

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Computational Methods in Systems and Control TheoryMax Planck Institute for Dynamics of Complex Technical SystemsMagdeburgGermany
  2. 2.Faculty of MathematicsOtto von Guericke University MagdeburgMagdeburgGermany

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