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Cross-Gramian-based dominant subspaces


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.


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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).

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The source code of the presented numerical examples can be obtained from: and is authored by: Christian Himpe.

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Correspondence to Christian Himpe.

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Communicated by: Anthony Nouy

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Benner, P., Himpe, C. Cross-Gramian-based dominant subspaces. Adv Comput Math 45, 2533–2553 (2019).

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  • Controllability
  • Observability
  • Cross Gramian
  • Model reduction
  • Dominant subspaces

Mathematics Subject Classification (2010)

  • 93A15
  • 93B11
  • 93B20