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Comparing (Empirical-Gramian-Based) Model Order Reduction Algorithms

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Model Reduction of Complex Dynamical Systems

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 171))

Abstract

In this work, the empirical-Gramian-based model reduction methods: Empirical poor man’s truncated balanced realization, empirical approximate balancing, empirical dominant subspaces, empirical balanced truncation, and empirical balanced gains are compared in a non-parametric and in two parametric variants, via ten error measures: Approximate Lebesgue \(L_0\), \(L_1\), \(L_2\), \(L_\infty \), Hardy \(H_2\), \(H_\infty \), Hankel, Hilbert-Schmidt-Hankel, modified induced primal, and modified induced dual norms, for variants of the thermal block model reduction benchmark. This comparison is conducted via a new meta-measure for model reducibility called MORscore.

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Notes

  1. 1.

    Specifically via: https://www.mathworks.com/help/matlab/ref/trapz.html.

  2. 2.

    Unstable ROMs are treated as relative error of one, \(\varepsilon = 1\), in the method’s MORscores.

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Acknowledgements

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 number: 0324019B).

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Authors and Affiliations

  1. Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106, Magdeburg, Germany

    Christian Himpe

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  1. Christian Himpe
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Correspondence to Christian Himpe .

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Editors and Affiliations

  1. Dynamics of Complex Technical Systems, Max Planck Institute, Magdeburg, Sachsen-Anhalt, Germany

    Peter Benner

  2. Institute of Mathematics, Technical University of Berlin, Berlin, Germany

    Tobias Breiten

  3. Institute for Numerical Analysis, TU Braunschweig, Braunschweig, Germany

    Heike Faßbender

  4. Mathematisches Institut, Universität Koblenz-Landau, Campus Koblenz, Koblenz, Germany

    Michael Hinze

  5. Institut für Mathematik, Universität Augsburg, Augsburg, Germany

    Tatjana Stykel

  6. Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark

    Ralf Zimmermann

Appendix

Appendix

1.1 Single Parameter Benchmark MORscores

Table 2 MORscore \(\mu (50,\epsilon _{\text {mach}}(\text {dp})\)) for the single parameter benchmark (\(L_1 \otimes X\))
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Table 3 MORscore \(\mu (50,\epsilon _{\text {mach}}(\text {dp})\)) for the single parameter benchmark (\(L_2 \otimes X\))
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Table 4 MORscore \(\mu (50,\epsilon _{\text {mach}}(\text {dp})\)) for the single parameter benchmark (\(L_\infty \otimes X\))
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1.2 Multi Parameter Benchmark MORscores

Table 5 MORscore \(\mu (50,\epsilon _{\text {mach}}(\text {dp})\)) for the multi parameter benchmark (\(L_1 \otimes X\))
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Table 6 MORscore \(\mu (50,\epsilon _{\text {mach}}(\text {dp})\)) for the multi parameter benchmark (\(L_2 \otimes X\))
Full size table
Table 7 MORscore \(\mu (50,\epsilon _{\text {mach}}(\text {dp})\)) for the multi parameter benchmark (\(L_\infty \otimes X\))
Full size table

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Himpe, C. (2021). Comparing (Empirical-Gramian-Based) Model Order Reduction Algorithms. In: Benner, P., Breiten, T., Faßbender, H., Hinze, M., Stykel, T., Zimmermann, R. (eds) Model Reduction of Complex Dynamical Systems. International Series of Numerical Mathematics, vol 171. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-72983-7_7

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