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Matrix Equations, Sparse Solvers: M-M.E.S.S.-2.0.1—Philosophy, Features, and Application for (Parametric) Model Order Reduction

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Model Reduction of Complex Dynamical Systems

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

Abstract

Matrix equations are omnipresent in (numerical) linear algebra and systems theory. Especially in model order reduction (MOR), they play a key role in many balancing-based reduction methods for linear dynamical systems. When these systems arise from spatial discretizations of evolutionary partial differential equations, their coefficient matrices are typically large and sparse. Moreover, the numbers of inputs and outputs of these systems are typically far smaller than the number of spatial degrees of freedom. Then, in many situations, the solutions of the corresponding large-scale matrix equations are observed to have low (numerical) rank. This feature is exploited by M-M.E.S.S. to find successively larger low-rank factorizations approximating the solutions. This contribution describes the basic philosophy behind the implementation and the features of the package, as well as its application in the MOR of large-scale linear time-invariant (LTI) systems and parametric LTI systems.

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Authors and Affiliations

  1. Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106, Magdeburg, Germany

    Peter Benner, Martin Köhler & Jens Saak

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  1. Peter Benner
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  2. Martin Köhler
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  3. Jens Saak
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Correspondence to Jens Saak .

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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

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Benner, P., Köhler, M., Saak, J. (2021). Matrix Equations, Sparse Solvers: M-M.E.S.S.-2.0.1—Philosophy, Features, and Application for (Parametric) Model Order Reduction. 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_18

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