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An Iterative Model Reduction Scheme for Quadratic-Bilinear Descriptor Systems with an Application to Navier–Stokes Equations

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Reduced-Order Modeling (ROM) for Simulation and Optimization

Abstract

We discuss an interpolatory model reduction framework for quadratic-bilinear (QB) descriptor systems, arising especially from the semi-discretization of the Navier–Stokes equations. Several recent results indicate that directly applying interpolatory model reduction frameworks, developed for systems of ordinary differential equations, to descriptor systems, may lead to an unbounded error between the original and reduced-order systems, e.g., in the \(\mathscr {H}_2\)-norm, due to an inappropriate treatment of the polynomial part of the original system. Thus, the main goal of this article is to extend the recently studied interpolation-based optimal model reduction framework for QB ordinary differential equations (QBODEs) to aforementioned descriptor systems while ensuring bounded error. For this, we first aim at transforming the descriptor system into an equivalent ODE system by means of projectors for which standard model reduction techniques can be applied. Subsequently, we discuss how to construct optimal reduced systems corresponding to an equivalent ODE, without requiring explicit computation of the expensive projection used in the analysis. The efficiency of the proposed algorithm is illustrated by means of a numerical example, obtained via semi-discretization of the Navier–Stokes equations.

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Acknowledgements

The authors would like to thank Dr. Jan Heiland for providing the lid-driven cavity model, used in the numerical experiment.

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Authors and Affiliations

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

    Peter Benner & Pawan Goyal

Authors
  1. Peter Benner
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  2. Pawan Goyal
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Correspondence to Pawan Goyal .

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Editors and Affiliations

  1. Department of Corporate Research, Robert Bosch GmbH, Renningen, Germany

    Winfried Keiper

  2. Interdisciplinary Center for Scientific Computing, Heidelberg University, Heidelberg, Germany

    Anja Milde

  3. Fachbereich Mathematik, Universität Konstanz, Konstanz, Germany

    Stefan Volkwein

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Benner, P., Goyal, P. (2018). An Iterative Model Reduction Scheme for Quadratic-Bilinear Descriptor Systems with an Application to Navier–Stokes Equations. In: Keiper, W., Milde, A., Volkwein, S. (eds) Reduced-Order Modeling (ROM) for Simulation and Optimization. Springer, Cham. https://doi.org/10.1007/978-3-319-75319-5_1

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