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A Robust Algorithm for Parametric Model Order Reduction Based on Implicit Moment Matching

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Reduced Order Methods for Modeling and Computational Reduction

Part of the book series: MS&A - Modeling, Simulation and Applications ((MS&A,volume 9))

Abstract

Parametric model order reduction (PMOR) has received a tremendous amount of attention in recent years. Among the first approaches considered, mainly in system and control theory as well as computational electromagnetics and nanoelectronics, are methods based on multi-moment matching. Despite numerous other successful methods, including the reduced-basis method (RBM), other methods based on (rational, matrix, manifold) interpolation, or Kriging techniques, multi-moment matching methods remain a reliable, robust, and flexible method for model reduction of linear parametric systems. Here we propose a numerically stable algorithm for PMOR based on multi-moment matching. Given any number of parameters and any number of moments of the parametric system, the algorithm generates a projection matrix for model reduction by implicit moment matching. The implementation of the method based on a repeated modified Gram-Schmidt-like process renders the method numerically stable. The proposed method is simple yet efficient. Numerical experiments show that the proposed algorithm is very accurate.

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Notes

  1. 1.

    1 See http://modelreduction.org.

  2. 2.

    2 Here we use a nonzero expansion point for the Laplace variable s, s 0 = 0.001, a zero expansion point for k, k 0 = 0, to ensure that the matrix \(\tilde E\) is nonsingular. For all the simulation results in Sect. 6.5.1, the same expansion points are taken for all the tested MOR methods: the non-parametric moment-matching MOR, the explicit multi-moment matching and the proposed Algorithm 6.1.

  3. 3.

    3 http://morwiki.mpi-magdeburg.mpg.de/morwiki/index.php/Scanning_Electrochemical_Microscopy.

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Authors and Affiliations

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

    Peter Benner & Lihong Feng

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  1. Peter Benner
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  2. Lihong Feng
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Correspondence to Lihong Feng .

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Editors and Affiliations

  1. CMCS-MATHICSE, Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland

    Alfio Quarteroni

  2. MOX, Department of Mathematics “F. Brioschi”, Politecnico di Milano, Milan, Italy

    Alfio Quarteroni

  3. SISSA mathLab, International School for Advanced Studies, Trieste, Italy

    Gianluigi Rozza

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Benner, P., Feng, L. (2014). A Robust Algorithm for Parametric Model Order Reduction Based on Implicit Moment Matching. In: Quarteroni, A., Rozza, G. (eds) Reduced Order Methods for Modeling and Computational Reduction. MS&A - Modeling, Simulation and Applications, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-02090-7_6

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