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Dynamic Mode Decomposition—A Numerical Linear Algebra Perspective

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The Koopman Operator in Systems and Control

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 484))

Abstract

Data-driven scenarios in analysis and modeling of complex dynamical systems pose formidable challenges to computational science and motivate development of new numerical methods with ever increasing level of sophistication and complexity. An excellent example is computational fluid dynamics (CFD), where the dynamic mode decomposition (DMD) and its enhancement, the sparsity promoting DMD (DMDSP) have emerged as tools of trade for analysis of flow field data, with a host of applications. The problems of high dimension, noisy data, theoretical foundation in connection with the Koopman operator and ergodic theory, and potential applicability in broad spectrum of engineering problems have triggered extensive research and resulted in many theoretical and computational advancements of the DMD framework. This chapter revisits the DMD from the core numerical linear algebra perspective. Recent results on improving numerical robustness and functionality of DMD are reviewed and supplemented with new insights. Further, a new variation of the algorithm is proposed and used as a case study for development of a numerical algorithm in the DMD framework.

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Notes

  1. 1.

    Both MATLAB and NumPy actually use LAPACK as a computing engine for most matrix operations, in particular for the SVD and computing eigenvalues and eigenvectors of general matrices.

  2. 2.

    In MATLAB, the function \(\texttt {sqrtm(.)}\) computes the matrix principal square root.

  3. 3.

    See Sect. 7.6, where we briefly discuss recently proposed scheme [22] that uses the inverse of \(\mathbb {V}_m\).

  4. 4.

    See [66] for a definition and basis properties of Krylov decomposition.

  5. 5.

    This is sometimes called mixed stability: backward error corresponds to slightly changed output.

  6. 6.

    For an earlier, extended version see [21].

  7. 7.

    In an example constructed by Kahan [47], the pivoting fails to reveal one small singular value.

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Acknowledgements

This research is supported by the DARPA Contract HR0011-16-C-0116 “On a Data-Driven, Operator-Theoretic Framework for Space–Time Analysis of Process Dynamics” and the DARPA Contract HR0011-18-9-0033 “The Physics of Artificial Intelligence”.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Zagreb, Zagreb, Croatia

    Zlatko Drmač

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  1. Zlatko Drmač
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  1. Department of Mathematics, University of Namur, Namur, Belgium

    Alexandre Mauroy

  2. Department of Mechanical Engineering, University of California Santa Barbara, Santa Barbara, CA, USA

    Igor Mezić

  3. Department of Electrical and Information Systems, Osaka Prefecture University, Sakai, Japan

    Yoshihiko Susuki

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Drmač, Z. (2020). Dynamic Mode Decomposition—A Numerical Linear Algebra Perspective. In: Mauroy, A., Mezić, I., Susuki, Y. (eds) The Koopman Operator in Systems and Control. Lecture Notes in Control and Information Sciences, vol 484. Springer, Cham. https://doi.org/10.1007/978-3-030-35713-9_7

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