Element contents
Latent Modes of Nonlinear Flows
A Koopman Theory Analysis
Published online by Cambridge University Press: 31 May 2023
Summary
Extracting the latent underlying structures of complex nonlinear local and nonlocal flows is essential for their analysis and modeling. In this Element the authors attempt to provide a consistent framework through Koopman theory and its related popular discrete approximation - dynamic mode decomposition (DMD). They investigate the conditions to perform appropriate linearization, dimensionality reduction and representation of flows in a highly general setting. The essential elements of this framework are Koopman eigenfunctions (KEFs) for which existence conditions are formulated. This is done by viewing the dynamic as a curve in state-space. These conditions lay the foundations for system reconstruction, global controllability, and observability for nonlinear dynamics. They examine the limitations of DMD through the analysis of Koopman theory and propose a new mode decomposition technique based on the typical time profile of the dynamics.
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- Online ISBN: 9781009323826Publisher: Cambridge University PressPrint publication: 29 June 2023
References
Andreu, Fuensanta, Ballester, Coloma, Caselles, Vicent, and Mazón, José. 2001. Minimizing total variation flow. Differential and Integral Equations, 14(3), 321–360.Google Scholar
Askham, Travis, and Kutz, J. Nathan. 2018. Variable projection methods for an optimized dynamic mode decomposition. SIAM Journal on Applied Dynamical Systems, 17(1), 380–416.Google Scholar
Aubry, Nadine, Guyonnet, Régis, and Lima, Ricardo. 1991. Spatiotemporal analysis of complex signals: Theory and applications. Journal of Statistical Physics, 64(3), 683–739.CrossRefGoogle Scholar
Balakrishnan, Alampallam. 1966. On the controllability of a nonlinear system. Proceedings of the National Academy of Sciences of the United States of America, 55(3), 465–468.Google Scholar
Bellettini, Giovanni, Caselles, Vicent, and Novaga, Matteo. 2002. The total variation flow in RN. Journal of Differential Equations, 184(2), 475–525.CrossRefGoogle Scholar
Bollt, Erik. 2021. Geometric considerations of a good dictionary for Koopman analysis of dynamical systems: Cardinality, “primary eigenfunction,” and efficient representation. Communications in Nonlinear Science and Numerical Simulation, 100, 105833.Google Scholar
Brezis, Haim. 1973. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Elsevier.Google Scholar
Brunton, Steven, Budišić, Marko, Kaiser, Eurika, and Kutz, J. Nathan. 2021. Modern Koopman theory for dynamical systems. SIAM Review, 64(2), 229–340. arXiv:2102.12086.Google Scholar
Brunton, Steven, and Kutz, J. Nathan. 2019. Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control. Cambridge University Press.Google Scholar
Brunton, Steven, Proctor, Joshua, and Kutz, J. Nathan. 2016. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15), 3932–3937.Google Scholar
Bungert, Leon, and Burger, Martin. 2019. Asymptotic profiles of nonlinear homogeneous evolution equations of gradient flow type. Journal of Evolution Equations, 20, 1061–1092.Google Scholar
Bungert, Leon, Burger, Martin, and Tenbrinck, Daniel. 2019a. Computing nonlinear eigenfunctions via gradient flow extinction. Pages 291–302 of International Conference on Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science book series, volume 11603. Springer. DOI: https://doi.org/10.1007/978-3-030-22368-7_23.Google Scholar
Bungert, Leon, Burger, Martin, Chambolle, Antonin, and Novaga, Matteo. 2019b. Nonlinear spectral decompositions by gradient flows of one-homogeneous functionals. Analysis & PDE, 14(3), 823–860. arXiv:1901.06979.Google Scholar
Burger, Martin, Gilboa, Guy, Moeller, Michael, Eckardt, Lina, and Cremers, Daniel. 2016. Spectral decompositions using one-homogeneous functionals. SIAM Journal on Imaging Sciences, 9(3), 1374–1408.Google Scholar
Chuaqui, Martin. 2018. General criteria for curves to be simple. Journal of Mathematical Analysis and Applications, 464(1), 955–963.CrossRefGoogle Scholar
Cohen, Ido, and Gilboa, Guy. 2018. Shape preserving flows and the p–Laplacian spectra. HAL Id: hal-01870019. https://hal.science/hal-01870019.Google Scholar
Cohen, Ido, and Gilboa, Guy. 2020. Introducing the p-Laplacian spectra. Signal Processing, 167, 107281. DOI: https://doi.org/10.1016/j.sigpro.2019.107281.Google Scholar
Cohen, Ido, Azencot, Omri, Lifshits, Pavel, and Gilboa, Guy. 2021a. Modes of homogeneous gradient flows. SIAM Journal on Imaging Sciences, 14(3), 913–945.Google Scholar
Cohen, Ido, Berkov, Tom, and Gilboa, Guy. 2021b. Total-variation mode decomposition. Pages 52–64 of Scale Space and Variational Methods in Computer Vision, edited by Elmoataz, Abderrahim, Fadili, Jalal, Quéau, Yvain, Rabin, Julien, and Loїc Simon. Springer.Google Scholar
Cohen, Ido, Falik, Adi, and Gilboa, Guy. 2019. Stable explicit p-Laplacian flows based on nonlinear eigenvalue analysis. Pages 315–327 of International Conference on Scale Space and Variational Methods in Computer Vision, edited by Lellmann, Jan, Modersitzki, Jan, and Burger, Martin. Lecture Notes in Computer Science, volume 11603. Springer.Google Scholar
Courant, Richard, and John, Fritz. 2012. Introduction to Calculus and Analysis I. Springer.Google Scholar
Dawson, Scott, Hemati, Maziar, Williams, Matthew, and Rowley, Clarence. 2016. Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition. Experiments in Fluids, 57(3), 42. DOI: https://doi.org/10.1007/s00348-016-2127-7.Google Scholar
Elad, Michael. 2010. Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing. Springer.CrossRefGoogle Scholar
Evangelisti, E. 2011. Controllability and Observability: Lectures Given at a Summer School of the Centro Internazionale Matematico Estivo (CIME) held in Pontecchio (Bologna), Italy, July 1 –9,1968. CIME Summer Schools, volume 46. Springer.CrossRefGoogle Scholar
Gavish, Matan, and Donoho, David. 2014. The optimal hard threshold for singular values is
. IEEE Transactions on Information Theory, 60(8), 5040–5053.Google Scholar
Giannakis, Dimitrios. 2019. Data-driven spectral decomposition and forecasting of ergodic dynamical systems. Applied and Computational Harmonic Analysis, 47(2), 338–396.CrossRefGoogle Scholar
Giannakis, Dimitrios, and Majda, Andrew. 2012. Nonlinear Laplacian spectral analysis for time series with intermittency and low-frequency variability. Proceedings of the National Academy of Sciences, 109(7), 2222–2227.CrossRefGoogle ScholarPubMed
Gilboa, Guy. 2014. A total variation spectral framework for scale and texture analysis. SIAM Journal on Imaging Sciences, 7(4), 1937–1961.CrossRefGoogle Scholar
Gilboa, Guy. 2018. Nonlinear Eigenproblems in Image Processing and Computer Vision. Springer.Google Scholar
Gilboa, Guy, and Osher, Stanley. 2009. Nonlocal operators with applications to image processing. Multiscale Modeling & Simulation, 7(3), 1005–1028.Google Scholar
Hemati, Maziar, Rowley, Clarence, Deem, Eric, and Cattafesta, Louis. 2017. De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets. Theoretical and Computational Fluid Dynamics, 31(4), 349–368.Google Scholar
Kaiser, Eurika, Kutz, J. Nathan, and Brunton, Steven. 2018. Discovering conservation laws from data for control. Pages 6415–6421 of 2018 IEEE Conference on Decision and Control (CDC). Institute of Electrical and Electronics Engineers.Google Scholar
Kaiser, Eurika, Kutz, J. Nathan, and Brunton, Steven. 2021. Data-driven discovery of Koopman eigenfunctions for control. Machine Learning: Science and Technology, 2(3), 035023. DOI: https://doi.org/10.1088/2632-2153/abf0f5.Google Scholar
Katzir, Oren. 2017. On the scale-space of filters and their applications. M.Phil. thesis, Technion – Israel Institute of Technology, Haifa.Google Scholar
Kawahara, Yoshinobu. 2016. Dynamic mode decomposition with reproducing kernels for Koopman spectral analysis. Advances in Neural Information Processing Systems, 29, 911–919.Google Scholar
Koopman, Bernard. 1931. Hamiltonian systems and transformation in Hilbert space. Proceedings of the National Academy of Sciences of the United States of America, 17(5), 315–318.CrossRefGoogle ScholarPubMed
Korda, Milan, and Mezić, Igor. 2018. Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control. Automatica, 93, 149–160.CrossRefGoogle Scholar
Kou, Shauying, Elliott, David, and Tarn, Tzyh Jong. 1973. Observability of nonlinear systems. Information and Control, 22(1), 89–99.CrossRefGoogle Scholar
Kutz, J. Nathan, Brunton, Steven, Brunton, Bingni, and Proctor, Joshua. 2016a. Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems. SIAM.Google Scholar
Kutz, J. Nathan, Proctor, Joshua, and Brunton, Steven. 2018. Applied Koopman theory for partial differential equations and data-driven modeling of spatio-temporal systems. Complexity, 2018, 6010634.Google Scholar
Kutz, J. Nathan, Proctor, Joshua, and Brunton, Steven. 2016b. Koopman theory for partial differential equations. arXiv:1607.07076.Google Scholar
Langley, Pat, Bradshaw, Gary, and Simon, Herbert. 1981. BACON. 5: The discovery of conservation laws. Pages 121–126 of International Joint Conference on Artificial Intelligence, vol. 1. Citeseer.Google Scholar
Li, Qianxiao, Dietrich, Felix, Bollt, Erik, and Kevrekidis, Ioannis. 2017. Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator. Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(10), 103111.Google Scholar
Lu, Hannah, and Tartakovsky, Daniel. 2020. Prediction accuracy of dynamic mode decomposition. SIAM Journal on Scientific Computing, 42(3), A1639–A1662.Google Scholar
Mairal, Julien, Bach, Francis, Ponce, Jean et al. 2014a. Spams: A sparse modeling software, v2.6. http://spams-devel.gforge.inria.fr/downloads.html.Google Scholar
Mairal, Julien, Bach, Francis, Ponce, Jean, et al. 2014b. Sparse modeling for image and vision processing. Foundations and Trends® in Computer Graphics and Vision, 8(2–3), 85–283.Google Scholar
Mauroy, Alexandre. 2021. Koopman operator theory for infinite-dimensional systems: Extended dynamic mode decomposition and identification of nonlinear PDEs. Mathematics, 9(19), 2495. arXiv:2103.12458.Google Scholar
Mauroy, Alexandre, Mezić, Igor, and Moehlis, Jeff. 2013. Isostables, isochrons, and Koopman spectrum for the action–angle representation of stable fixed point dynamics. Physica D: Nonlinear Phenomena, 261, 19–30.Google Scholar
Mauroy, Alexandre, Susuki, Y, and Mezić, I. 2020. The Koopman Operator in Systems and Control. Springer.Google Scholar
Mezić, Igor. 2005. Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dynamics, 41(1), 309–325. https://doi.org/10.1007/s11071-005-2824-x.Google Scholar
Nakao, Hiroya, and Mezić, Igor. 2020. Spectral analysis of the Koopman operator for partial differential equations. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(11), 113131.CrossRefGoogle ScholarPubMed
Otto, Samuel, and Rowley, Clarence. 2021. Koopman operators for estimation and control of dynamical systems. Annual Review of Control, Robotics, and Autonomous Systems, 4, 59–87.CrossRefGoogle Scholar
Pan, Shaowu, Arnold-Medabalimi, Nicholas, and Duraisamy, Karthik. 2021. Sparsity-promoting algorithms for the discovery of informative Koopman-invariant subspaces. Journal of Fluid Mechanics, 917, A18. doi:https://doi.org/10.1017/jfm.2021.271.Google Scholar
Rudy, Samuel, Brunton, Steven, Proctor, Joshua, and Kutz, J. Nathan. 2017. Data-driven discovery of partial differential equations. Science Advances, 3(4), e1602614.Google Scholar
Schmid, Peter. 2010. Dynamic mode decomposition of numerical and experimental data. Journal of Fluid Mechanics, 656, 5–28.Google Scholar
Schmidt, Michael, and Lipson, Hod. 2009. Distilling free-form natural laws from experimental data. Science, 324(5923), 81–85.CrossRefGoogle ScholarPubMed
Tu, Jonathan, Rowley, Clarence, Luchtenburg, Dirk, Brunton, Steven, and Kutz, J. Nathan. 2013. On dynamic mode decomposition: Theory and applications. Journal of Computational Dynamics, 1(2), 391–421. arXiv:1312.0041.CrossRefGoogle Scholar
Valmorbida, Giorgio, and Anderson, James. 2017. Region of attraction estimation using invariant sets and rational Lyapunov functions. Automatica, 75(January), 37–45.Google Scholar
Venturi, Daniele, and Dektor, Alec. 2021. Spectral methods for nonlinear functionals and functional differential equations. Research in the Mathematical Sciences, 8(2), 1–39.Google Scholar
Williams, Matthew, Hemati, Maziar, Dawson, Scott, Kevrekidis, Ioannis, and Rowley, Clarence. 2016. Extending data-driven Koopman analysis to actuated systems. IFAC-PapersOnLine, 49(18), 704–709.Google Scholar
Williams, Matthew, Kevrekidis, Ioannis, and Rowley, Clarence 2015a. A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition. Journal of Nonlinear Science, 25(6), 1307–1346.Google Scholar
Williams, Matthew, Rowley, Clarence, Mezić, Igor, and Kevrekidis, Ioannis. 2015b. Data fusion via intrinsic dynamic variables: An application of data-driven Koopman spectral analysis. Europhysics Letters, 109(4), 40007.CrossRefGoogle Scholar
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