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A kernel-based method for data-driven koopman spectral analysis

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  • A data-driven, kernel-based method for approximating the leading Koopman eigenvalues, eigenfunctions, and modes in problems with high-dimensional state spaces is presented. This approach uses a set of scalar observables (functions that map a state to a scalar value) that are defined implicitly by the feature map associated with a user-defined kernel function. This circumvents the computational issues that arise due to the number of functions required to span a ``sufficiently rich'' subspace of all possible scalar observables in such applications. We illustrate this method on two examples: the first is the FitzHugh-Nagumo PDE, a prototypical one-dimensional reaction-diffusion system, and the second is a set of vorticity data computed from experimentally obtained velocity data from flow past a cylinder at Reynolds number 413. In both examples, we use the output of Dynamic Mode Decomposition, which has a similar computational cost, as the benchmark for our approach.
    Mathematics Subject Classification: Primary: 37M10, 65P99, 47B33.

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  • [1]

    S. Bagheri, Koopman-mode decomposition of the cylinder wake, Journal of Fluid Mechanics, 726 (2013), 596-623.doi: 10.1017/jfm.2013.249.

    [2]

    S. Bagheri, Effects of weak noise on oscillating flows: Linking quality factor, Floquet modes, and Koopman spectrum, Physics of Fluids, 26 (2014), 094104.doi: 10.1063/1.4895898.

    [3]

    G. Baudat and F. Anouar, Kernel-based methods and function approximation, In Proceedings of the International Joint Conference on Neural Networks, IEEE, 2 (2001), 1244-1249.doi: 10.1109/IJCNN.2001.939539.

    [4]

    C. M. Bishop et al, Pattern Recognition and Machine Learning, Springer, 2006.doi: 10.1007/978-0-387-45528-0.

    [5]

    J. P. Boyd, Chebyshev and Fourier Spectral Methods, Courier Dover Publications, Mineola, NY, 2001.

    [6]

    M. Budišić, R. Mohr and I. Mezić, Applied Koopmanism, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22 (2012), 047510, 33pp.doi: 10.1063/1.4772195.

    [7]

    C. J. Burges, A tutorial on support vector machines for pattern recognition, Data Mining and Knowledge Discovery, 2 (1998), 121-167.

    [8]

    A. Chatterjee, An introduction to the proper orthogonal decomposition, Current Science, 78 (2000), 808-817.

    [9]

    K. K. Chen, J. H. Tu and C. W. Rowley, Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses, Journal of Nonlinear Science, 22 (2012), 887-915.doi: 10.1007/s00332-012-9130-9.

    [10]

    R. R. Coifman and S. Lafon, Diffusion maps, Applied and Computational Harmonic Analysis, 21 (2006), 5-30.doi: 10.1016/j.acha.2006.04.006.

    [11]

    N. Cristianini and J. Shawe-Taylor, An Introduction to Support Vector Machines and Other Kernel-Based Learning Methods, Cambridge University Press, 2000.doi: 10.1017/CBO9780511801389.

    [12]

    C. E. Elmer and E. S. Van Vleck, Spatially discrete FitzHugh-Nagumo equations, SIAM Journal on Applied Mathematics, 65 (2005), 1153-1174.doi: 10.1137/S003613990343687X.

    [13]

    P. Gaspard and S. Tasaki, Liouvillian dynamics of the {Hopf} bifurcation, Physical Review E, 64 (2001), 056232, 17pp.doi: 10.1103/PhysRevE.64.056232.

    [14]

    M. S. Hemati, M. O. Williams and C. W. Rowley, Dynamic mode decomposition for large and streaming datasets, Physics of Fluids, 26 (2014), 111701.doi: 10.1063/1.4901016.

    [15]

    P. Holmes, J. L. Lumley, G. Berkooz and C. W. Rowley, Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, 2nd edition, 2012.doi: 10.1017/CBO9780511919701.

    [16]

    M. R. Jovanović, P. J. Schmid and J. W. Nichols, Sparsity-promoting dynamic mode decomposition, Physics of Fluids, 26 (2014), 024103.

    [17]

    J.-N. Juang, Applied System Identification, Prentice Hall, 1994.

    [18]

    B. O. Koopman and J. von Neumann, Dynamical systems of continuous spectra, Proceedings of the National Academy of Sciences of the United States of America, 18 (1932), 255-163.doi: 10.1073/pnas.18.3.255.

    [19]

    B. O. Koopman, Hamiltonian systems and transformation in Hilbert space, Proceedings of the National Academy of Sciences of the United States of America, 17 (1931), 315-318.doi: 10.1073/pnas.17.5.315.

    [20]

    J. A. Lee and M. Verleysen, Nonlinear Dimensionality Reduction, Springer, 2007.doi: 10.1007/978-0-387-39351-3.

    [21]

    R. B. Lehoucq, D. C. Sorensen and C. Yang, ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, volume 6. SIAM, 1998.doi: 10.1137/1.9780898719628.

    [22]

    A. Mauroy and I. Mezic, A spectral operator-theoretic framework for global stability, In 52nd IEEE Conference on Decision and Control, (2013), 5234-5239.doi: 10.1109/CDC.2013.6760712.

    [23]

    I. Mezić, Spectral properties of dynamical systems, model reduction and decompositions, Nonlinear Dynamics, 41 (2005), 309-325.doi: 10.1007/s11071-005-2824-x.

    [24]

    I. Mezić, Analysis of fluid flows via spectral properties of the Koopman operator, Annual Review of Fluid Mechanics, 45 (2013), 357-378.doi: 10.1146/annurev-fluid-011212-140652.

    [25]

    C. E. Rasmussen, Gaussian Processes for Machine Learning, MIT Press, 2006.

    [26]

    C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter and D. S. Henningson, Spectral analysis of nonlinear flows, Journal of Fluid Mechanics, 641 (2009), 115-127.doi: 10.1017/S0022112009992059.

    [27]

    P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, Journal of Fluid Mechanics, 656 (2010), 5-28.doi: 10.1017/S0022112010001217.

    [28]

    P. J. Schmid, D. Violato and F. Scarano, Decomposition of time-resolved tomographic PIV, Experiments in Fluids, 52 (2012), 1567-1579.doi: 10.1007/s00348-012-1266-8.

    [29]

    P. J. Schmid, L. Li, M. P. Juniper and O. Pust, Applications of the dynamic mode decomposition, Theoretical and Computational Fluid Dynamics, 25 (2011), 249-259.doi: 10.1007/s00162-010-0203-9.

    [30]

    B. Scholkopf, The kernel trick for distances, Advances in Neural Information Processing Systems, (2001), 301-307.

    [31]

    L. Sirovich, Turbulence and the dynamics of coherent structures. part I: Coherent structures, Quarterly of applied mathematics, 45 (1987), 561-571.

    [32]

    G. Tissot, L. Cordier, N. Benard and B. R. Noack, Model reduction using dynamic mode decomposition, Comptes Rendus Mćcanique, 342 (2014), 410-416.doi: 10.1016/j.crme.2013.12.011.

    [33]

    J. H. Tu, C. W. Rowley, J. N. Kutz and J. K. Shang, Toward compressed DMD: Spectral analysis of fluid flows using sub-Nyquist-rate PIV data, Experiments in Fluids, 55 (2014), 1-13.

    [34]

    J. H. Tu, C. W. Rowley, D. M. Luchtenburg, S. L. Brunton and J. N. Kutz, On dynamic mode decomposition: Theory and applications, Journal of Computational Dynamics, 1 (2014), 391-421.doi: 10.3934/jcd.2014.1.391.

    [35]

    M. O. Williams, I. G. Kevrekidis and C. W. Rowley, A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition, Journal of Nonlinear Science, 25 (2015), 1307-1346.doi: 10.1007/s00332-015-9258-5.

    [36]

    A. Wynn, D. S. Pearson, B. Ganapathisubramani and P. J. Goulart, Optimal mode decomposition for unsteady flows, Journal of Fluid Mechanics, 733 (2013), 473-503.doi: 10.1017/jfm.2013.426.

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