American Institute of Mathematical Sciences

Advanced
June  2015, 2(2): 247-265. doi: 10.3934/jcd.2015005

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

Matthew O. Williams 1, , Clarence W. Rowley 2, and  Ioannis G. Kevrekidis 3,

1. 

United Technologies Research Center, 411 Silver Lane, East Hartford, CT 06118, United States
2. 

Dept. of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544
3. 

Department of Chemical and Biological Engineering & PACM, Princeton University, Princeton, NJ 08544, United States

Received  February 2015 Revised  February 2016 Published  May 2016

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.
Keywords: time series analysis, machine learning., kernel methods, Koopman operator, dynamic mode decomposition.
Mathematics Subject Classification: Primary: 37M10, 65P99, 47B3.
Citation: Matthew O. Williams, Clarence W. Rowley, Ioannis G. Kevrekidis. A kernel-based method for data-driven koopman spectral analysis. Journal of Computational Dynamics, 2015, 2 (2) : 247-265. doi: 10.3934/jcd.2015005
References:
[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.

show all references

References:
[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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