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Journal of Nonlinear Science

, Volume 25, Issue 6, pp 1307–1346 | Cite as

A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

  • Matthew O. Williams
  • Ioannis G. Kevrekidis
  • Clarence W. Rowley
Article

Abstract

The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a “black box” integrator. We will show that this approach is, in effect, an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the “stochastic Koopman operator” (Mezic in Nonlinear Dynamics 41(1–3): 309–325, 2005). Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data and two that show potential applications of the Koopman eigenfunctions.

Keywords

Data mining Koopman spectral analysis Set oriented methods Spectral methods Reduced order models 

Mathematics Subject Classification

Primary: 65P99 37M25 Secondary: 47B33 

Notes

Acknowledgments

The authors would like to thank Igor Mezić, Jonathan Tu, Maziar Hemati, and Scott Dawson for interesting and useful discussions on dynamic mode decomposition and the Koopman operator. M.O.W. gratefully acknowledges support from NSF DMS-1204783. I.G.K acknowledges support from AFOSR FA95550-12-1-0332 and NSF CMMI-1310173. C.W.R acknowledges support from AFOSR FA9550-12-1-0075.

References

  1. Bagheri, S.: Effects of weak noise on oscillating flows: linking quality factor, Floquet modes and Koopman spectrum. Phys. Fluids. 26, 094104 (2014)Google Scholar
  2. Bagheri, S.: Koopman-mode decomposition of the cylinder wake. J. Fluid Mech. 726, 596–623 (2013)MATHMathSciNetCrossRefGoogle Scholar
  3. Belytschko, T., Krongauz, Y., Organ, D., Fleming, M., Krysl, P.: Meshless methods: an overview and recent developments. Comput. Methods Appl. Mech. Eng. 139(1), 3–47 (1996)MATHCrossRefGoogle Scholar
  4. Bishop, C.M., et al.: Pattern Recognition and Machine Learning (Information Science and Statistics), Springer-Verlag, New York (2006)Google Scholar
  5. Bollt, E.M., Santitissadeekorn, N.: Applied and Computational Measurable Dynamics. SIAM, Philadelphia (2013)MATHCrossRefGoogle Scholar
  6. Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Courier Dover Publications, New York (2013)Google Scholar
  7. Budišić, M., Mohr, R., Mezić, I.: Applied Koopmanism. Chaos Interdiscip. J. Nonlinear Sci. 22(4), 047510 (2012)CrossRefGoogle Scholar
  8. Budisic, M., Mezic, I.: Geometry of the ergodic quotient reveals coherent structures in flows. Phys. D 241, 1255–1269 (2012)MATHMathSciNetCrossRefGoogle Scholar
  9. Chen, K.K., Tu, J.H., Rowley, C.W.: Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. J. Nonlinear Sci. 22(6), 887–915 (2012)MATHMathSciNetCrossRefGoogle Scholar
  10. Coifman, R.R., Lafon, S.: Diffusion maps. Appl. Comput. Harmonic Anal. 21(1), 5–30 (2006)MATHMathSciNetCrossRefGoogle Scholar
  11. Dean, J., Ghemawat, S.: MapReduce: simplified data processing on large clusters. Commun. ACM 51(1), 107–113 (2008)CrossRefGoogle Scholar
  12. Dellnitz, M., Froyland, G., Junge, O.: The algorithms behind GAIO—set oriented numerical methods for dynamical systems. In: Bernold Fiedler (ed.) Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, pp. 145–174, Springer, Berlin (2001)Google Scholar
  13. Dsilva, C.J., Talmon, R., Rabin, N., Coifman, R.R., Kevrekidis, I.G.: Nonlinear intrinsic variables and state reconstruction in multiscale simulations. J. Chem. Phys. 139(18), 184109 (2013)CrossRefGoogle Scholar
  14. Eisenhower, B., Maile, T., Fischer, M., Mezić, I.: Decomposing building system data for model validation and analysis using the Koopman operator. In: Proceedings of the National IBPSAUSA Conference, New York, USA (2010)Google Scholar
  15. Erban, R., Frewen, T.A., Wang, X., Elston, T.C., Coifman, R., Nadler, B., Kevrekidis, I.G.: Variable-free exploration of stochastic models: a gene regulatory network example. J. Chem. Phys. 126(15), 155103 (2007)CrossRefGoogle Scholar
  16. Froyland, G., Gottwald, G.A., Hammerlindl, A.: A computational method to extract macroscopic variables and their dynamics in multiscale systems. SIAM J. Appl. Dyn. Sys. 13(4), 1816–1846 (2014)Google Scholar
  17. Froyland, G.: Statistically optimal almost-invariant sets. Phys. D Nonlinear Phenom. 200(3), 205–219 (2005)MATHMathSciNetCrossRefGoogle Scholar
  18. Froyland, G., Padberg, K., England, M.H., Treguier, A.M.: Detection of coherent oceanic structures via transfer operators. Phys. Rev. Lett. 98(22), 224503 (2007)CrossRefGoogle Scholar
  19. Froyland, G., Padberg, K.: Almost-invariant sets and invariant manifolds—connecting probabilistic and geometric descriptions of coherent structures in flows. Phys. D Nonlinear Phenom. 238(16), 1507–1523 (2009)MATHMathSciNetCrossRefGoogle Scholar
  20. Gaspard, P., Nicolis, G., Provata, A., Tasaki, S.: Spectral signature of the pitchfork bifurcation: Liouville equation approach. Phys. Rev. E 51(1), 74 (1995)MathSciNetCrossRefGoogle Scholar
  21. Gaspard, P., Tasaki, S.: Liouvillian dynamics of the Hopf bifurcation. Phys. Rev. E 64(5), 056232 (2001)MathSciNetCrossRefGoogle Scholar
  22. Givon, D., Kupferman, R., Stuart, A.: Extracting macroscopic dynamics: model problems and algorithms. Nonlinearity 17(6), R55 (2004)MATHMathSciNetCrossRefGoogle Scholar
  23. Hansen, P.C.: Truncated singular value decomposition solutions to discrete ill-posed problems with ill-determined numerical rank. SIAM J. Sci. Stat. Comput. 11(3), 503–518 (1990)MATHCrossRefGoogle Scholar
  24. Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)MATHMathSciNetCrossRefGoogle Scholar
  25. Hirsch, C.: Numerical computation of internal and external flows: The fundamentals of computational fluid dynamics vol. 1, Butterworth-Heinemann (2007)Google Scholar
  26. Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1998)MATHGoogle Scholar
  27. Jovanović, M.R., Schmid, P.J., Nichols, J.W.: Sparsity-promoting dynamic mode decomposition. Phys. Fluids 26, 024103 (2014)Google Scholar
  28. Juang, J.-N.: Applied System Identification. Prentice Hall, Englewood Cliffs (1994)MATHGoogle Scholar
  29. Karniadakis, G., Sherwin, S.: Spectral/Hp Element Methods for Computational Fluid Dynamics. Oxford University Press, Oxford (2013)MATHGoogle Scholar
  30. Kloeden, P.E., Platen, E.: Numerical solution of stochastic differential equations. Springer, Berlin (1992)Google Scholar
  31. Koopman, B.O.: Hamiltonian systems and transformation in Hilbert space. Proc. Natl. Acad. Sci. U. S. A. 17(5), 315 (1931)CrossRefGoogle Scholar
  32. Koopman, B.O., Neumann, J.V.: Dynamical systems of continuous spectra. Proc. Natl. Acad. Sci. U. S. A. 18(3), 255 (1932)CrossRefGoogle Scholar
  33. Lee, J.A., Verleysen, M.: Nonlinear Dimensionality Reduction. Springer, Berlin (2007)MATHCrossRefGoogle Scholar
  34. Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK users’ guide: Solution of large-scale eigenvalue problems. SIAM, Philadelphia (1998)CrossRefGoogle Scholar
  35. Liu, G.-R.: Meshfree methods: Moving beyond the finite element method. CRC Press, Boca Raton (2010)Google Scholar
  36. Matkowsky, B., Schuss, Z.: Eigenvalues of the Fokker-Planck operator and the approach to equilibrium for diffusions in potential fields. SIAM J. Appl. Math. 40(2), 242–254 (1981)MATHMathSciNetCrossRefGoogle Scholar
  37. Mauroy, A., Mezic, I.: A spectral operator-theoretic framework for global stability. In: Decision and Control (CDC), 2013 IEEE 52nd Annual Conference on, pp. 5234–5239 (2013)Google Scholar
  38. Mauroy, A., Mezić, I., Moehlis, J.: Isostables, isochrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics. Phys. D Nonlinear Phenom. 261, 19–30 (2013)MATHCrossRefGoogle Scholar
  39. Mauroy, A., Mezić, I.: On the use of Fourier averages to compute the global isochrons of (quasi) periodic dynamics. Chaos Interdiscip. J. Nonlinear Sci. 22(3), 033112 (2012)CrossRefGoogle Scholar
  40. Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41(1–3), 309–325 (2005)MATHGoogle Scholar
  41. Monaghan, J.J.: Smoothed particle hydrodynamics. Annu. Rev. Astron. Astrophys. 30, 543–574 (1992)CrossRefGoogle Scholar
  42. Muld, T.W., Efraimsson, G., Henningson, D.S.: Flow structures around a high-speed train extracted using proper orthogonal decomposition and dynamic mode decomposition. Comput. Fluids 57, 87–97 (2012)MathSciNetCrossRefGoogle Scholar
  43. Nadler, B., Lafon, S., Kevrekidis, I.G., Coifman, R.R.: Diffusion maps, spectral clustering and eigenfunctions of Fokker-Planck operators. Adv Neural Inf Process Syst. 18, 955–962 (2005)Google Scholar
  44. Nadler, B., Lafon, S., Coifman, R.R., Kevrekidis, I.G.: Diffusion maps, spectral clustering and reaction coordinates of dynamical systems. Appl. Comput. Harmonic Anal. 21(1), 113–127 (2006)MATHMathSciNetCrossRefGoogle Scholar
  45. Rowley, C.W., Mezić, I., Bagheri, S., Schlatter, P., Henningson, D.S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009)MATHMathSciNetCrossRefGoogle Scholar
  46. Santitissadeekorn, N., Bollt, E.: The infinitesimal operator for the semigroup of the Frobenius-Perron operator from image sequence data: vector fields and transport barriers from movies. Chaos Interdiscip. J. Nonlinear Sci. 17(2), 023126 (2007)MathSciNetCrossRefGoogle Scholar
  47. Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 65(6), 5–28 (2010)CrossRefGoogle Scholar
  48. Schmid, P., Li, L., Juniper, M., Pust, O.: Applications of the dynamic mode decomposition. Theor. Comput. Fluid Dyn. 25(1–4), 249–259 (2011)MATHCrossRefGoogle Scholar
  49. Schmid, P.J., Violato, D., Scarano, F.: Decomposition of time-resolved tomographic PIV. Exp. Fluids 52(6), 1567–1579 (2012)CrossRefGoogle Scholar
  50. Seena, A., Sung, H.J.: Dynamic mode decomposition of turbulent cavity flows for self-sustained oscillations. Int. J. Heat Fluid Flow 32(6), 1098–1110 (2011)CrossRefGoogle Scholar
  51. Sirisup, S., Karniadakis, G.E., Xiu, D., Kevrekidis, I.G.: Equation-free/Galerkin-free POD-assisted computation of incompressible flows. J. Comput. Phys. 207(2), 568–587 (2005)MATHMathSciNetCrossRefGoogle Scholar
  52. Stengel, R.F.: Optimal Control and Estimation. Courier Dover Publications, New York (2012)Google Scholar
  53. Susuki, Y., Mezic, I.: Nonlinear Koopman modes and power system stability assessment without models. IEEE Trans. Power Syst. 29(2), 899–907 (2014)Google Scholar
  54. Susuki, Y., Mezić, I.: Nonlinear Koopman modes and coherency identification of coupled swing dynamics. IEEE Trans. Power Syst. 26(4), 1894–1904 (2011)CrossRefGoogle Scholar
  55. Susuki, Y., Mezić, I.: Nonlinear Koopman modes and a precursor to power system swing instabilities. Power Syst. IEEE Trans. 27(3), 1182–1191 (2012)CrossRefGoogle Scholar
  56. Todorov, E.: Optimal control theory. In: Bayesian brain: Probabilistic approaches to neural coding, Kenji Doya (Editor), pp. 269–298. MIT Press, Cambridge (2007)Google Scholar
  57. Trefethen, L.N.: Spectral Methods in MATLAB, vol. 10. SIAM, Philadelphia (2000)MATHCrossRefGoogle Scholar
  58. Tu, J.H., Rowley, C.W., Luchtenburg, D.M., Brunton, S.L., Kutz, J.N.: On dynamic mode decomposition: Theory and applications. J Comput Dyn. 1(2), 391–421 (2014)Google Scholar
  59. Wendland, H.: Meshless Galerkin methods using radial basis functions. Math. Comput. Am. Math. Soc. 68(228), 1521–1531 (1999)MATHMathSciNetCrossRefGoogle Scholar
  60. Wynn, A., Pearson, D., Ganapathisubramani, B., Goulart, P.: Optimal mode decomposition for unsteady flows. J. Fluid Mech. 733, 473–503 (2013)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Program in Applied and Computational Mathematics (PACM)Princeton UniversityPrincetonUSA
  2. 2.Chemical and Biological Engineering Department & PACMPrinceton UniversityPrincetonUSA
  3. 3.Mechanical and Aerospace Engineering DepartmentPrinceton UniversityPrincetonUSA

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