On dynamic mode decomposition: Theory and applications

Journal of Computational Dynamics

Volumn 1, Issue 2, 2014, Pages 391-421

  Tu, Jonathan H ^a   Rowley, Clarence W ^a   Luchtenburg, Dirk M ^a   Brunton, Steven L ^b   Kutz, J Nathan ^b  

^a Princeton University   (United States)

^b UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Dynamic mode decomposition; Koopman operator; Reduced order models; Spectral analysis; Time series analysis

Indexed keywords

EID: 85009209557     PISSN: None     EISSN: 21582505     Source Type: Journal    
DOI: 10.3934/jcd.2014.1.391     Document Type: Article

References (46)

1. S. Bagheri, Koopman-mode decomposition of the cylinder wake, J. Fluid Mech., 726 (2013), 596-623
2. B. A. Belson, J. H. Tu and C. W. Rowley, A Parallelized Model Reduction Library, ACM T. Math. Software, 2013 (accepted)
3. M. B. Blumenthal, Predictability of a coupled ocean-atmosphere model, J. Climate, 4 (1991), 766-784
4. K. K. Chen, J. H. Tu and C. W. Rowley, Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses, J. Nonlinear Sci., 22 (2012), 887-915
5. T. Colonius and K. Taira, A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions, Comput. Method Appl. M., 197 (2008), 2131-2146
6. D. Duke, D. Honnery and J. Soria, Experimental investigation of nonlinear instabilities in annular liquid sheets, J. Fluid Mech., 691 (2012), 594-604
7. D. Duke, J. Soria and D. Honnery, An error analysis of the dynamic mode decomposition, Exp. Fluids, 52 (2012), 529-542
8. P. J. Goulart, A. Wynn and D. Pearson, Optimal mode decomposition for high dimensional systems, In Proceedings of the 51st IEEE Conference on Decision and Control, 2012 (2012), 4965-4970
9. M. Grilli, P. J. Schmid, S. Hickel and N. A. Adams, Analysis of unsteady behaviour in shockwave turbulent boundary layer interaction, J. Fluid Mech., 700 (2012), 16-28
10. K. Hasselmann, PIPs and POPs: The reduction of complex dynamical-systems using Principal Interaction and Oscillation Patterns, J. Geophys. Res.-Atmos., 93 (1988), 11015-11021
11. B. L. Ho and R. E. Kalman, Effective construction of linear state-variable models from input/ output data, Proceedings of the Third Annual Allerton Conference on Circuit and System Theory, (1965), 449-459
12. P. J. Holmes, J. L. Lumley, G. Berkooz and C. W. Rowley, Turbulence, Coherent Struc-tures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, UK, 2nd edition, 2012
13. H. Hotelling, Analysis of a complex of statistical variables into principal components, J. Educ. Psychol., 24 (1933), 417-441
14. H. Hotelling, Analysis of a complex of statistical variables into principal components, J. Educ. Psychol., 24 (1933), 498-520
15. M. R. Jovanović, P. J. Schmid and J. W. Nichols, Sparsity-promoting dynamic mode decomposition, Phys. Fluids, 26 (2014), 024103, arXiv:1309.4165v1
16. J. N. Juang and R. S. Pappa, An eigensystem realization-algorithm for modal parameteridentification and model-reduction, J. Guid. Control Dynam., 8 (1985), 620-627
17. E. N. Lorenz, Empirical orthogonal functions and statistical weather prediction, Technical report, Massachusetts Institute of Technology, Dec. 1956
18. Z. Ma, S. Ahuja and C. W. Rowley, Reduced-order models for control of fluids using the eigensystem realization algorithm, Theor. Comp. Fluid Dyn., 25 (2011), 233-247
19. L. Massa, R. Kumar and P. Ravindran, Dynamic mode decomposition analysis of detonation waves, Phys. Fluids, 24, June 2012
20. I. Mezić, Spectral properties of dynamical systems, model reduction and decompositions, Nonlin. Dynam., 41 (2005), 309-325
21. I. Mezić, Analysis of fluid flows via spectral properties of the Koopman operator, Annu. Rev. Fluid Mech., 45 (2013), 357-378
22. T. W. Muld, G. Efraimsson and D. S. Henningson, Flow structures around a high-speed train extracted using proper orthogonal decomposition and dynamic mode decomposition, Comput. Fluids, 57 (2012), 87-97
23. B. R. Noack, K. Afanasiev, M. Morzyński, G. Tadmor and F. Thiele, A hierarchy of lowdimensional models for the transient and post-transient cylinder wake, J. Fluid Mech., 497 (2003), 335-363
24. B. R. Noack, G. Tadmor and M. Morzyński, Actuation models and dissipative control in empirical Galerkin models of fluid flows, In Proceedings of the American Control Conference, (2004), 5722-5727
25. K. Pearson, LIII. on lines and planes of closest fit to systems of points in space, Philos. Mag., 2 (1901), 559-572
26. C. Penland, Random forcing and forecasting using Principal Oscillation Pattern analysis, Mon. Weather Rev., 117 (1989), 2165-2185
27. C. Penland and T. Magorian, Prediction of Ni~no 3 sea-surface temperatures using linear inverse modeling, J. Climate, 6 (1993), 1067-1076
28. C. W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition, Int. J. Bifurcat. Chaos, 15 (2005), 997-1013
29. C. W. Rowley, I. Mezic, S. Bagheri, P. Schlatter and D. S. Henningson, Spectral analysis of nonlinear flows, J. Fluid Mech., 641 (2009), 115-127
30. P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech., 656 (2010), 5-28
31. P. J. Schmid, Application of the dynamic mode decomposition to experimental data, Exp. Fluids, 50 (2011), 1123-1130
32. P. J. Schmid, L. Li, M. P. Juniper and O. Pust, Applications of the dynamic mode decomposition, Theor. Comp. Fluid Dyn., 25 (2011), 249-259
33. P. J. Schmid and J. Sesterhenn, Dynamic mode decomposition of numerical and experimental data, Journal of Fluid Mechanics, 656 (2010), 5-28
34. P. J. Schmid, D. Violato and F. Scarano, Decomposition of time-resolved tomographic PIV, Exp. Fluids, 52 (2012), 1567-1579
35. A. Seena and H. J. Sung, Dynamic mode decomposition of turbulent cavity flows for selfsustained oscillations, Int. J. Heat Fluid Fl., 32 (2011), 1098-1110
36. O. Semeraro, G. Bellani and F. Lundell, Analysis of time-resolved PIV measurements of a confined turbulent jet using POD and Koopman modes, Exp. Fluids, 53 (2012), 1203-1220
37. J. R. Singler, Optimality of balanced proper orthogonal decomposition for data reconstruction, Numerical Functional Analysis and Optimization, 31 (2010), 852-869
38. L. Sirovich, Turbulence and the dynamics of coherent structures. 2. Symmetries and transformations, Q. Appl. Math., 45 (1987), 573-582
39. K. Taira and T. Colonius, The immersed boundary method: A projection approach, J. Com-put. Phys., 225 (2007), 2118-2137
40. L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, Philadelphia, 1997
41. J. H. Tu, J. Griffin, A. Hart, C. W. Rowley, L. N. Cattafesta III and L. S. Ukeiley, Integration of non-time-resolved PIV and time-resolved velocity point sensors for dynamic estimation of velocity fields, Exp. Fluids, 54 (2013), pp1429
42. J. H. Tu and C. W. Rowley, An improved algorithm for balanced POD through an analytic treatment of impulse response tails, J. Comput. Phys., 231 (2012), 5317-5333
43. J. H. Tu, C. W. Rowley, E. Aram and R. Mittal, Koopman spectral analysis of separated flow over a finite-thickness at plate with elliptical leading edge, AIAA Paper 2011-38, 49th AIAA Aerospace Sciences Meeting and Exhibit, Jan. 2011
44. H. von Storch, G. Bürger, R. Schnur and J. S. von Storch, Principal oscillation patterns: A review, J. Climate, 8 (1995), 377-400
45. M. O. Williams, I. G. Kevrekidis and C. W. Rowley, A data-driven approximation of the Koopman operator: extending dynamic mode decomposition, arXiv:1408.4408, 2014
46. A. Wynn, D. Pearson, B. Ganapathisubramani and P. J. Goulart, Optimal mode decomposition for unsteady flows, J. Fluid Mech., 733 (2013), 473-503