June  2014, 1(2): 391-421. doi: 10.3934/jcd.2014.1.391

On dynamic mode decomposition: Theory and applications

1. 

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

2. 

Dept. of Applied Mathematics, University of Washington, Seattle, WA 98195, United States, United States

Received  November 2013 Revised  November 2014 Published  December 2014

Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken. We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator. This generalizes DMD to a larger class of datasets, including nonsequential time series. We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively. We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples. Such computations are not considered in the existing literature but can be understood using our more general framework. In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory. It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science. We show that under certain conditions, DMD is equivalent to LIM.
Citation: Jonathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton, J. Nathan Kutz. On dynamic mode decomposition: Theory and applications. Journal of Computational Dynamics, 2014, 1 (2) : 391-421. doi: 10.3934/jcd.2014.1.391
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show all references

References:
[1]

J. Fluid Mech., 726 (2013), 596-623. doi: 10.1017/jfm.2013.249.  Google Scholar

[2]

ACM T. Math. Software, 2013 (accepted). Google Scholar

[3]

J. Climate, 4 (1991), 766-784. doi: 10.1175/1520-0442(1991)004<0766:POACOM>2.0.CO;2.  Google Scholar

[4]

J. Nonlinear Sci., 22 (2012), 887-915. doi: 10.1007/s00332-012-9130-9.  Google Scholar

[5]

Comput. Method Appl. M., 197 (2008), 2131-2146. doi: 10.1016/j.cma.2007.08.014.  Google Scholar

[6]

J. Fluid Mech., 691 (2012), 594-604. doi: 10.1017/jfm.2011.516.  Google Scholar

[7]

Exp. Fluids, 52 (2012), 529-542. doi: 10.1007/s00348-011-1235-7.  Google Scholar

[8]

In Proceedings of the 51st IEEE Conference on Decision and Control, 2012 (2012), 4965-4970. doi: 10.1109/CDC.2012.6426995.  Google Scholar

[9]

J. Fluid Mech., 700 (2012), 16-28. doi: 10.1017/jfm.2012.37.  Google Scholar

[10]

J. Geophys. Res.-Atmos., 93 (1988), 11015-11021. doi: 10.1029/JD093iD09p11015.  Google Scholar

[11]

Proceedings of the Third Annual Allerton Conference on Circuit and System Theory, (1965), 449-459.  Google Scholar

[12]

Cambridge University Press, Cambridge, UK, 2nd edition, 2012. doi: 10.1017/CBO9780511919701.  Google Scholar

[13]

J. Educ. Psychol., 24 (1933), 417-441. doi: 10.1037/h0071325.  Google Scholar

[14]

J. Educ. Psychol., 24 (1933), 498-520. Google Scholar

[15]

Phys. Fluids, 26 (2014), 024103, arXiv:1309.4165v1. doi: 10.1063/1.4863670.  Google Scholar

[16]

J. Guid. Control Dynam., 8 (1985), 620-627. doi: 10.2514/3.20031.  Google Scholar

[17]

Technical report, Massachusetts Institute of Technology, Dec. 1956. Google Scholar

[18]

Theor. Comp. Fluid Dyn., 25 (2011), 233-247. doi: 10.1007/s00162-010-0184-8.  Google Scholar

[19]

Phys. Fluids, 24, June 2012. Google Scholar

[20]

Nonlin. Dynam., 41 (2005), 309-325. doi: 10.1007/s11071-005-2824-x.  Google Scholar

[21]

Annu. Rev. Fluid Mech., 45 (2013), 357-378. doi: 10.1146/annurev-fluid-011212-140652.  Google Scholar

[22]

Comput. Fluids, 57 (2012), 87-97. doi: 10.1016/j.compfluid.2011.12.012.  Google Scholar

[23]

J. Fluid Mech., 497 (2003), 335-363. doi: 10.1017/S0022112003006694.  Google Scholar

[24]

In Proceedings of the American Control Conference, (2004), 5722-5727. Google Scholar

[25]

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[26]

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[27]

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[28]

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[29]

J. Fluid Mech., 641 (2009), 115-127. doi: 10.1017/S0022112009992059.  Google Scholar

[30]

J. Fluid Mech., 656 (2010), 5-28. doi: 10.1017/S0022112010001217.  Google Scholar

[31]

Exp. Fluids, 50 (2011), 1123-1130. doi: 10.1007/s00348-010-0911-3.  Google Scholar

[32]

Theor. Comp. Fluid Dyn., 25 (2011), 249-259. doi: 10.1007/s00162-010-0203-9.  Google Scholar

[33]

Journal of Fluid Mechanics, 656 (2010), 5-28. doi: 10.1017/S0022112010001217.  Google Scholar

[34]

Exp. Fluids, 52 (2012), 1567-1579. doi: 10.1007/s00348-012-1266-8.  Google Scholar

[35]

Int. J. Heat Fluid Fl., 32 (2011), 1098-1110. doi: 10.1016/j.ijheatfluidflow.2011.09.008.  Google Scholar

[36]

Exp. Fluids, 53 (2012), 1203-1220. doi: 10.1007/s00348-012-1354-9.  Google Scholar

[37]

Numerical Functional Analysis and Optimization, 31 (2010), 852-869. doi: 10.1080/01630563.2010.500022.  Google Scholar

[38]

Q. Appl. Math., 45 (1987), 573-582.  Google Scholar

[39]

J. Comput. Phys., 225 (2007), 2118-2137. doi: 10.1016/j.jcp.2007.03.005.  Google Scholar

[40]

SIAM, Philadelphia, 1997. doi: 10.1137/1.9780898719574.  Google Scholar

[41]

Exp. Fluids, 54 (2013), pp1429. doi: 10.2514/6.2012-33.  Google Scholar

[42]

J. Comput. Phys., 231 (2012), 5317-5333. doi: 10.1016/j.jcp.2012.04.023.  Google Scholar

[43]

AIAA Paper 2011-38, 49th AIAA Aerospace Sciences Meeting and Exhibit, Jan. 2011. doi: 10.2514/6.2011-38.  Google Scholar

[44]

J. Climate, 8 (1995), 377-400. Google Scholar

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arXiv:1408.4408, 2014. Google Scholar

[46]

J. Fluid Mech., 733 (2013), 473-503. doi: 10.1017/jfm.2013.426.  Google Scholar

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