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
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]

Philos. Mag., 2 (1901), 559-572. doi: 10.1080/14786440109462720.  Google Scholar

[26]

Mon. Weather Rev., 117 (1989), 2165-2185. Google Scholar

[27]

J. Climate, 6 (1993), 1067-1076. Google Scholar

[28]

Int. J. Bifurcat. Chaos, 15 (2005), 997-1013. doi: 10.1142/S0218127405012429.  Google Scholar

[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

[45]

arXiv:1408.4408, 2014. Google Scholar

[46]

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

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]

Philos. Mag., 2 (1901), 559-572. doi: 10.1080/14786440109462720.  Google Scholar

[26]

Mon. Weather Rev., 117 (1989), 2165-2185. Google Scholar

[27]

J. Climate, 6 (1993), 1067-1076. Google Scholar

[28]

Int. J. Bifurcat. Chaos, 15 (2005), 997-1013. doi: 10.1142/S0218127405012429.  Google Scholar

[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

[45]

arXiv:1408.4408, 2014. Google Scholar

[46]

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

[1]

Luigi C. Berselli, Tae-Yeon Kim, Leo G. Rebholz. Analysis of a reduced-order approximate deconvolution model and its interpretation as a Navier-Stokes-Voigt regularization. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1027-1050. doi: 10.3934/dcdsb.2016.21.1027

[2]

Carsten Hartmann, Juan C. Latorre, Wei Zhang, Grigorios A. Pavliotis. Addendum to "Optimal control of multiscale systems using reduced-order models". Journal of Computational Dynamics, 2017, 4 (1&2) : 167-167. doi: 10.3934/jcd.2017006

[3]

Carsten Hartmann, Juan C. Latorre, Wei Zhang, Grigorios A. Pavliotis. Optimal control of multiscale systems using reduced-order models. Journal of Computational Dynamics, 2014, 1 (2) : 279-306. doi: 10.3934/jcd.2014.1.279

[4]

Hao Sun, Shihua Li, Xuming Wang. Output feedback based sliding mode control for fuel quantity actuator system using a reduced-order GPIO. Discrete & Continuous Dynamical Systems - S, 2021, 14 (4) : 1447-1464. doi: 10.3934/dcdss.2020375

[5]

Mojtaba F. Fathi, Ahmadreza Baghaie, Ali Bakhshinejad, Raphael H. Sacho, Roshan M. D'Souza. Time-resolved denoising using model order reduction, dynamic mode decomposition, and kalman filter and smoother. Journal of Computational Dynamics, 2020, 7 (2) : 469-487. doi: 10.3934/jcd.2020019

[6]

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

[7]

Steven L. Brunton, Joshua L. Proctor, Jonathan H. Tu, J. Nathan Kutz. Compressed sensing and dynamic mode decomposition. Journal of Computational Dynamics, 2015, 2 (2) : 165-191. doi: 10.3934/jcd.2015002

[8]

Hao Zhang, Scott T. M. Dawson, Clarence W. Rowley, Eric A. Deem, Louis N. Cattafesta. Evaluating the accuracy of the dynamic mode decomposition. Journal of Computational Dynamics, 2020, 7 (1) : 35-56. doi: 10.3934/jcd.2020002

[9]

Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665

[10]

Heikki Haario, Leonid Kalachev, Marko Laine. Reduction and identification of dynamic models. Simple example: Generic receptor model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 417-435. doi: 10.3934/dcdsb.2013.18.417

[11]

Mustaffa Alfatlawi, Vaibhav Srivastava. An incremental approach to online dynamic mode decomposition for time-varying systems with applications to EEG data modeling. Journal of Computational Dynamics, 2020, 7 (2) : 209-241. doi: 10.3934/jcd.2020009

[12]

Jae-Hong Pyo, Jie Shen. Normal mode analysis of second-order projection methods for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 817-840. doi: 10.3934/dcdsb.2005.5.817

[13]

Yu-Ting Lin, John Malik, Hau-Tieng Wu. Wave-shape oscillatory model for nonstationary periodic time series analysis. Foundations of Data Science, 2021  doi: 10.3934/fods.2021009

[14]

Rafael Tiedra De Aldecoa. Spectral analysis of time changes of horocycle flows. Journal of Modern Dynamics, 2012, 6 (2) : 275-285. doi: 10.3934/jmd.2012.6.275

[15]

Lingling Lv, Wei He, Xianxing Liu, Lei Zhang. A robust reduced-order observers design approach for linear discrete periodic systems. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2799-2812. doi: 10.3934/jimo.2019081

[16]

Rakesh Pilkar, Erik M. Bollt, Charles Robinson. Empirical mode decomposition/Hilbert transform analysis of postural responses to small amplitude anterior-posterior sinusoidal translations of varying frequencies. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1085-1097. doi: 10.3934/mbe.2011.8.1085

[17]

Yuri Nechepurenko, Michael Khristichenko, Dmitry Grebennikov, Gennady Bocharov. Bistability analysis of virus infection models with time delays. Discrete & Continuous Dynamical Systems - S, 2020, 13 (9) : 2385-2401. doi: 10.3934/dcdss.2020166

[18]

Annalisa Pascarella, Alberto Sorrentino, Cristina Campi, Michele Piana. Particle filtering, beamforming and multiple signal classification for the analysis of magnetoencephalography time series: a comparison of algorithms. Inverse Problems & Imaging, 2010, 4 (1) : 169-190. doi: 10.3934/ipi.2010.4.169

[19]

Zhendong Luo. A reduced-order SMFVE extrapolation algorithm based on POD technique and CN method for the non-stationary Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1189-1212. doi: 10.3934/dcdsb.2015.20.1189

[20]

Chuang Peng. Minimum degrees of polynomial models on time series. Conference Publications, 2005, 2005 (Special) : 720-729. doi: 10.3934/proc.2005.2005.720

 Impact Factor: 

Metrics

  • PDF downloads (2676)
  • HTML views (0)
  • Cited by (341)

[Back to Top]