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Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses

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Abstract

Dynamic mode decomposition (DMD) is an Arnoldi-like method based on the Koopman operator. It analyzes empirical data, typically generated by nonlinear dynamics, and computes eigenvalues and eigenmodes of an approximate linear model. Without explicit knowledge of the dynamical operator, it extracts frequencies, growth rates, and spatial structures for each mode. We show that expansion in DMD modes is unique under certain conditions. When constructing mode-based reduced-order models of partial differential equations, subtracting a mean from the data set is typically necessary to satisfy boundary conditions. Subtracting the mean of the data exactly reduces DMD to the temporal discrete Fourier transform (DFT); this is restrictive and generally undesirable. On the other hand, subtracting an equilibrium point generally preserves the DMD spectrum and modes. Next, we introduce an “optimized” DMD that computes an arbitrary number of dynamical modes from a data set. Compared to DMD, optimized DMD is superior at calculating physically relevant frequencies, and is less numerically sensitive. We test these decomposition methods on data from a two-dimensional cylinder fluid flow at a Reynolds number of 60. Time-varying modes computed from the DMD variants yield low projection errors.

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Notes

  1. In ergodic theory, we assume the map f is measure preserving, making the Koopman operator unitary (Petersen 1983). Here, we make no such assumption, allowing for the analysis of dynamics not lying on an attractor. As such, the growth rates given by |λ j | may differ from unity.

References

  • Åkervik, E., Brandt, L., Henningson, D.S., Hœpffner, J., Marxen, O., Schlatter, P.: Steady solutions of the Navier–Stokes equations by selective frequency damping. Phys. Fluids 18(6), 068102 (2006)

    Article  Google Scholar 

  • Colonius, T., Taira, K.: A fast immersed boundary method using a nullspace approach and multi-domain far-field boundary conditions. Comput. Methods Appl. Mech. Eng. 197(25–28), 2131–2146 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Giannetti, F., Luchini, P.: Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Gillies, E.A.: Low-dimensional control of the circular cylinder wake. J. Fluid Mech. 371, 157–178 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  • Holmes, P., Lumley, J.L., Berkooz, G.: Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  • Ilak, M., Rowley, C.W.: Modeling of transitional channel flow using balanced proper orthogonal decomposition. Phys. Fluids 20(3), 034103 (2008)

    Article  Google Scholar 

  • Marquet, O., Sipp, D., Jacquin, L.: Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221–252 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dyn. 41(1–3), 309–325 (2005)

    MATH  Google Scholar 

  • Mezić, I., Banaszuk, A.: Comparison of systems with complex behavior. Physica D 197(1–2), 101–133 (2004)

    MathSciNet  MATH  Google Scholar 

  • Moore, B.C.: Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26(1), 17–32 (1981)

    Article  MATH  Google Scholar 

  • Noack, B.R., Afanasiev, K., Morzyński, M., Tadmor, G., Thiele, F.: A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Noack, B.R., Papas, P., Monkewitz, P.A.: The need for a pressure-term representation in empirical Galerkin models of incompressible shear flow. J. Fluid Mech. 523, 339–365 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Petersen, K.: Ergodic Theory. Cambridge University Press, Cambridge (1983)

    MATH  Google Scholar 

  • Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes, 3rd edn. Cambridge University Press, Cambridge (2007)

    MATH  Google Scholar 

  • Provansal, M., Mathis, C., Boyer, L.: Bénard–von Kármán instability: transient and forced regimes. J. Fluid Mech. 182, 1–22 (1987)

    Article  MATH  Google Scholar 

  • Roshko, A.: On the development of turbulent wakes from vortex streets. Technical report NACA 1191, National Advisory Committee for Aeronautics (1954)

  • Roussopoulos, K.: Feedback control of vortex shedding at low Reynolds numbers. J. Fluid Mech. 248, 267–296 (1993)

    Article  Google Scholar 

  • Rowley, C.W.: Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifurc. Chaos 15(3), 997–1013 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Rowley, C.W., Mezić, I., Bagheri, S., Schlatter, P., Henningson, D.S.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Schmid, P.J.: Application of the dynamic mode decomposition to experimental data. Exp. Fluids 50(4), 1123–1130 (2011)

    Article  Google Scholar 

  • Schmid, P.J., Henningson, D.S.: Stability and Transition in Shear Flows. Springer, Berlin (2000)

    Google Scholar 

  • Schmid, P.J., Li, L., Juniper, M.P., Pust, O.: Applications of the dynamic mode decomposition. Theor. Comput. Fluid Dyn. 25(1–4), 249–259 (2011)

    Article  Google Scholar 

  • Tadmor, G., Lehmann, O., Noack, B.R., Morzyński, M.: Mean field representation of the natural and actuated cylinder wake. Phys. Fluids 22(3), 034102 (2010)

    Article  Google Scholar 

  • Taira, K., Colonius, T.: The immersed boundary method: a projection approach. J. Comput. Phys. 225(2), 2118–2137 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Williamson, C.H.K.: Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech. 28, 477–539 (1996)

    Article  Google Scholar 

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Acknowledgements

This work was supported by the Department of Defense National Defense Science & Engineering Graduate (DOD NDSEG) Fellowship, the National Science Foundation Graduate Research Fellowship Program (NSF GRFP), and AFOSR grant FA9550-09-1-0257.

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Correspondence to Kevin K. Chen.

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Communicated by P. Newton.

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Chen, K.K., Tu, J.H. & Rowley, C.W. Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses. J Nonlinear Sci 22, 887–915 (2012). https://doi.org/10.1007/s00332-012-9130-9

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  • DOI: https://doi.org/10.1007/s00332-012-9130-9

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