Journal of Nonlinear Science

, Volume 22, Issue 6, pp 887–915

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

  • Kevin K. Chen
  • Jonathan H. Tu
  • Clarence W. Rowley
Article

DOI: 10.1007/s00332-012-9130-9

Cite this article as:
Chen, K.K., Tu, J.H. & Rowley, C.W. J Nonlinear Sci (2012) 22: 887. doi:10.1007/s00332-012-9130-9

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.

Keywords

Koopman operatorDynamic mode decompositionTime seriesBoundary conditionsDiscrete Fourier transformApproximate eigenvalues and eigenvectorsNavier–Stokes equations

Mathematics Subject Classification

35G6135P1537M1037N1047B33

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Kevin K. Chen
    • 1
  • Jonathan H. Tu
    • 1
  • Clarence W. Rowley
    • 1
  1. 1.Department of Mechanical & Aerospace EngineeringPrinceton UniversityPrincetonUSA