Variants of dynamic mode decomposition: Boundary condition, Koopman, and fourier analyses

Kevin K. Chen, Jonathan H. Tu, Clarence Worth Rowley

Research output: Contribution to journalArticlepeer-review

551 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)887-915
Number of pages29
JournalJournal of Nonlinear Science
Issue number6
StatePublished - Dec 2012

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • General Engineering
  • Applied Mathematics


  • Approximate eigenvalues and eigenvectors
  • Boundary conditions
  • Discrete Fourier transform
  • Dynamic mode decomposition
  • Koopman operator
  • Navier-Stokes equations
  • Time series


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