Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses
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.
KeywordsKoopman operator Dynamic mode decomposition Time series Boundary conditions Discrete Fourier transform Approximate eigenvalues and eigenvectors Navier–Stokes equations
Mathematics Subject Classification35G61 35P15 37M10 37N10 47B33
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.