Variants of dynamic mode decomposition: connections between Koopman and Fourier analyses

K. K. Chen, J. H. Tu, and C. W. Rowley

Journal of Nonlinear Science (online first), April 2012.

Abstract:

Dynamic mode decomposition (DMD) is an Arnoldi-like method based on the Koopman operator that analyzes empirical data, typically generated by nonlinear dynamics, and computes eigenvalues and eigenvectors 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, and that DMD on mean-subtracted data is equivalent to the temporal discrete Fourier transform (DFT). The temporal DFT does not yield frequencies or growth rates, and modal energies exhibit a slow rolloff for non-periodic data. However, the mean flow mode satisfies linear boundary conditions, and all other DFT modes have homogenous boundary conditions. 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 two-dimensional cylinder flow data at a Reynolds number of 60. Time-varying temporal DFT and optimized DMD modes both yield low projection errors.

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