TY - JOUR
T1 - Variants of dynamic mode decomposition
T2 - Boundary condition, Koopman, and fourier analyses
AU - Chen, Kevin K.
AU - Tu, Jonathan H.
AU - Rowley, Clarence Worth
N1 - Funding Information:
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.
Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.
PY - 2012/12
Y1 - 2012/12
N2 - 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.
AB - 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.
KW - Approximate eigenvalues and eigenvectors
KW - Boundary conditions
KW - Discrete Fourier transform
KW - Dynamic mode decomposition
KW - Koopman operator
KW - Navier-Stokes equations
KW - Time series
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U2 - 10.1007/s00332-012-9130-9
DO - 10.1007/s00332-012-9130-9
M3 - Article
AN - SCOPUS:84878537308
SN - 0938-8974
VL - 22
SP - 887
EP - 915
JO - Journal of Nonlinear Science
JF - Journal of Nonlinear Science
IS - 6
ER -