JCD
On dynamic mode decomposition: Theory and applications
Jonathan H. Tu Clarence W. Rowley Dirk M. Luchtenburg Steven L. Brunton J. Nathan Kutz
Journal of Computational Dynamics 2014, 1(2): 391-421 doi: 10.3934/jcd.2014.1.391
Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken. We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator. This generalizes DMD to a larger class of datasets, including nonsequential time series. We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively. We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples. Such computations are not considered in the existing literature but can be understood using our more general framework. In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory. It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science. We show that under certain conditions, DMD is equivalent to LIM.
keywords: spectral analysis Dynamic mode decomposition Koopman operator time series analysis reduced-order models.
JCD
A kernel-based method for data-driven koopman spectral analysis
Matthew O. Williams Clarence W. Rowley Ioannis G. Kevrekidis
Journal of Computational Dynamics 2015, 2(2): 247-265 doi: 10.3934/jcd.2015005
A data-driven, kernel-based method for approximating the leading Koopman eigenvalues, eigenfunctions, and modes in problems with high-dimensional state spaces is presented. This approach uses a set of scalar observables (functions that map a state to a scalar value) that are defined implicitly by the feature map associated with a user-defined kernel function. This circumvents the computational issues that arise due to the number of functions required to span a ``sufficiently rich'' subspace of all possible scalar observables in such applications. We illustrate this method on two examples: the first is the FitzHugh-Nagumo PDE, a prototypical one-dimensional reaction-diffusion system, and the second is a set of vorticity data computed from experimentally obtained velocity data from flow past a cylinder at Reynolds number 413. In both examples, we use the output of Dynamic Mode Decomposition, which has a similar computational cost, as the benchmark for our approach.
keywords: time series analysis machine learning. kernel methods Koopman operator dynamic mode decomposition

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