Abstract
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.
Highlights
Fluid flows often exhibit low-dimensional behavior, despite the fact that they are governed by infinite-dimensional partial differential equations
We have presented a new definition in which dynamic mode decomposition (DMD) is defined to be the eigendecomposition of an approximating linear operator
Our framework can be considered to be an extension of existing DMD theory to a more general class of datasets
Summary
Fluid flows often exhibit low-dimensional behavior, despite the fact that they are governed by infinite-dimensional partial differential equations (the Navier–Stokes equations). Koopman operator, spectral analysis, time series analysis, reduced-order models. We present DMD as an analysis of pairs of n-dimensional data vectors (xk, yk), in contrast to the sequential time series that are typically considered From these data we construct a particular linear operator A and define DMD as the eigendecomposition of that operator (see Definition 1). There is no guarantee that analyzing this particular approximating operator is meaningful for data generated by nonlinear dynamics To this end, we show that our definition strengthens the connections between DMD and Koopman operator theory, extending those connections to include more general sampling strategies. We show that our definition strengthens the connections between DMD and Koopman operator theory, extending those connections to include more general sampling strategies This is important, as it allows us to maintain the interpretion of DMD as an approximation to Koopman spectral analysis. Where U is n × r, Σ is diagonal and r × r, V is m × r, and r is the rank of X
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
