The Kernel Perspective on Dynamic Mode Decomposition
Abstract: The purpose of the new DMD algorithm developed in this paper is to show that DMD methods very similar to KDMD emerge naturally out of a finite rank representation of the Koopman operator. It should be noted that the developed algorithm, while derived in a different way than traditional KDMD, involves computations that are nearly identical to KDMD, and as such, is not expected to offer any performance benefits over KDMD. Moreover, the algorithmic development of the present method does not invoke feature space representations and infinite matrices as in Williams et al., rather this method uses directly the properties of Koopman (or composition) operators and kernel functions. By doing so, this makes the theoretical dependencies of kernel based DMD methods transparent as densely defined operators over infinite dimensional kernel spaces. In order to present this new kernel perspective of Koopman analysis, the manuscript first introduces reproducing kernel Hilbert spaces (RKHSs) and examines the properties of Koopman operators over said spaces. Additionally, the examination of these properties led to the proof that the Koopman operator over the Gaussian RBF's native space is only bounded when it corresponds to discrete dynamics that are affine.
Submission Length: Long submission (more than 12 pages of main content)
Changes Since Last Submission: The first 8 pages represent the author's responses to the comments made by the reviewers. The final submission will not include these pages.
The following lists some of the changes made to the manuscript:
A new abstract
More details regarding the DMD algorithm presented in the manuscript
Addition of pseudocode
An additional numerical experiment
Comparison to a previous DMD method
Addition of a more formal metric for comparing the snapshots that were obtained from the new DMD algorithm to those from the original data set.
Assigned Action Editor: ~Ivan_Oseledets1
Submission Number: 1492
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