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Center for Nonlinear Studies |
Abstract: The purpose of this series of talks is to introduce the kernel perspective on Dynamic Mode Decompositions, Koopman Operators, and the Generators. This perspective sidesteps the usual ergodic theoretic and dynamical systems take on Koopman operators, and instead leverages function theoretic operator theory, which illuminates many theoretical challenges that DMD faces and which are rarely acknowledged. This series of talks are given in four pieces, which begins with the study of Koopman operators over reproducing kernel Hilbert spaces, and then becomes progressively more general, examining DMD methods for continuous time systems directly using occupation kernels and avoiding a discretization, and then we will look at how we can integrate these ideas into studying control affine dynamical systems and higher order dynamical systems. The fourth talk will then discuss how we can use this framework to resolve inverse problems using different operators.
The first part of this series will establish a language about operator theory and Koopman operators that will be leveraged throughout the four talks. We will introduce reproducing kernel HIlbert spaces, discuss function theoretic operators over these spaces, and then discuss Koopman operators in general. It turns out that bounded and compact Koopman (composition) operators are very rare with regard to kernel spaces, and we will develop a DMD algorithm that leverages the theory of densely defined operators to give a theoretically motivated numerical method for DMD.
Host: Humberto C Godinez (CCS-2) and Nishant Panda (CCS-3)