Applied and Computational Math Seminar: Koopman spectra in reproducing kernel Hilbert spaces

Speaker:Suddhasattwa Das, Courant Institute of Mathematical Sciences, New York University
Title: Koopman spectra in reproducing kernel Hilbert spaces

Abstract: Every invertible dynamical system induces a Koopman operator, which is a linear, unitary operator acting on the space of observables. Koopman eigenfunctions represent the periodic or non-mixing component of the dynamics. The extraction of these eigenfunctions from a given time-series is a non-trivial problem when the underlying system has a continuous spectrum, which behaves like a strong noisy component to the signal. Of particular significance are the eigenfunctions of the Koopman operator, one among many of their physical significance is that they correspond to stable spatio-temporal patterns in the dynamics. This paper describes methods for identifying Koopman eigenfrequencies and eigenfunctions from a discretely sampled time-series generated by an unknown dynamical system. Given the values of a function at these time samples, our main result gives necessary and sufficient conditions under which these values can be extended to a functional space called a reproducing kernel Hilbert space or RKHS. An RKHS is a dense subset of the space of continuous functions and is very useful for out-of-sample extensions. We take a data-driven approach, which means inferring properties of the system from a time-series which could be of dimension much smaller than the underlying system, and without having any prior knowledge of the system, or a model or equations to start with, or parameters to tune/fit. Given a sequence of N time samples of the dynamical system through some observation map, one fits an RKHS function for every candidate eigenfrequency, omega, and calculate its RKHS norm w_N(\omega). We use the limit $\lim_{N\to\infty} w_N(\omega)$ to derive necessary and sufficient conditions for omega to be an eigenfrequency.

Time: Friday, March 2, 2018, 1:30-2:30pm

Place: Exploratory Hall, Room 4106