DEPARTMENT OF MATHEMATICAL SCIENCES

APPLIED AND COMPUTATIONAL MATHEMATICS SEMINAR

**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

Department of Mathematical Sciences

George Mason University

4400 University Drive, MS 3F2

Fairfax, VA 22030-4444

http://math.gmu.edu/

Tel. 703-993-1460, Fax. 703-993-1491