# American Institute of Mathematical Sciences

June  2015, 2(2): 247-265. doi: 10.3934/jcd.2015005

## A kernel-based method for data-driven koopman spectral analysis

 1 United Technologies Research Center, 411 Silver Lane, East Hartford, CT 06118, United States 2 Dept. of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544 3 Department of Chemical and Biological Engineering & PACM, Princeton University, Princeton, NJ 08544, United States

Received  February 2015 Revised  February 2016 Published  May 2016

A data-driven, kernel-based method for approximating the leading Koopman eigenvalues, eigenfunctions, and modes in problems with high-dimensional state spaces is presented. This approach uses a set of scalar observables (functions that map a state to a scalar value) that are defined implicitly by the feature map associated with a user-defined kernel function. This circumvents the computational issues that arise due to the number of functions required to span a sufficiently rich'' subspace of all possible scalar observables in such applications. We illustrate this method on two examples: the first is the FitzHugh-Nagumo PDE, a prototypical one-dimensional reaction-diffusion system, and the second is a set of vorticity data computed from experimentally obtained velocity data from flow past a cylinder at Reynolds number 413. In both examples, we use the output of Dynamic Mode Decomposition, which has a similar computational cost, as the benchmark for our approach.
Citation: Matthew O. Williams, Clarence W. Rowley, Ioannis G. Kevrekidis. A kernel-based method for data-driven koopman spectral analysis. Journal of Computational Dynamics, 2015, 2 (2) : 247-265. doi: 10.3934/jcd.2015005
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