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

Matthew O.  Williams; Clarence W. Rowley; Ioannis G.  Kevrekidis

doi:10.3934/jcd.2015005

Article Contents

2015, Volume 2, Issue 2: 247-265. Doi: 10.3934/jcd.2015005

This issue Previous Article Computing continuous and piecewise affine lyapunov functions for nonlinear systems Next Article

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

1.
United Technologies Research Center, 411 Silver Lane, East Hartford, CT 06118
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

Received: January 31, 2015

Revised: January 31, 2016

Published: May 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 37M10, 65P99, 47B33.

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