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

Matthew O. Williams, Clarence Worth Rowley, Yannis Kevrekidis

Research output: Contribution to journalArticlepeer-review

54 Scopus citations

Abstract

A data-driven, kernel-based method for approximating the leading Koopman eigenvalues, eigenfunctions, and modes in problems with highdimensional 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.

Original languageEnglish (US)
Pages (from-to)247-265
Number of pages19
JournalJournal of Computational Dynamics
Volume2
Issue number2
DOIs
StatePublished - Dec 1 2015

All Science Journal Classification (ASJC) codes

  • Computational Mechanics
  • Computational Mathematics

Keywords

  • Dynamic mode decomposition
  • Kernel methods
  • Koopman operator
  • Machine learning
  • Time series analysis

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