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

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Dynamic mode decomposition Scalar (mathematics) Eigenfunction Observable Kernel (algebra) Mathematics Representer theorem Vorticity Algorithm Subspace topology Applied mathematics Scalar field Computer science Kernel embedding of distributions Kernel method Mathematical analysis Artificial intelligence Discrete mathematics Geometry Physics Eigenvalues and eigenvectors Support vector machine Machine learning Quantum mechanics Mathematical physics Vortex Thermodynamics

Matthew O. Williams , Clarence W. Rowley , Ioannis G. Kevrekidis ·

YOU? · · 2015 · Open Access · · DOI: https://doi.org/10.3934/jcd.2015005 · OA: W2395962256

A data-driven, kernel-based method for approximating the leading Koopmaneigenvalues, eigenfunctions, and modes in problems with high-dimensional statespaces 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 offunctions 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-diffusionsystem, and the second is a set of vorticity data computed from experimentally obtainedvelocity 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.