TY - JOUR
T1 - A kernel-based method for data-driven Koopman spectral analysis
AU - Williams, Matthew O.
AU - Rowley, Clarence Worth
AU - Kevrekidis, Yannis
N1 - Funding Information:
The authors acknowledge support from NSF (awards DMS-1204783 and CDSE-1310173) and AFOSR
Publisher Copyright:
© American Institute of Mathematical Sciences.
PY - 2015/12/1
Y1 - 2015/12/1
N2 - 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.
AB - 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.
KW - Dynamic mode decomposition
KW - Kernel methods
KW - Koopman operator
KW - Machine learning
KW - Time series analysis
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U2 - 10.3934/jcd.2015005
DO - 10.3934/jcd.2015005
M3 - Article
AN - SCOPUS:85018029552
SN - 2158-2505
VL - 2
SP - 247
EP - 265
JO - Journal of Computational Dynamics
JF - Journal of Computational Dynamics
IS - 2
ER -