TY - JOUR

T1 - Reproducing kernel Hilbert space compactification of unitary evolution groups

AU - Das, Suddhasattwa

AU - Giannakis, Dimitrios

AU - Slawinska, Joanna

N1 - Funding Information:
Dimitrios Giannakis acknowledges support by ONR YIP grant N00014-16-1-2649 , ONR MURI grant N00014-19-1-2421 , NSF grants DMS-1521775 , DMS-1854383 , 1842538 , and DARPA grant HR0011-16-C-0116 . Suddhasattwa Das was supported as a postdoctoral research fellow from the first two grants. Joanna Slawinska acknowledges support from NSF grants 1551489 and 1842538 . The authors would like to thank Igor Mezić for pointing out a possible connection between the results of Theorem 2 and the pseudospectrum of the Koopman operator, which led to Corollary 3 .
Publisher Copyright:
© 2021 The Author(s)

PY - 2021/9

Y1 - 2021/9

N2 - A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator Wτ on a reproducing kernel Hilbert space (RKHS). The operator Wτ is skew-adjoint, and thus can be represented by a projection-valued measure, discrete by compactness, with an associated orthonormal basis of eigenfunctions. These eigenfunctions are ordered in terms of a Dirichlet energy, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, Wτ generates a unitary evolution group {etWτ}t∈R on the RKHS, which approximates the unitary Koopman group of the system. We establish convergence results for the spectrum and Borel functional calculus of Wτ as τ→0+, as well as an associated data-driven formulation utilizing time series data. Numerical applications to ergodic systems with atomic and continuous spectra, namely a torus rotation, the Lorenz 63 system, and the Rössler system, are presented.

AB - A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator Wτ on a reproducing kernel Hilbert space (RKHS). The operator Wτ is skew-adjoint, and thus can be represented by a projection-valued measure, discrete by compactness, with an associated orthonormal basis of eigenfunctions. These eigenfunctions are ordered in terms of a Dirichlet energy, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, Wτ generates a unitary evolution group {etWτ}t∈R on the RKHS, which approximates the unitary Koopman group of the system. We establish convergence results for the spectrum and Borel functional calculus of Wτ as τ→0+, as well as an associated data-driven formulation utilizing time series data. Numerical applications to ergodic systems with atomic and continuous spectra, namely a torus rotation, the Lorenz 63 system, and the Rössler system, are presented.

KW - Ergodic dynamical systems

KW - Koopman operators

KW - Perron-Frobenius operators

KW - Reproducing kernel Hilbert spaces

KW - Spectral theory

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U2 - 10.1016/j.acha.2021.02.004

DO - 10.1016/j.acha.2021.02.004

M3 - Article

AN - SCOPUS:85102896468

SN - 1063-5203

VL - 54

SP - 75

EP - 136

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

ER -