FN Archimer Export Format
PT J
TI Eigenvalues of autocovariance matrix: A practical method to identify the Koopman eigenfrequencies
BT 
AF Zhen, Yicun
   Chapron, Bertrand
   Mémin, Etienne
   Peng, Lin
AS 1:1;2:1;3:2;4:3;
FF 1:PDG-ODE-LOPS-SIAM;2:PDG-ODE-LOPS-SIAM;3:;4:;
C1 Institut Franais de Recherche pour l'Exploitation de la Mer, 29280 Plouzané, France
   INRIA/IRMAR, Campus universitaire de Beaulieu, Rennes, 35042 Cedex, France
   Ocean University of China, 266100 Qingdao, China
C2 IFREMER, FRANCE
   INRIA, FRANCE
   UNIV OCEAN CHINA, CHINA
SI BREST
SE PDG-ODE-LOPS-SIAM
UM LOPS
IN WOS Ifremer UMR
   copubli-france
   copubli-int-hors-europe
   copubli-sud
IF 2.4
TC 4
UR https://archimer.ifremer.fr/doc/00764/87649/93099.pdf
LA English
DT Article
AB To infer eigenvalues of the infinite-dimensional Koopman operator, we study the leading eigenvalues of the autocovariance matrix associated with a given observable of a dynamical system. For any observable f for which all the time-delayed autocovariance exist, we construct a Hilbert space Hf and a Koopman-like operator K that acts on Hf. We prove that the leading eigenvalues of the autocovariance matrix has one-to-one correspondence with the energy of f that is represented by the eigenvectors of K. The proof is associated to several representation theorems of isometric operators on a Hilbert space, and the weak-mixing property of the observables represented by the continuous spectrum. We also provide an alternative proof of the weakly mixing property. When f is an observable of an ergodic dynamical system which has a finite invariant measure μ, Hf coincides with closure in L2(X,dμ) of Krylov subspace generated by f, and K coincides with the classical Koopman operator. The main theorem sheds light to the theoretical foundation of several semi-empirical methods, including singular spectrum analysis (SSA), data-adaptive harmonic analysis (DAHD), Hankel DMD, and Hankel alternative view of Koopman analysis (HAVOK). It shows that, when the system is ergodic and has finite invariant measure, the leading temporal empirical orthogonal functions indeed correspond to the Koopman eigenfrequencies. A theorem-based practical methodology is then proposed to identify the eigenfrequencies of K from a given time series. It builds on the fact that the convergence of the renormalized eigenvalues of the Gram matrix is a necessary and sufficient condition for the existence of K−eigenfrequencies. Numerical illustrating results on simple low dimensional systems and real interpolated ocean sea-surface height data are presented and discussed. 
PY 2022
PD MAR
SO Physical Review E
SN 2470-0045
PU American Physical Society (APS)
VL 105
IS 3
UT 000779822800001
DI 10.1103/PhysRevE.105.034205
ID 87649
ER

EF