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Lookup NU author(s): Professor Mihai Putinar
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND).
© 2018 Elsevier Inc. Starting from measured data, we develop a method to compute the fine structure of the spectrum of the Koopman operator with rigorous convergence guarantees. The method is based on the observation that, in the measure-preserving ergodic setting, the moments of the spectral measure associated to a given observable are computable from a single trajectory of this observable. Having finitely many moments available, we use the classical Christoffel–Darboux kernel to separate the atomic and absolutely continuous parts of the spectrum, supported by convergence guarantees as the number of moments tends to infinity. In addition, we propose a technique to detect the singular continuous part of the spectrum as well as two methods to approximate the spectral measure with guaranteed convergence in the weak topology, irrespective of whether the singular continuous part is present or not. The proposed method is simple to implement and readily applicable to large-scale systems since the computational complexity is dominated by inverting an N×N Hermitian positive-definite Toeplitz matrix, where N is the number of moments, for which efficient and numerically stable algorithms exist; in particular, the complexity of the approach is independent of the dimension of the underlying state-space. We also show how to compute, from measured data, the spectral projection on a given segment of the unit circle, allowing us to obtain a finite approximation of the operator that explicitly takes into account the point and continuous parts of the spectrum. Finally, we describe a relationship between the proposed method and the so-called Hankel Dynamic Mode Decomposition, providing new insights into the behavior of the eigenvalues of the Hankel DMD operator. A number of numerical examples illustrate the approach, including a study of the spectrum of the lid-driven two-dimensional cavity flow.
Author(s): Korda M, Putinar M, Mezic I
Publication type: Article
Publication status: Published
Journal: Applied and Computational Harmonic Analysis
Year: 2020
Volume: 48
Issue: 2
Pages: 599-629
Print publication date: 01/03/2020
Online publication date: 22/08/2018
Acceptance date: 03/08/2018
Date deposited: 09/11/2018
ISSN (print): 1063-5203
ISSN (electronic): 1096-603X
Publisher: Academic Press Inc.
URL: https://doi.org/10.1016/j.acha.2018.08.002
DOI: 10.1016/j.acha.2018.08.002
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