EDMD for expanding circle maps and their complex perturbations

We show that spectral data of the Koopman operator arising from an analytic expanding circle map \(\tau\) can be effectively calculated using an EDMD-type algorithm combining a collocation method of order m with a Galerkin method of order n. The main result is that if \(m \geq \delta n\), where \(\d...

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Bibliographic Details
Published in:arXiv.org 2023-08
Main Authors: Bandtlow, Oscar F, Just, Wolfram, Slipantschuk, Julia
Format: Article
Language:English
Subjects:
Algorithms
Annuli
Collocation methods
Galerkin method
Perturbation
Online Access:Get full text
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Summary:We show that spectral data of the Koopman operator arising from an analytic expanding circle map \(\tau\) can be effectively calculated using an EDMD-type algorithm combining a collocation method of order m with a Galerkin method of order n. The main result is that if \(m \geq \delta n\), where \(\delta\) is an explicitly given positive number quantifying by how much \(\tau\) expands concentric annuli containing the unit circle, then the method converges and approximates the spectrum of the Koopman operator, taken to be acting on a space of analytic hyperfunctions, exponentially fast in n. Additionally, these results extend to more general expansive maps on suitable annuli containing the unit circle.
ISSN:2331-8422