On the approximation of Koopman spectra of measure-preserving flows

TL;DR

This paper extends periodic approximation methods to measure-preserving flows, showing how to accurately approximate Koopman spectra through discretization, with conditions ensuring convergence, supported by numerical examples.

Contribution

It introduces a generalized approach for approximating Koopman spectra of measure-preserving flows using discretization, establishing convergence conditions and contrasting with classical numerical schemes.

Findings

Spectral convergence occurs when spatial refinements outpace temporal refinements.

A sufficient condition links time and space discretizations for weak spectral convergence.

Numerical results demonstrate the method's effectiveness on benchmark flows.

Abstract

The method of using periodic approximations to compute the spectral decomposition of the Koop- man operator is generalized to the class of measure-preserving flows on compact metric spaces. It is shown that the spectral decomposition of the continuous one-parameter unitary group can be approximated from an intermediate time discretization of the flow. A sufficient condition is established between the time-discretization of the flow and the spatial discretization of the periodic approximation, so that weak convergence of spectra will occur in the limit. This condition effectively translates to the requirement that the spatial refinements must occur at a faster pace than the temporal refinements. This result is contrasted with the well-known CLF condition of finite difference schemes for advection equations. Numerical results of spectral computations are shown for some benchmark examples of volume-preserving flows.