On Numerical Approximations of the Koopman Operator

TL;DR

This paper investigates numerical methods for approximating the spectral properties of the Koopman operator, introducing convergence results and analyzing the effectiveness of finite section and Krylov subspace methods.

Contribution

It characterizes Koopman Modes in Banach spaces, relates DMD to finite section theory, and provides convergence rates and error analysis for these numerical approximations.

Findings

Finite section method can fail for certain mixing maps.

Convergence rate of finite section approximation under sample size increase.

Krylov subspace methods converge in pseudospectral sense for operators with pure point spectrum.

Abstract

We study numerical approaches to computation of spectral properties of composition operators. We provide a characterization of Koopman Modes in Banach spaces using Generalized Laplace Analysis. We cast the Dynamic Mode-Decomposition type methods in the context of Finite Section theory of infinite dimensional operators, and provide an example of a mixing map for which the finite section method fails. Under assumptions on the underlying dynamics, we provide the first result on the convergence rate under sample size increase in the finite-section approximation. We study the error in the Krylov subspace version of the finite section method and prove convergence in pseudospectral sense for operators with pure point spectrum. This result indicates that Krylov sequence-based approximations can have low error without an exponential-in-dimension increase in the number of functions needed for approximation.