On the numerical approximation of the Perron-Frobenius and Koopman operator

TL;DR

This paper reviews numerical methods for approximating the Perron-Frobenius and Koopman operators, highlighting their similarities, differences, and applications in dynamical systems analysis.

Contribution

It provides a comprehensive comparison of methods like Ulam's and EDMD for finite-dimensional approximation of these operators.

Findings

Comparison of Ulam's method and EDMD

Applications to stochastic differential equations

Illustrations with molecular dynamics

Abstract

Information about the behavior of dynamical systems can often be obtained by analyzing the eigenvalues and corresponding eigenfunctions of linear operators associated with a dynamical system. Examples of such operators are the Perron-Frobenius and the Koopman operator. In this paper, we will review different methods that have been developed over the last decades to compute finite-dimensional approximations of these infinite-dimensional operators - e.g. Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and differences between these approaches. The results will be illustrated using simple stochastic differential equations and molecular dynamics examples.