American Institute of Mathematical Sciences

January  2016, 3(1): 51-79. doi: 10.3934/jcd.2016003

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

 1 Department of Mathematics and Computer Science, Freie Universität Berlin, Germany, Germany 2 Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin

Received  December 2015 Revised  July 2016 Published  September 2016

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 di erent methods that have been developed over the last decades to compute nite-dimensional approximations of these in nite-dimensional operators - in particular Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - and highlight the similarities and di erences between these approaches. The results will be illustrated using simple stochastic di erential equations and molecular dynamics examples.
Citation: Stefan Klus, Péter Koltai, Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (1) : 51-79. doi: 10.3934/jcd.2016003
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