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

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  • 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.
    Mathematics Subject Classification: Primary: 37M10, 37M25; Secondary: 34L16, 37L65, 37N25.


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