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A data-driven approximation of the koopman operator: extending dynamic mode decomposition. (English) Zbl 1329.65310

Summary: The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data-driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a “black box” integrator. We show that this approach is, in effect, an extension of dynamic mode decomposition, which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the “stochastic Koopman operator” (cf. [I. Mezić, Nonlinear Dyn. 41, No. 1–3, 309–325 (2005; Zbl 1098.37023)]). Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data and two that show potential applications of the Koopman eigenfunctions.

MSC:

65P99 Numerical problems in dynamical systems
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
37H10 Generation, random and stochastic difference and differential equations
47B33 Linear composition operators

Citations:

Zbl 1098.37023
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References:

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