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

By Stefan Klus, Péter Koltai, Christof Schutte. 2015

332 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/1e7af7f01b1fece668ec65fb7ed990dc91b4eca0

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In Atomic and Molecular Physics, and Optics

In Statistical and Nonlinear Physics

Citations in the corpus

Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control (2016, background): “The authors would also like to thank Péter Koltai for bringing to our attantion reference [8].”
Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator (2016, methodology): “algorithm is closely related to the Prony approximation of KMD [42], and it can interpreted as a variation of extended DMD [48] on ergodic trajectories (also see [17]).”
Data-driven approximation of the Koopman generator: Model reduction, system identification, and control (2019, methodology): “The derivation of the modes is equivalent to the standard EDMD case, see [4, 38] for more details.”
Variational Approach for Learning Markov Processes from Time Series Data (2017, background): “, [61,62,23,24]), and the eigenfunctions form an invariant subspace of the Koopman operator”
On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator (2017, background): “It was proven in [11] that as M → ∞, the operator KN,M converges to KN , the orthogonal projection of the action of the Koopman operator on the span of the observables used in EDMD.”
Data-driven discovery of Koopman eigenfunctions for control (2017, methodology): “Current data-driven methods to approximate the Koopman operator include extended DMD (EDMD) [70,71,29] and the variational approach of conformation dynamics (VAC) [49,50].”
Data-Driven Model Reduction and Transfer Operator Approximation (2017, methodology): “nsionsoftheaforementionedmethodscalledthevariationalapproachofconformation dynamics (VAC)[13,14,15]developedmainlyforreversiblemoleculardynamicsproblems and extended dynamic mode decomposition (EDMD) [16,17,18] can be used to compute eigenfunctions,eigenvalues,andeigenmodesoftheKoopmanoperator(andit”
On Numerical Approximations of the Koopman Operator (2020, methodology): “This result has been obtained in [30], without the use of the dual basis, relying on the Moore-Penrose pseudoinverse, the connection with which we discuss next.”