Numerical methods to evaluate Koopman matrix from system equations

Abstract

The Koopman operator is beneficial for analyzing nonlinear and stochastic dynamics; it is linear but infinite-dimensional, and it governs the evolution of observables. The extended dynamic mode decomposition (EDMD) is one of the famous methods in the Koopman operator approach. The EDMD employs a data set of snapshot pairs and a specific dictionary to evaluate an approximation for the Koopman operator, i.e., the Koopman matrix. In this study, we focus on stochastic differential equations, and a method to obtain the Koopman matrix is proposed. The proposed method does not need any data set, which employs the original system equations to evaluate some of the targeted elements of the Koopman matrix. The proposed method comprises combinatorics, an approximation of the resolvent, and extrapolations. Comparisons with the EDMD are performed for a noisy van der Pol system. The proposed method yields reasonable results even in cases wherein the EDMD exhibits a slow convergence behavior.

arXiv comment22 pages, 4 figures

References in corpus (11)

• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control
• Applied Koopmanism
• Extended dynamic mode decomposition with dictionary learning: a data-driven adaptive spectral decomposition of the Koopman operator
• On the numerical approximation of the Perron-Frobenius and Koopman operator
• Duality and hidden symmetries in interacting particle systems
• On the Koopman operator of algorithms
• Extended duality relations between birth-death processes and partial differential equations
• Duality in stochastic processes from the viewpoint of basis expansions
• Making Birth-Death Processes from Backward Fokker-Planck Equations for Computing Expectations in Langevin Systems
• Connection among stochastic Hamilton-Jacobi-Bellman equation, path-integral, and Koopman operator on nonlinear stochastic optimal control

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