Numerical methods to evaluate Koopman matrix from system equations
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A method that is employed to evaluate a Koopman matrix from a data set of snapshot pairs is the extended dynamical mode decomposition (EDMD). The Koopman operator is a linear but infinite-dimensional operator that governs the evolution of observables, and is beneficial when employed in the analysis of dynamics. The Koopman matrix corresponds to an approximation of the Koopman operator, requiring a specific dictionary to represent the operator. In this study, an alternative approach for evaluating the Koopman matrix for stochastic differential equations has been proposed. Using the system equations the Koopman matrix can be directly derived without any sampling. Hence, this approach is complementary to a data-driven approach provided a prior knowledge of the system equations is available. The proposed method comprises combinatorics, an approximation of the resolvent, and extrapolations. Comparisons with the EDMD have also been demonstrated considering a noisy van der Pol system. The proposed method yields reasonable results even in cases wherein the EDMD exhibits a slow convergence behavior.
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