Extended dynamic mode decomposition with Fourier dictionaries: Error bounds and fast implementation

Abstract: The Koopman operator has gained considerable attention in dynamical systems due to its capability to provide a linear viewpoint for nonlinear systems using data-driven methods such as extended dynamic mode decomposition (EDMD). In this work, we suggest an EDMD-variant with a Fourier dictionary on the $d$-dimensional torus, where the data are sampled on an equispaced tensor grid. In this setting, the EDMD regression problem admits a unique closed-form solution, which we show to coincide with trigonometric interpolation of the Koopman image. This identification has two consequences. First, using Koopman invariance of Sobolev spaces, we transfer approximation-theoretic results for trigonometric interpolation to derive error bounds on approximations of the Koopman operator. In particular, the established bounds are of optimal order with explicit constants. Second, the EDMD matrix never has to be assembled, since its action reduces to (nonequispaced) fast Fourier transforms such that the EDMD-approximation may be evaluated matrix-free with quasi-linear cost in the dictionary size. We illustrate the results for the Kuramoto model of coupled oscillators with dictionaries of up to $10^7$ modes.