On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator

Abstract

Extended dynamic mode decomposition (EDMD) (Williams et al. in J Nonlinear Sci 25(6):1307–1346, 2015) is an algorithm that approximates the action of the Koopman operator on an N-dimensional subspace of the space of observables by sampling at M points in the state space. Assuming that the samples are drawn either independently or ergodically from some measure $\mu$, it was shown in Klus et al. (J Comput Dyn 3(1):51–79, 2016) that, in the limit as $M\rightarrow \infty$, the EDMD operator $\mathcal {K}_{N,M}$ converges to $\mathcal {K}_N$, where $\mathcal {K}_N$ is the $L_2(\mu )$-orthogonal projection of the action of the Koopman operator on the finite-dimensional subspace of observables. We show that, as $N \rightarrow \infty$, the operator $\mathcal {K}_N$ converges in the strong operator topology to the Koopman operator. This in particular implies convergence of the predictions of future values of a given observable over any finite time horizon, a fact important for practical applications such as forecasting, estimation and control. In addition, we show that accumulation points of the spectra of $\mathcal {K}_N$ correspond to the eigenvalues of the Koopman operator with the associated eigenfunctions converging weakly to an eigenfunction of the Koopman operator, provided that the weak limit of the eigenfunctions is nonzero. As a by-product, we propose an analytic version of the EDMD algorithm which, under some assumptions, allows one to construct $\mathcal {K}_N$ directly, without the use of sampling. Finally, under additional assumptions, we analyze convergence of $\mathcal {K}_{N,N}$ (i.e., $M=N$), proving convergence, along a subsequence, to weak eigenfunctions (or eigendistributions) related to the eigenmeasures of the Perron–Frobenius operator. No assumptions on the observables belonging to a finite-dimensional invariant subspace of the Koopman operator are required throughout.