On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator

摘要

Extended Dynamic Mode Decomposition (EDMD) is an algorithm that approximatesthe action of the Koopman operator on an $N$-dimensional subspace of the spaceof observables by sampling at $M$ points in the state space. Assuming that thesamples are drawn either independently or ergodically from some measure $\mu$,it was shown 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 thefinite-dimensional subspace of observables. In this work, we show that, as $N\rightarrow \infty$, the operator $\mathcal{K}_N$ converges in the strongoperator topology to the Koopman operator. This in particular impliesconvergence of the predictions of future values of a given observable over anyfinite time horizon, a fact important for practical applications such asforecasting, estimation and control. In addition, we show that accumulationpoints of the spectra of $\mathcal{K}_N$ correspond to the eigenvalues of theKoopman operator with the associated eigenfunctions converging weakly to aneigenfunction of the Koopman operator, provided that the weak limit ofeigenfunctions is nonzero. As a by-product, we propose an analytic version ofthe EDMD algorithm which, under some assumptions, allows one to construct$\mathcal{K}_N$ directly, without the use of sampling. Finally, underadditional assumptions, we analyze convergence of $\mathcal{K}_{N,N}$ (i.e.,$M=N$), proving convergence, along a subsequence, to weak eigenfunctions (oreigendistributions) related to the eigenmeasures of the Perron-Frobeniusoperator. No assumptions on the observables belonging to a finite-dimensionalinvariant subspace of the Koopman operator are required throughout.