On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator

Milan Korda, Igor Mezić

Published 2017-03-14Version 1

Extended Dynamic Mode Decomposition (EDMD) \cite{Williamsetal:2015} is an algorithm that represents the action of the Koopman operator on an $N$-dimensional subspace of the space of observables using a sample of $M$ points in the state space. Under the assumption that the samples are drawn either independently or ergodically from some measure $\mu$, we show 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. Next, we show that, as $N \rightarrow \infty$, the operator $\mathcal{K}_N$ converges in the strong operator topology to the Koopman operator and in addition 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 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.