Special CMX Seminar
Koopman operators globally linearize nonlinear dynamical systems, and their spectral information can be a powerful tool for the analysis and decomposition of nonlinear dynamical systems. However, Koopman operators are infinite-dimensional, and computing their spectral information is a considerable challenge. We introduce measure-preserving extended dynamic mode decomposition (mpEDMD), the first Galerkin method whose eigendecomposition converges to the spectral quantities of Koopman operators for general measure-preserving dynamical systems. mpEDMD is a data-driven and structure-preserving algorithm based on an orthogonal Procrustes problem that enforces measure-preserving truncations of Koopman operators using a general dictionary of observables. It is flexible and easy to use with any pre-existing DMD-type method and with different data types. We prove the convergence of mpEDMD for projection-valued and scalar-valued spectral measures, spectra, and Koopman mode decompositions. For the case of delay embedding (Krylov subspaces), our results include convergence rates of the approximation of spectral measures as the size of the dictionary increases. We demonstrate mpEDMD on a range of challenging examples, its increased robustness to noise compared with other DMD-type methods, and its ability to capture the energy conservation and cascade of a turbulent boundary layer flow with Reynolds number > 60000 and state-space dimension >100000. Finally, if time permits, we discuss how this algorithm forms part of a broader program on the foundations of infinite-dimensional spectral computations.