Recent work in the study of dynamic systems has focused on data-driven decomposition techniques that approximate the action of the Koopman operator on observable functions of the underlying phenomena. 

In particular, the data-driven method of dynamic mode decomposition (DMD) has been explored, with multiple variants of the algorithm in existence, including extended DMD, DMD in reproducing kernel Hilbert spaces, a Bayesian framework, a variant for stochastic dynamical systems, and a variant that uses deep neural networks.  In this talk, I shall briefly summarize the large existing work on data-driven learning of Koopman operator models, and then describe new sampling-based sketching approaches (SketchyCoreSVD, SketchyCoreTucker)  together with matrix-valued Kernels, to achieve accelerated Koopman operator approximations of dynamic observable data. Examples are drawn from remote sensing and FTIR  hyperspectral tensor images, bio-medical cardiac magnetic resonance video, and time series reactive flow simulations of a single ejector combustion process.