In this talk we are going to discuss a methodology for achieving convergent Dynamic Mode Decomposition (DMD) algorithms in the kernel setting. DMD methods attempt to model an unknown dynamical system using approximations of certain dynamic operators, such as Koopman and Liouville operators. Most existent convergence theories rely on ergodic theorems, which provide at best probability one results.

Recent progress in kernel-based methods enable algorithms that converge everywhere, owing to the dominance of the supremum norm by the Hilbert space norm in a reproducing kernel Hilbert space (RKHS). However, the rigidity of the structure of the functions in and operators over a RKHS can create difficulties in finding dynamics that lead to well behaved operators (i.e. compact).

We will outline a methodology that allows for compact dynamic operators and leads everywhere convergent algorithms.