Section: New Results

Low–rank dynamic mode decomposition: optimal solution in polynomial time

Participant : Patrick Héas.

This is a collaboration with Cédric Herzet (EPI FLUMINANCE, Inria Rennes–Bretagne Atlantique)

The works [15] and  [41] study the linear approximation of high–dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data–driven approach can be formalised as attempting to solve a low-rank constrained optimisation problem. This problem is non–convex and state–of–the–art algorithms are all sub–optimal. We show that there exists a closed-form solution, which can be computed in polynomial time, and characterises the ${\ell }_2$–norm of the optimal approximation error. The theoretical results serve to design low–complexity algorithms building reduced models from the optimal solution, based on singular value decomposition or low–rank DMD. The algorithms are evaluated by numerical simulations using synthetic and physical data benchmarks.