Low-rank dynamic mode decomposition: an exact and tractable solution. (English) Zbl 1484.65086

Summary: This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition. Searching this approximation in a data-driven approach is formalized as attempting to solve a low-rank constrained optimization problem. This problem is non-convex, and state-of-the-art algorithms are all sub-optimal. This paper shows that there exists a closed-form solution, which is computed in polynomial time, and characterizes the \(\ell_2\)-norm of the optimal approximation error. The paper also proposes low-complexity algorithms building reduced models from this optimal solution, based on singular value decomposition or eigenvalue decomposition. The algorithms are evaluated by numerical simulations using synthetic and physical data benchmarks.


65F55 Numerical methods for low-rank matrix approximation; matrix compression
65K05 Numerical mathematical programming methods


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