Optimal Low-Rank Dynamic Mode Decomposition
This provides an incremental improvement for researchers and practitioners in fields like climate prediction and molecular dynamics by offering a more efficient algorithm for reduced-order modeling.
The paper tackles the problem of finding an optimal solution for low-rank Dynamic Mode Decomposition, a non-convex optimization problem previously addressed with sub-optimal algorithms, by proving a closed-form optimal solution exists and designing an effective SVD-based algorithm, with a toy example showing performance gains over state-of-the-art techniques.
Dynamic Mode Decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of non-linear systems from experimental datasets. Recently, several attempts have extended DMD to the context of low-rank approximations. This extension is of particular interest for reduced-order modeling in various applicative domains, e.g. for climate prediction, to study molecular dynamics or micro-electromechanical devices. This low-rank extension takes the form of a non-convex optimization problem. To the best of our knowledge, only sub-optimal algorithms have been proposed in the literature to compute the solution of this problem. In this paper, we prove that there exists a closed-form optimal solution to this problem and design an effective algorithm to compute it based on Singular Value Decomposition (SVD). A toy-example illustrates the gain in performance of the proposed algorithm compared to state-of-the-art techniques.