最优低秩动态模式分解
Optimal Low-Rank Dynamic Mode Decomposition
摘要：
动态模式分解（DMD）已成为一个强大的工具，从实验数据分析非线性系统的动力学。最近，一些尝试已经将DMD扩展到低秩近似的上下文中。这种扩展是特别感兴趣的降阶建模在各种应用领域，例如气候预测，研究分子动力学或微机电设备。这种低秩扩展采用非凸优化问题的形式。据我们所知，只有次优算法已被提出在文献中计算这个问题的解决方案。在本文中，我们证明了存在一个封闭形式的最优解的问题，并设计了一个有效的算法来计算它的基础上奇异值分解（SVD）。一个玩具的例子说明了所提出的算法相比，国家的最先进的技术的性能增益。
英文摘要：
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.
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