Department of Aeronautics and Astronautics, The University of Tokyo, Bunkyo, Tokyo 113-8656, Japan.
The Institute of Scientific and Industrial Research, Osaka University, Ibaraki, Osaka 567-0047, Japan.
Phys Rev E. 2017 Sep;96(3-1):033310. doi: 10.1103/PhysRevE.96.033310. Epub 2017 Sep 18.
The analysis of nonlinear dynamical systems based on the Koopman operator is attracting attention in various applications. Dynamic mode decomposition (DMD) is a data-driven algorithm for Koopman spectral analysis, and several variants with a wide range of applications have been proposed. However, popular implementations of DMD suffer from observation noise on random dynamical systems and generate inaccurate estimation of the spectra of the stochastic Koopman operator. In this paper, we propose subspace DMD as an algorithm for the Koopman analysis of random dynamical systems with observation noise. Subspace DMD first computes the orthogonal projection of future snapshots to the space of past snapshots and then estimates the spectra of a linear model, and its output converges to the spectra of the stochastic Koopman operator under standard assumptions. We investigate the empirical performance of subspace DMD with several dynamical systems and show its utility for the Koopman analysis of random dynamical systems.
基于 Koopman 算子的非线性动力系统分析在各个应用领域引起了关注。动态模态分解 (DMD) 是一种用于 Koopman 谱分析的数据驱动算法，已经提出了几种变体，具有广泛的应用。然而，DMD 的流行实现对随机动力系统的观测噪声敏感，并对随机 Koopman 算子的谱进行不准确的估计。在本文中，我们提出了子空间 DMD，作为一种用于具有观测噪声的随机动力系统的 Koopman 分析的算法。子空间 DMD 首先计算未来快照到过去快照空间的正交投影，然后估计线性模型的谱，并且在标准假设下，其输出收敛到随机 Koopman 算子的谱。我们研究了子空间 DMD 在几个动力系统中的经验性能，并展示了其在随机动力系统的 Koopman 分析中的实用性。
