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On Robust Computation of Koopman Operator and Prediction in Random Dynamical Systems
Journal of Nonlinear Science ( IF 3 ) Pub Date : 2019-11-13 , DOI: 10.1007/s00332-019-09597-6
Subhrajit Sinha , Bowen Huang , Umesh Vaidya

In the paper, we consider the problem of robust approximation of transfer Koopman and Perron–Frobenius (P–F) operators from noisy time-series data. In most applications, the time-series data obtained from simulation or experiment are corrupted with either measurement or process noise or both. The existing results show the applicability of algorithms developed for the finite-dimensional approximation of the deterministic system to a random uncertain case. However, these results hold only in asymptotic and under the assumption of infinite data set. In practice, the data set is finite, and hence it is important to develop algorithms that explicitly account for the presence of uncertainty in data set. We propose a robust optimization-based framework for the robust approximation of the transfer operators, where the uncertainty in data set is treated as deterministic norm bounded uncertainty. The robust optimization leads to a min–max type optimization problem for the approximation of transfer operators. This robust optimization problem is shown to be equivalent to regularized least-square problem. This equivalence between robust optimization problem and regularized least-square problem allows us to comment on various interesting properties of the obtained solution using robust optimization. In particular, the robust optimization formulation captures inherent trade-offs between the quality of approximation and complexity of approximation. These trade-offs are necessary to balance for the proposed application of transfer operators, for the design of optimal predictor. Simulation results demonstrate that our proposed robust approximation algorithm performs better than some of the existing algorithms like extended dynamic mode decomposition (EDMD), subspace DMD, noise-corrected DMD, and total DMD for systems with process and measurement noise.



中文翻译:

随机动力系统中Koopman算子的鲁棒计算与预测

在本文中,我们考虑了从嘈杂的时间序列数据中转移Koopman和Perron-Frobenius(PF)算子的鲁棒逼近问题。在大多数应用中,从仿真或实验中获得的时间序列数据会因测量噪声或过程噪声或两者同时受到破坏。现有结果表明,为确定性系统的有限维近似开发的算法对随机不确定情况的适用性。但是,这些结果仅在渐近和无限数据集的假设下成立。实际上,数据集是有限的,因此开发明确考虑数据集不确定性存在的算法非常重要。我们为转移算子的鲁棒逼近提出了一个基于鲁棒优化的框架,其中数据集的不确定性被视为确定性范数有界不确定性。健壮的优化导致最小-最大类型优化问题,近似于转移算子。该鲁棒性优化问题显示为等效于正则化最小二乘问题。鲁棒优化问题和正则化最小二乘问题之间的等价关系使我们能够使用鲁棒优化对获得的解决方案的各种有趣特性进行评论。特别地,鲁棒的优化公式捕获了近似质量和近似复杂性之间的固有权衡。这些折衷对于平衡转移算子的拟议应用和最佳预测器的设计是必要的。

更新日期:2019-11-13
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