Multiscale Modeling &Simulation, Volume 19, Issue 2, Page 758-774, January 2021.
This article addresses the problem of estimating the Koopman generator of a Markov process. The direct computation of the infinitesimal generator is not easy because of the discretization of the state space, in particular because of the trade-off inherent in the choice of the best lag time to study the process. Short lag times implies a strong discretization of the state space and a consequent loss of Markovianity. Large lag times bypass events on fast timescales. We propose a method to approximate the generator with the computation of the Newton polynomial extrapolation. This technique is a multistep approach which uses as its input Koopman transfer operators evaluated for a series of lag times. Thus, the estimated infinitesimal generator combines information from different time resolutions and does not bias only fast- or slow-decaying dynamics. We show that the multi-scale Newton method can improve the estimation of the generator in comparison to the computation using finite difference or matrix logarithm methods.

中文翻译：

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用牛顿外推法估计 Koopman 生成器

多尺度建模与仿真，第 19 卷，第 2 期，第 758-774 页，2021 年 1 月。
本文解决了估计马尔可夫过程的 Koopman 生成器的问题。由于状态空间的离散化，特别是因为选择最佳滞后时间来研究过程中固有的权衡，因此无穷小生成器的直接计算并不容易。短滞后时间意味着状态空间的强烈离散化和随之而来的马尔可夫性损失。大滞后时间绕过快速时间尺度上的事件。我们提出了一种通过计算牛顿多项式外推来近似生成器的方法。该技术是一种多步方法，它使用对一系列滞后时间进行评估的 Koopman 转移算子作为其输入。因此，估计的无穷小生成器结合了来自不同时间分辨率的信息，并且不会只偏置快速或慢速衰减的动态。