标题： 马尔可夫矩阵扰动以优化动态和熵泛函 显示英文标题

标题： Markov matrix perturbations to optimize dynamical and entropy functionals

摘要： 在应用动力系统中，一个重要的问题是计算引发所需可观测量最大响应的外部扰动。 为此，我们研究了马尔可夫矩阵的扰动理论，并将其与统计物理中的线性响应理论联系起来。 我们使用微扰展开来推导出优化物理相关量的线性算法，例如：马尔可夫矩阵及其相关概率向量的熵、Kullback-Liebler散度和熵产生。 这些优化算法被应用于离散和连续流动的马尔可夫链表示，包括平衡内外的情况。 我们考虑源自转移算子的Ulam型近似和基于不稳定周期轨道理论的湍流流动的降阶模型的马尔可夫矩阵表示。 我们还提出了一种数值协议，将矩阵扰动重新表述为矢量场扰动。 结果允许以数据驱动的方式物理地解释获得的优化扰动，而无需了解底层方程。 显示英文摘要

摘要： An important problem in applied dynamical systems is to compute the external forcing that provokes the largest response of a desired observable quantity. For this, we investigate the perturbation theory of Markov matrices in connection with linear response theory in statistical physics. We use perturbative expansions to derive linear algorithms to optimize physically relevant quantities such as: entropy, Kullback-Liebler-divergence and entropy production of Markov matrices and their related probability vectors. These optimization algorithms are applied to Markov chain representations of discrete and continuous flows in and out of equilibrium. We consider Markov matrix representations originating from Ulam-type approximations of transfer operators and a reduced order model of a turbulent flow based on unstable periodic orbits theory. We also propose a numerical protocol to recast matrix perturbations into vector field perturbations. The results allow to physically interpret the obtained optimizing perturbations without knowledge of the underlying equations, in a data-driven way.