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Random Dynamical Systems and Limit Theorems for Optimal Tracking

随机动力系统和最优跟踪的极限定理

基本信息

批准号：
1613261
负责人：
Kevin McGoff
金额：
$ 22.5万
依托单位：
University of North Carolina at Charlotte
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-06-01 至 2020-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1613261&HistoricalAwards=false
关键词：
Random Dynamical Systems Limit Theorems

项目摘要

Dynamical systems serve as important mathematical models for a wide variety of physical phenomena, arising in such areas as weather modeling, systems biology, and statistical physics. A dynamical system consists of a state space, in which a point represents a complete description of the state of the system, and a rule governing the evolution of the system from one state to another. This project focuses on the long-term behavior of such systems from two complementary points of view. From the first point of view, the project seeks to describe the behavior of typical systems when the rules of evolution are chosen at random. Such results shed light on what properties one might expect to find in disordered systems. The second point of view, the "inverse problem," concerns the statistical problem of recovering some information from the observation of a dynamical system. While there are many examples of dynamical systems being used as mathematical models, and there is a large statistical literature regarding inference and estimation, the performance of statistical procedures when applied to data generated by nonlinear dynamical systems is poorly understood. This project focuses on characterizing when traditional statistical procedures may be effectively applied in the context of dynamical systems. Beyond the very fertile potential applications, the project will also have broader impact on training of graduate students who will acquire invaluable skills in sound probabilistic modeling and statistical inference by working on the project's research topics.Symbolic dynamical systems, which may be defined in terms of discrete constraints on the possible trajectories, serve as prototypical models of systems that evolve over time. Random ensembles of these systems may be produced by selecting the constraints at random. Ideas from both discrete probability and dynamical systems may then be used to analyze the structural properties of the resulting systems with high probability. For the inverse problem, this project seeks to evaluate the performance of several statistical inference procedures, both frequentist and Bayesian, when the models involved are dynamical systems. Fundamental questions about convergence and consistency of the procedures may be addressed using tools from ergodic theory, such as joinings and the thermodynamic formalism.

动力学系统作为各种物理现象的重要数学模型，出现在天气建模，系统生物学和统计物理学等领域。一个动态系统由一个状态空间和一个规则组成，在状态空间中，一个点代表了系统状态的完整描述，而规则控制着系统从一个状态到另一个状态的演化。该项目从两个互补的角度关注此类系统的长期行为。从第一个角度来看，该项目旨在描述随机选择进化规则时典型系统的行为。这样的结果揭示了人们可能期望在无序系统中发现的性质。第二个观点，“逆问题”，涉及从动态系统的观测中恢复一些信息的统计问题。虽然有许多动态系统被用作数学模型的例子，并且有大量关于推断和估计的统计文献，但当应用于由非线性动态系统生成的数据时，统计过程的性能知之甚少。该项目的重点是表征传统的统计程序可以有效地应用于动态系统的背景下。除了非常丰富的潜在应用外，该项目还将对研究生的培训产生更广泛的影响，这些研究生将通过研究项目的研究课题获得可靠的概率建模和统计推断方面的宝贵技能。符号动力系统可以根据可能轨迹的离散约束来定义，可以作为随时间演化的系统的原型模型。这些系统的随机系综可以通过随机选择约束来产生。从离散概率和动力系统的想法，然后可以用来分析所产生的系统的结构特性与高概率。对于逆问题，这个项目旨在评估几个统计推断程序的性能，包括频率论和贝叶斯，当涉及的模型是动态系统。关于程序的收敛性和一致性的基本问题可以使用遍历理论的工具来解决，例如连接和热力学形式主义。