Analysis, Algorithm Design, and Computation for Stochastic Systems and Optimization

新兴的应用程序的动机，这一建议包括几个研究课题的系统理论和随机优化方法。(1)它旨在开发新的随机近似算法（具有延迟，分布式处理器和代表随机环境的切换过程）。 这些算法的渐近性质和相关的极限结果将被建立。结果可以应用于共识控制问题等。(2)将研究具有随机延迟（可能是由于通信延迟）的系统的稳定性。得到了非线性系统稳定的充分条件和随机时滞泛函微分系统的判别准则。预期的结果将揭示更多的光处理系统的稳定性是延迟依赖。 (3)将获得使用常规和量化观测的系统识别的大偏差形式的误差估计。通过考虑量化方面的空间复杂性和数据窗口大小方面的时间复杂性，本研究的重点是提供一个更好的理解概率误差和资源，代表算法中的数据大小，分析中的样本大小，和通信中的信道带宽之间的基本关系。(4)为了近似扩散的第一个出口时间，将开发马尔可夫链近似方法，并获得它们的收敛速度。为了处理具有连续状态依赖切换的随机微分方程的数值解，将使用在扩大的概率空间中具有与原始系统相同分布的重新嵌入的数值解序列来确定路径收敛率。本项目旨在桥接系统理论、随机优化方法和应用。提出的研究课题包括开发迭代算法，使用并行处理器和考虑随机环境，调查系统的稳定性，包括随机延迟，获得下，不同的观测模式下的系统识别的估计误差的上下界，和设计和分析的数值解的问题，涉及某些微分方程的随机不确定性。所要研究的问题是从真实的应用中提取出来的。预计这项研究的结果将是有用的应用在网络系统，无线通信，金融工程，系统识别与定期和量化的意见，并涉及随机微分方程的某些问题的数值解。要构建的模型，系统的内在属性，以及要开发的数值方法和算法将导致潜在的转移的随机系统理论和优化方法的进步，上述应用。拟议的研究将涉及研究生的参与，它也将包括本科生的研究项目。通过将本研究与教学和学生培养相结合，计划的工作将有助于数学系统理论的进一步发展和数学教育的改进。

Motivated by emerging applications, this proposal encompasses several research topics in systems theory and stochastic optimization methods. (1) It aims to develop new stochastic approximation algorithms (with delays, distributed processors, and a switching process representing the random environment). Asymptotic properties of these algorithms and related limit results will be established. The results can be applied to consensus control problems among others. (2) Stability of systems with random delays (possibly due to communication latency) will be investigated. Sufficient conditions for stability of nonlinear systems and criteria for functional differential systems with random delays will be obtained. The expected results will shed more lights on treating stability of systems that are delay dependent. (3) Error estimates in the form of large deviations for system identification using regular and quantized observations will be obtained. By considering both space complexity in terms of quantization and time complexity with respect to data window sizes, this study focuses on providing a better understanding to the fundamental relationship between probabilistic errors and resources that represent data sizes in algorithms, sample sizes in analysis, and channel bandwidths in communications. (4) To approximate the first exit time for diffusions, Markov chain approximation methods will be developed and their rates of convergence will be obtained. To treat numerical solutions to stochastic differential equations with continuous-state-dependent switching, pathwise rates of convergence will be ascertained using a sequence of re-embedded numerical solutions having the same distribution as the original systems in an enlarged probability space.This project aims to bridge systems theory, stochastic optimization methods, and applications. The research topics proposed include developing iterative algorithms using parallel processors and taking random environment into consideration, investigating stability of systems involving random delays, obtaining lower and upper estimation error bounds for system identification under different observation patterns, and designing and analyzing numerical solutions of problem involving certain differential equations with random uncertainty. The problems to be studied have been extracted from or motivated by real applications. It is anticipated that the results of this research will be useful for applications in networked systems, wireless communication, financial engineering, system identification with regular and quantized observations, and numerical solutions of certain problems involving random differential equations. The models to be constructed, the intrinsic properties of the systems, and the numerical methods and algorithms to be developed will lead to potential transfer of advances in stochastic systems theory and optimization methods to the aforementioned applications. Theproposed research will involve participation of graduate students; it will also include undergraduate student research projects. By integrating the proposed research with teaching and student training, the planned work will contribute to the further development of mathematical systems theory and the improvement of mathematics education.