Modeling, Analysis, Optimization, Computation, and Applications of Stochastic Systems

本项目旨在研究随机干扰起重要作用的随机系统。这项研究将包括数学模型的动态行为研究，以及在生态和生物系统、无线通信、金融工程、网络系统和控制工程系统等领域的应用。研究将集中在随机影响下的模型系统、不同构型之间的切换以及复杂结构下的模型系统。结果将提供对这种建模系统的基本属性和基本特征的理解。该项目将为研究生和本科生提供培训机会。这项研究将促进多样性和包容性，提高公众科学素养，加强跨学科合作和STEM劳动力。这个项目将包括分析和计算几个重要的主题，这些主题来自网络系统、控制工程、系统优化、无线通信、生物学、生态学、经济学和社会网络等新兴和现有的应用。(1)建立了一种新的分析切换跳扩散型Kolmogorov系统的方法。研究的新特征包括由跳跃引起的非局部行为，以及使用随机切换的不确定环境建模。将处理长期存在的基本问题，例如人口动态中持续和灭绝所需的最低条件。(2)在随机逼近算法中，处理迭代中的不连续和极限中的非光滑动态是至关重要的。这个项目将从一个新的角度来关注这个问题。将首次获得随机微分包含极限，并将其用于确定收敛速度和提高渐近效率。(3)虽然非线性滤波是一个被认为是很发达的领域，但由于其无限大的维度，计算仍然是主要的挑战。该项目旨在开发一种基于机器学习和神经网络的方法，采用自适应学习率递归的新方法，从而产生潜在的更有效的计算方法。(4)数值求解非线性随机微分方程的关键是处理高非线性和数值有限时间爆炸。这个项目将开发一类算法来处理这个问题。一个新颖的想法是使用随机生成的增长截断边界。将讨论收敛性和收敛率。(5)针对网络中耦合方程处理的迫切需要，本项目将重点研究耦合开关跳扩散。利用动态系统的思想和概率论中的耦合方法，获得对网络系统有影响的稳定性和稳定化。将进行广泛的数值实验和模拟，以补充数学分析和算法设计。它将为数学的进一步研究开辟一个新的领域，具有更广泛的应用范围。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project aims to study stochastic systems in which random disturbances play a significant role. This research will encompass the study of the dynamic behavior of mathematical models and applications in areas of ecological and biological systems, wireless communication, financial engineering, networked systems, and systems in control engineering. The research will focus on model systems under the random influence, switching among different configurations, and complex structures. The results will provide an understanding of the fundamental properties and the basic features of such modeling systems. This project will provide training opportunities for graduate and undergraduate students. The research will promote diversity and inclusion, increase public scientific literacy, and enhance interdisciplinary collaborations and the STEM workforce.This project will encompass analysis and computation of several important topics from emerging and existing applications in networked systems, control engineering, optimization of systems, wireless communications, biology, ecology, economics, and social networks. (1) It aims to develop a new methodology for analyzing switching jump-diffusion type Kolmogorov systems. Novel features to be studied include non-local behavior due to the jumps, and uncertain environment modeling using random switching. Long-standing fundamental issues such as minimal conditions needed for persistence and extinction in population dynamics will be addressed. (2) Treating discontinuity in the iterates and non-smooth dynamics in the limits for stochastic approximation algorithms is vitally important. This project will focus on this issue from a new angle. Stochastic differential inclusion limits will be obtained and used to ascertain rates of convergence and to improve asymptotic efficiency for the first time. (3) Although nonlinear filtering is an area deemed to be well developed, computation remains to be the main challenge because of the infinite dimensionality. This project aims to develop a methodology based on machine learning and neural networks with a new approach using adaptive learning rate recursion, leading to potentially more efficient computational methods. (4) A key in numerically solving nonlinear stochastic differential equations is to treat high nonlinearity and numerical finite time explosion. This project will develop a class of algorithms to handle the problem. A novel idea will be the use of randomly generated growing truncation bounds. Convergence and rates of convergence will be developed. (5) In response to the urgent need to handle coupled equations in networks, this project will focus on the study of coupled switching jump diffusions. By using ideas from dynamic systems and coupling methods in probability, this project aims to obtain stability and stabilization with impact on networked systems. Extensive numerical experiments and simulations will be performed to complement the mathematical analysis and algorithm design. It will open a new domain for further research in mathematics with a broader range of applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.