Stochastic Models with Random Times: Long-Time Behavior and Large Population Limit

具有随机时间的随机模型：长时间行为和大群体限制

基本信息

批准号：
2206038
负责人：
Wenpin Tang
金额：
$ 19.32万
依托单位：
Columbia University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-08-15 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2206038&HistoricalAwards=false
关键词：
Stochastic Models Random Times Long

项目摘要

Large stochastic systems and other interacting processes are increasingly conspicuous in scientific discoveries and decision making. Scientists in a variety of disciplines have unprecedented access to massive data which hinge on structures with complex interactions. Decision makers also need to optimize social welfare involving a large population full of randomness. Traditional models of low dimensional nature are no longer adequate for these scientific problems and as a basis for decision making. The project will address these problems by developing mathematical and computational tools for analyzing complex interacting systems that will have far-reaching public health, economic and scientific implications. The project will focus on developing several innovations in mathematical theory and algorithms, which will inform basic science and policy questions arising in diverse disciplines. The results will be disseminated broadly across diverse scientific and social communities. The project will provide training opportunities for graduate students.The project will investigate long-time behavior and large population limit of stochastic processes. The project will address three specific topics. The first topic will concern stochastic models involving hitting times to understand the long-time behavior on the mean-field limit and design an optimal strategy to control the large complex system. The main tools that the investigator plans to develop will be from probability theory and partial differential equations. The second topic will involve accelerating gradient methods for escaping from saddle points of a non-convex high-dimensional objective function. The third topic will investigate the sensitivity of some probabilistic ranking models when the number of observations is large. In both the second and third topics, the investigator plans to develop tools from probability theory and combinatorics.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.

大型随机系统和其他相互作用过程在科学发现和决策中越来越引人注目。各学科的科学家可以前所未有地获取大量数据，这些数据取决于具有复杂相互作用的结构。决策者还需要优化涉及大量充满随机性的人口的社会福利。传统的低维模型不再足以解决这些科学问题并作为决策的基础。该项目将通过开发数学和计算工具来分析复杂的相互作用系统来解决这些问题，这将对公共卫生、经济和科学产生深远的影响。该项目将重点开发数学理论和算法方面的多项创新，这将为不同学科中出现的基础科学和政策问题提供信息。研究结果将在不同的科学界和社会界广泛传播。该项目将为研究生提供培训机会。该项目将研究随机过程的长期行为和大群体极限。该项目将讨论三个具体主题。第一个主题将涉及涉及命中时间的随机模型，以了解平均场极限上的长期行为并设计控制大型复杂系统的最佳策略。研究人员计划开发的主要工具将来自概率论和偏微分方程。第二个主题将涉及逃离非凸高维目标函数鞍点的加速梯度方法。第三个主题将研究当观察数量很大时一些概率排名模型的敏感性。在第二个和第三个主题中，研究人员计划开发概率论和组合学的工具。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。