Methods for Analysis and Optimization of Stochastic Systems with Model Uncertainty and Related Monte Carlo Schemes

具有模型不确定性的随机系统的分析和优化方法及相关蒙特卡罗方案

• 批准号: 1904992
• 负责人: Paul Dupuis
• 金额: $ 48.29万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-09-01 至 2023-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1904992&HistoricalAwards=false
• 关键词: Methods; Analysis; Optimization; Stochastic; Systems

Mathematical models are used in every area of science, engineering, and policy to design systems or to understand physical or social phenomena. In every instance, the issue of model error is important. In general, it is not possible for practical reasons (such as limited amounts of data, or the need to maintain computational feasibility) to work with a perfectly accurate model. Hence it is important to identify those aspects of the model that are uncertain, quantify their impact on predictions, and perhaps even account for this uncertainty while using the model, for example as an engineering tool. The models of interest in this project are probabilistic. In this setting we acknowledge that the system is random, and the model error is due to an imperfect understanding of the parameters that describe the probability distribution. To assess how mathematical predictions based on the model change as the model itself changes, one needs metrics to compare the outcome based on different distributions (e.g., the distribution that is used for "design," and an ideal but not available "true" distribution). The topic of this research is the development of the theory and application of such metrics. In contrast to prior work, here we focus on situations where the quantities of interest are tied to rare events, such as a catastrophic system failure. Graduate students participate in the research of the project.The main theme of this project is the use of divergences and metrics on probability measures to study model uncertainty, and optimization and control in the presence of model uncertainty. The probability measures are typically on high-dimensional or complicated spaces, and typically on a path space to model stochastic dynamics. An important aspect of the work is to establish useful qualitative properties, such as scaling limits and chain rule-type formulas. In contrast to prior work, the focus here is on situations where (a) one wishes to consider differing models that are not absolutely continuous, and (b) performance measures and quantities of interest are largely determined by rare events and tail properties. The main mathematical tools used are convex duality or variational formulas that relate the divergences to exponential integrals. To implement the theory, one needs to evaluate such exponential integrals, which for example may take the form of a moment-generating function with respect to the stationary distribution of some Markov process. The project also considers the design and analysis of Monte Carlo methods for this class of problems. Graduate students participate in the research of the project.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.

数学模型用于设计系统或了解物理或社会现象的科学，工程和政策的每个领域。 在每种情况下，模型错误的问题都很重要。 通常，出于实际原因（例如有限的数据或维持计算可行性的需求）不可能使用完全准确的模型工作。 因此，重要的是要确定不确定的模型的那些方面，量化其对预测的影响，甚至可以在使用模型时考虑到这种不确定性，例如作为工程工具。 该项目感兴趣的模型是概率的。 在这种情况下，我们确认系统是随机的，模型误差是由于对描述概率分布的参数的不了解。 为了评估基于模型本身变化的数学预测如何根据模型的变化，需要指标来基于不同的分布（例如，用于“设计”的分布以及理想但不可用的“真实”分布）比较结果。 这项研究的主题是这种指标的理论和应用的发展。 与先前的工作相反，在这里，我们专注于与罕见事件相关的兴趣数量的情况，例如灾难性系统故障。 研究生参加了该项目的研究。该项目的主题是使用差异和指标对概率措施的使用来研究模型不确定性，以及在存在模型不确定性的情况下优化和控制。 概率度量通常位于高维或复杂的空间上，通常是在模拟随机动力学的路径空间上。 工作的一个重要方面是建立有用的定性属性，例如缩放限制和链条规则类型公式。 与先前的工作相反，这里的重点是（a）希望考虑并非绝对连续的不同模型，并且（b）绩效指标和兴趣的数量在很大程度上取决于罕见事件和尾部特性。 使用的主要数学工具是将差异与指数积分相关联的凸双重性或变分公式。 为了实施该理论，需要评估这种指数积分，例如，对于某些马尔可夫过程的固定分布，可能采用矩造函数的形式。 该项目还考虑了此类问题的蒙特卡洛方法的设计和分析。 研究生参加了该项目的研究。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评论标准来评估值得支持的。

项目成果

The Large Deviation Principle for Interacting Dynamical Systems on Random Graphs

随机图上相互作用动力系统的大偏差原理

• DOI: 10.1007/s00220-022-04312-1
• 发表时间: 2022
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Dupuis, Paul;Medvedev, Georgi S.
• 通讯作者: Medvedev, Georgi S.

Large Deviation Properties of the Empirical Measure of a Metastable Small Noise Diffusion

亚稳态小噪声扩散经验测量的大偏差特性

• DOI: 10.1007/s10959-020-01072-3
• 发表时间: 2022
• 期刊: Journal of Theoretical Probability
• 影响因子: 0.8
• 作者: Dupuis, Paul;Wu, Guo-Jhen
• 通讯作者: Wu, Guo-Jhen

Formulation and properties of a divergence used to compare probability measures without absolute continuity

用于比较没有绝对连续性的概率度量的散度的公式和性质

• DOI: 10.1051/cocv/2022002
• 发表时间: 2022
• 期刊: Optimisation and Calculus of Variations
• 作者: Dupuis, Paul;Mao, Yixiang
• 通讯作者: Mao, Yixiang

Analysis and Optimization of Certain Parallel Monte Carlo Methods in the Low Temperature Limit

• DOI: 10.1137/21m1402029
• 发表时间: 2020-11
• 期刊: Multiscale Model. Simul.
• 作者: P. Dupuis;Guo-Jhen Wu

Quasistationary distributions and ergodic control problems

准平稳分布和遍历控制问题

• DOI: 10.1016/j.spa.2021.12.004
• 发表时间: 2022
• 期刊: Stochastic Processes and their Applications
• 影响因子: 1.4
• 作者: Budhiraja, Amarjit;Dupuis, Paul;Nyquist, Pierre;Wu, Guo-Jhen
• 通讯作者: Wu, Guo-Jhen