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.

数学模型用于科学、工程和政策的各个领域，以设计系统或理解物理或社会现象。 在每一种情况下，模型误差的问题都很重要。 一般来说，由于实际原因（例如数据量有限，或需要保持计算可行性），不可能使用完全准确的模型。 因此，重要的是要确定模型的不确定性，量化它们对预测的影响，甚至在使用模型时考虑这种不确定性，例如作为工程工具。 在这个项目中感兴趣的模型是概率的。 在这种情况下，我们承认系统是随机的，模型误差是由于对描述概率分布的参数的不完全理解。 为了评估基于模型的数学预测如何随着模型本身的变化而变化，需要度量来比较基于不同分布的结果（例如，用于“设计”的分布，以及理想但不可用的“真实”分布）。 本研究的主题是发展这种度量的理论和应用。 与以前的工作相比，在这里，我们专注于感兴趣的数量与罕见事件（如灾难性的系统故障）有关的情况。 研究生参与了本项目的研究，本项目的主要主题是利用概率测度上的分歧和度量来研究模型的不确定性，以及存在模型不确定性时的优化和控制。 概率测度通常在高维或复杂空间上，并且通常在路径空间上以对随机动态进行建模。 工作的一个重要方面是建立有用的定性性质，如标度限制和链式规则型公式。 与以前的工作相比，这里的重点是在以下情况下，（a）人们希望考虑不同的模型，不是绝对连续的，和（B）的性能指标和数量的利益在很大程度上是由罕见的事件和尾部属性。 所使用的主要数学工具是凸对偶或变分公式，这些公式将发散性与指数积分联系起来。 为了实现该理论，需要评估这样的指数积分，例如，它可以采取相对于某个马尔可夫过程的平稳分布的矩生成函数的形式。 该项目还考虑了这类问题的蒙特卡罗方法的设计和分析。 研究生参与该项目的研究。该奖项反映了NSF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。

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