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

数学模型用于科学、工程和政策的各个领域，用于设计系统或理解物理或社会现象。在每个实例中，模型误差的问题都很重要。通常，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，识别模型中不确定的方面，量化它们对预测的影响，甚至在使用模型（例如作为工程工具）时考虑这种不确定性是很重要的。本项目中感兴趣的模型是概率模型。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理解不完善。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（例如，用于“设计”的分布，以及理想但不可用的“真实”分布）的结果。本研究的主题是这些度量的理论和应用的发展。与之前的工作相反，这里我们关注的是与罕见事件相关的量，比如灾难性的系统故障。研究生参与该项目的研究。本课题的主题是利用概率度量上的散度和度量来研究模型不确定性，以及在模型不确定性存在下的优化和控制。概率测度通常在高维或复杂空间上，通常在路径空间上模拟随机动力学。工作的一个重要方面是建立有用的定性性质，如缩放极限和链式规则型公式。与之前的工作相反，这里的重点是(a)人们希望考虑不是绝对连续的不同模型，以及(b)性能度量和感兴趣的数量在很大程度上由罕见事件和尾部属性决定的情况。使用的主要数学工具是凸对偶或将散度与指数积分联系起来的变分公式。为了实现这一理论，我们需要计算这样的指数积分，例如，它可以采用关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。

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.