DMS-EPSRC Collaborative Research: Sharp Large Deviation Estimates of Fluctuations in Stochastic Hydrodynamic Systems

极端事件的影响可能非常大。它们通常是罕见的，如果它们的后果对社会是负面的，这是幸运的，但也使它们难以预测。这个项目的重点是开发计算工具，可以应用于了解极端事件如何在复杂的随机系统中发生。例如，用于预测极端天气事件的模型，如热带风暴和洪水，以及海洋石油泄漏时污染物扩散的模型。这些工具将使研究人员能够提出超出目前可能的问题。这将导致对现有预测模型的革命性改进，这对于有效管理自然灾害和人为灾害至关重要。进一步的应用包括在行为类似于流体的随机模型中描述极端事件，例如在流行病、交通和恒星形成的背景下。这个合作项目每年将资助一名纽约大学的研究生。在受控（数值或物理）实验中，即使是低维系统，也很难观察到罕见事件。难度随着自由度的增加而增加，这使得高维系统更难分析——用随机流体动力学模型描述的流体就是一个特别有趣的例子。因此，为了深入了解这些系统中的极端事件，研究人员可以提出的问题往往是有限的。这个项目的目标是通过开发新的数学和计算工具来分析复杂系统中罕见但重要的事件，以建立它们最可能的发生方式，并计算它们的概率和递归时间的尖锐渐近估计（包括前因子）。其目的是创建一个工具箱，适用于具有大量自由度的模型，这些模型由随机偏微分方程（PDEs）描述，如平流扩散方程和纳维-斯托克斯方程，并可跨越学科边界转移。这些工具将被应用于随机流体动力系统，以便对经典湍流有更深入的了解。此外，这种新方法的效率将在现实应用的背景下得到证明，特别是污染物的平流和船舶倾覆。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。

Extreme events can be highly impactful. They are typically rare, which is fortunate if their consequences are negative on society, but also makes them difficult to predict. The focus of this project is to develop computational tools that can be applied to gain understanding of how extreme events occur in complex stochastic systems. Examples are models for the forecasting of extreme weather-related events like tropical storms and flooding as well as the spread of pollutants in case of ocean oil spills. These tools will enable researchers to ask questions beyond what is currently possible. This will lead to transformative improvement of current predictive models, which is essential for efficient management of natural and man-made disasters. Further applications include the characterization of extreme events in stochastic models that behave similar to fluids, for example in the context of epidemics, traffic, and star formation. This collaborative project will support one graduate student per year at NYU.Rare events are difficult to observe in controlled (numerical or physical) experiments, even for low-dimensional systems. The difficulty increases with the number of degrees of freedom, which makes high-dimensional systems even harder to analyze — fluids described by stochastic hydrodynamic models are a particular example of interest. As a result the questions that researchers can ask in order to gain insights about extreme events in these systems are often limited. The goal of this project is to analyze rare but important events in complex systems by developing new mathematical and computational tools to establish their most likely way of occurrence and calculate sharp asymptotic estimates (with prefactor included) of their probability and recurrence time. The aim is to create a toolbox applicable to a wide range of models with a large number of degrees of freedom described by stochastic partial differential equations (PDEs), like advection-diffusion equations and Navier-Stokes equations, and transferable across disciplinary borders. These tools will be applied to stochastic hydrodynamic systems in order to gain deeper insights of classical turbulence. In addition, the efficiency of this novel approach will be demonstrated in the context of real-world applications, in particular the advection of pollutants and the capsizing of ships.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.