Multiscale Effects and Tail Events for Infinite-Dimensional Processes and Interacting Particle Systems

概率模型通常用于表示物理、生物和金融现象，这些现象通常过于复杂，无法在计算机上解决、近似甚至模拟。应用数学和概率论面临的挑战之一是如何利用概率模型获得精确且可证明有效的方法来近似和模拟一系列复杂系统。本研究的主要目的是严谨地探讨与多尺度系统、罕见事件及相关模拟方法有关的问题。首席研究者（PI）感兴趣的是研究可能具有不同时间尺度的随机动力系统，并量化在给定时间尺度上可能罕见但对系统本身具有重要影响的相关事件。本项目中感兴趣的问题是由基础数学问题和其他科学分支中的广泛问题所激发的。例子从化学物理、流体力学、耦合化学反应与空间依赖扩散的概率估计到群体遗传学和舆论动力学。该研究项目与一个教育项目相结合，旨在帮助培养应用数学、物理、工程和化学领域的本科生和研究生，帮助他们探索罕见事件、多尺度过程及其分析和模拟。PI对大偏差区（尾部事件）和中等偏差区（典型中心和分布尾部之间的间隙）都感兴趣。由于缺乏显式解，人们不得不依靠近似和模拟方法，因此严格发展可证明有效的近似方法和模拟蒙特卡罗方法是必不可少的。在一个密切相关的方向上，PI为无限维动力系统发展了一个严谨的亚稳态理论，这些系统可能具有多个尺度，与感兴趣的罕见事件相互作用。此外，PI对与相互作用的粒子系统相关的多尺度尾事件的影响感兴趣。相互作用扩散系统出现在许多科学领域，随机矩阵理论，数学生物学，机器学习和优化中的神经网络，Kahler-Einstein度量的构建，舆论动力学，金融和工程。提出的工作导致了一个严格的数学框架的发展，允许设计与罕见事件建模相关的可证明的有效算法。它将使一些尚未被很好理解的概念和方法结晶化，例如亚稳性对蒙特卡罗方法的影响，多尺度对大偏差的影响，以及用于无限维随机动力系统和相互作用粒子系统的蒙特卡罗方法。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。

Probabilistic models are commonly used to represent physical, biological, and financial phenomena that are often too complex to solve, approximate or even simulate on computers. One of the challenges facing applied mathematics and probability is to obtain accurate and provably efficient methods to approximate and simulate a range of complex systems using probabilistic models. The primary purpose of this research is to rigorously investigate problems related to multiscale systems, rare events, and related simulation methods. The principal investigator (PI) is interested in studying stochastic dynamical systems that may have different time scales and quantifying related events that may be rare on a given time scale but can have important consequences for the system itself. The questions of interest in this project are motivated both by fundamental mathematical questions and by a broad array of questions in other branches of science. Examples range from estimation of rare event probabilities in chemical physics, hydrodynamics, coupled chemical reactions with spatially-dependent diffusion, to population genetics and opinion dynamics. This research project is integrated with an educational program that is designed to help in the training of undergraduate and graduate students in applied mathematics, physics, engineering, and chemistry in the exploration of rare events, multiscale processes and their analysis and simulation.The PI is interested in the large deviations regime (tail events) as well as in the moderate deviations regime (the gap between the typical center and the tail of the distribution). Due to the lack of explicit solutions, one has to rely on approximation and simulation methods and for this reason rigorous development of provably efficient approximation methods and simulation Monte Carlo methods is essential. In a closely related direction, the PI develops a rigorous theory of metastability for infinite dimensional dynamical systems that may have multiple scales, interacting with the rare events of interest. Moreover, the PI is interested in the effect of multiple scales on tail events associated to interacting particle systems. Systems of interacting diffusions arise in many areas of science, theory of random matrices, mathematical biology, neural networks in machine learning and optimization, construction of Kahler-Einstein metrics, opinion dynamics, finance, and engineering. The proposed work leads to the development of a rigorous mathematical framework that allows to design provably efficient algorithms associated to modeling of rare events. It will crystalize concepts and methods that are not well understood such as the effect of metastability on Monte Carlo methods and of multiple scales on large deviations and Monte Carlo methods for stochastic dynamical systems in infinite dimensions and for interacting particle systems.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.