CAREER: Applicable Kinetic Computation with Boundaries and Rough Media

动力学理论以统计的方式描述大量粒子的动力学。该理论是统计力学的核心，自然地将牛顿定律描述的微观世界和纳维-斯托克斯方程描述的宏观世界联系起来。它的应用来自航空航天工程、机械工程、核工程和大气科学。在所有这些工程领域中，稀薄气体、光子、中子和许多其他类型的粒子都是通过动力学方程来建模的。该研究项目旨在以抽象和实用的方式推进动力学理论，开发广泛适用于科学和工程中核心利益系统的新理论。本课题主要研究动力学理论中的三个主要问题：1.从理论上和数值上理解粒子与边界/界面的相互作用，重点放在理解边界层行为上; 2.分析和数值放松均匀性假设的介质，与高度振荡和粗糙的介质考虑; 3.以正向和反向方式研究问题的不确定性和敏感性，即跟踪正向设置中扰动的传播，以了解相关逆问题中误差如何扩大。由于动力学理论的多尺度多物理性质，理论结果将在很大程度上通过形式水平上的渐近分析和严格水平上的正则性理论来获得。为了应对计算方面的挑战，PI打算结合偏微分方程分析给出的解结构，并探索在数据科学和压缩感知领域取得成功的随机解算器。在整个项目中，PI打算与工程师合作，以确保正在开发的技术是真正适用的。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Kinetic theory describes the dynamics of a large number of particles in a statistical manner. The theory is central to statistical mechanics and naturally connects the micro-world described by Newton's law and the macro-world described by the Navier-Stokes equations. Its applications arise from aerospace engineering, mechanical engineering, nuclear engineering, and atmospheric science. In all these engineering fields, rarified gas, photons, neutrons, and many other types of particles are modeled by kinetic equations. This research project aims to advance kinetic theory in both abstract and practical ways, developing novel theory that is widely applicable to systems of central interest in science and engineering. This research project concentrates on three main questions in kinetic theory: 1. theoretically and numerically understanding particles' interactions with boundaries/interfaces, with the focus placed on understanding the boundary layer behavior; 2. analytically and numerically relaxing homogeneity assumptions for the media, with both highly oscillatory and rough media considered; 3. studying uncertainties and sensitivities of the problem in both forward and backward manner, that is, tracing the propagation of perturbations in the forward setting to understand how error gets enlarged in the associated inverse problem. Due to the multi-scale multi-physics nature of kinetic theory, the theoretical results will largely be obtained from performing asymptotic analysis on the formal level and regularity theory on the rigorous level. To tackle the challenges in computation, the PI intends to incorporate solution structure given by partial-differential-equation analysis, and to explore randomized solvers that have seen success in data science and compressed sensing. Throughout the project, the PI intends to collaborate with engineers to ensure the techniques under development are truly applicable.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.