CAREER: Applicable Kinetic Computation with Boundaries and Rough Media

动力学理论以统计的方式描述了大量粒子的动力学。这一理论是统计力学的核心，并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在一起。它的应用领域包括航天工程、机械工程、核工程和大气科学。在所有这些工程领域，稀有气体、光子、中子和许多其他类型的粒子都是用动力学方程来模拟的。这项研究项目的目的是在抽象和实际两方面推进动力学理论，开发出广泛适用于科学和工程中的核心兴趣系统的新理论。本研究集中于运动论中的三个主要问题：1.从理论和数值上了解粒子与边界/界面的相互作用，重点是了解边界层行为；2.在解析和数值上放宽介质的均匀性假设，同时考虑高度振荡和粗糙介质；3.研究问题的不确定性和敏感性，包括正向和反向两种方式，即追踪扰动在正向背景下的传播，以了解在相关反问题中误差是如何放大的。由于动力学理论的多尺度、多物理性质，理论结果在很大程度上是通过在形式水平上进行渐近分析和在严格水平上进行正则性理论来获得的。为了解决计算中的挑战，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传感方面取得成功的随机求解器。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真正适用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.