CAREER: Computational Methods for Multiscale Kinetic Systems: Uncertainty, Non-Locality, and Variational Formulation

动力学理论已经成为研究具有随机运动的多粒子系统的重要工具，这种系统广泛地出现在等离子体物理、半导体、动物群、核工程等许多领域。它在微观粒子系统和宏观连续介质描述之间架起了一座桥梁，因此是多尺度建模的核心。除了它的多尺度性质，这个项目还打算在涉及不确定性、非局部性和变分公式的新的、新兴的方面促进对动力学理论的理解和计算。一个平行的教育目标是通过高级课程、专题研讨会和暑期计划为所有层次的学生进行多学科研究做准备和培训。该项目的具体目标包括：(1)利用宏观和动力学方程的变分公式，通过先进的优化技术，开发可伸缩的、结构保持的、数学上合理的方法；(2)设计非局部相互作用动力学系统的多尺度计算方法，重点是非局部碰撞和与分数扩散的联系；(3)开发具有不确定性的双曲型方程的稳健算法，特别是在处理不连续解方面；(4)研究了非线性动力学系统的反问题，包括变尺度稳定性分析、数值正则化和算法。拟议的活动涉及一个跨学科的主题，计算数学家和其他领域的科学家普遍感兴趣。变分方法为克服目前大多数偏微分方程(PDE)模型所共有的多尺度、高维和保留物理量的必要性提供了一个新的视角。研究成果将对其他学科产生影响，包括计算最优运输、最优控制理论、平均场博弈和机器学习。分数扩散解算器将同样适用于宇宙尘埃或大气中的光子传输、电子束剂量计算以及材料科学、金融和等离子体物理中出现的其他非局部偏微分方程组。动力学方程中无处不在的不确定性对解的行为有深远的影响，必须仔细量化。通过这个项目调查的分析和算法，无论是在正向和反向设置，都将有助于理解系统在随机扰动下的敏感性，并在很大程度上促进具有最佳性能的设备的现代设计。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 has emerged as a critical tool in studying many-particle systems with random motion, which arise widely in plasma physics, semiconductors, animal swarms, nuclear engineering, among many others. It bridges the gap between microscopic particle system and macroscopic continuum description, and therefore is at the core of multiscale modeling. In addition to its multiscale nature, this project intends to advance the understanding and computation of kinetic theory in new, emerging aspects that involve uncertainties, non-localities, and variational formulations. A parallel educational objective is to prepare and train students at all levels for multi-disciplinary research through advanced courses, topic seminars, and summer programs.The specific aims of the project include: (1) utilize the variational formulation of macroscopic and kinetic equations to develop scalable, structure preserving, mathematically justifiable methods via advanced optimization techniques; (2) design multiscale computational methods for nonlocal interacting kinetic systems, with emphases on nonlocal collision and connection to fractional diffusion; (3) develop robust algorithms for hyperbolic equations with uncertainty, especially in treating discontinuous solutions; (4) study the inverse problem for nonlinear kinetic systems, including stability analysis with varying scales, numerical regularization and algorithms. The proposed activity is on an interdisciplinary topic and of general interest to both computational mathematicians and scientists from other areas. The variational methods provide a new perspective in overcoming difficulties that are shared among most partial differential equation (PDE) models nowadays: multiple scales, high dimensionality and necessity in preserving physical quantities. The research outcome will have an impact on other disciplines including computational optimal transport, optimal control theory, mean field games, and machine learning. The fractional diffusion solvers will be equally applicable to photon transport through cosmic dust or atmosphere, electron beam dose calculation, and other nonlocal PDEs arising in material science, finance, and plasma physics. Uncertainties that are omnipresent in kinetic equations have a profound influence on the solution behavior and must be carefully quantified. The analysis and algorithms investigated through this project, in both forward and inverse setting, will facilitate the understanding of sensitivity in the system under random perturbations, and largely advance the modern design of device with optimal performance.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.