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

运动理论已成为研究随机运动的多粒子系统的关键工具，在等离子体物理、半导体、动物群、核工程等许多领域广泛出现。它在微观粒子系统和宏观连续体描述之间架起了桥梁，因此是多尺度建模的核心。除了它的多尺度性质外，该项目还打算在涉及不确定性、非局域性和变分公式的新方面推进动力学理论的理解和计算。另一个并行的教育目标是通过高级课程、专题研讨会和暑期项目，为各个层次的学生进行多学科研究做好准备和培训。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扩展，结构保存，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，重点考虑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鲁棒算法，特别是在处理不连续解方面；(4)研究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。提议的活动是一个跨学科的主题，计算数学家和其他领域的科学家都感兴趣。变分方法为克服当前大多数偏微分方程（PDE）模型所共有的困难提供了新的视角：多尺度、高维和必须保留物理量。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。分数扩散求解器同样适用于通过宇宙尘埃或大气的光子传输，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。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，必须仔细地量化。本课题所研究的分析和算法，无论是正态还是逆态，都将有助于理解系统在随机扰动下的灵敏度，并极大地促进了性能最优器件的现代设计。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。

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.