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

职业：多尺度动力学系统的计算方法：不确定性、非定域性和变分公式

• 批准号: 1846854
• 负责人: Li Wang
• 金额: $ 40万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1846854&HistoricalAwards=false
• 关键词: CAREER; Computational; Methods; Multiscale; Kinetic

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.

动力学理论已经成为研究随机运动的多粒子系统的关键工具，这些系统广泛存在于等离子体物理，半导体，动物群，核工程等许多领域。它弥补了微观粒子系统和宏观连续描述之间的差距，因此是多尺度模拟的核心。除了它的多尺度性质，这个项目旨在推进动力学理论的理解和计算，在新的，新兴的方面，涉及不确定性，非局部性和变分公式。同时，本计划亦透过进阶课程、专题研讨会及暑期课程，培养各阶层学生进行跨学科研究的能力，具体目标包括：（1）利用宏观及动力学方程的变分公式，发展可量测、结构保持及数学上合理的先进最佳化方法;（2）设计非局部相互作用动力学系统的多尺度计算方法，重点研究非局部碰撞及其与分数阶扩散的联系：（3）发展求解不确定双曲型方程的鲁棒算法，特别是在处理间断解方面;（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。拟议的活动是一个跨学科的主题和普遍感兴趣的计算数学家和科学家从其他领域。变分方法为克服偏微分方程（PDE）模型的多尺度性、高维性和保持物理量的必要性等问题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，平均场游戏和机器学习。分数扩散求解器将同样适用于光子传输通过宇宙尘埃或大气，电子束剂量计算，以及其他非局部偏微分方程中产生的材料科学，金融和等离子体物理。动力学方程中无处不在的不确定性对溶液行为有着深远的影响，必须仔细量化。通过该项目研究的分析和算法，在正向和反向设置中，将促进对系统在随机扰动下的灵敏度的理解，并极大地推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，并被认为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来支持。

项目成果

A proximal-gradient algorithm for crystal surface evolution

• DOI: 10.1007/s00211-022-01320-0
• 发表时间: 2020-06
• 期刊: Numerische Mathematik
• 影响因子: 2.1
• 作者: Katy Craig;Jianguo Liu;Jianfeng Lu;J. Marzuola;Li Wang

Bayesian Instability of Optical Imaging: Ill Conditioning of Inverse Linear and Nonlinear Radiative Transfer Equation in the Fluid Regime

• DOI: 10.3390/computation10020015
• 发表时间: 2022-01
• 期刊: Comput.
• 作者: Qin Li;Kit Newton;Li Wang

Hessian Informed Mirror Descent

黑森知情镜后裔

• DOI: 10.1007/s10915-022-01933-5
• 发表时间: 2022
• 期刊: Journal of Scientific Computing
• 影响因子: 2.5
• 作者: Wang, Li;Yan, Ming
• 通讯作者: Yan, Ming

Transfer learning enhanced DeepONet for long-time prediction of evolution equation

迁移学习增强 DeepONet 用于进化方程的长时间预测

• 发表时间: 2023
• 期刊: Proceedings of the AAAI Conference on Artificial Intelligence
• 作者: Xu, Wuzhe;Lu, Yulong;Wang, Li
• 通讯作者: Wang, Li

Solving multiscale steady radiative transfer equation using neural networks with uniform stability

• DOI: 10.1007/s40687-022-00345-z
• 发表时间: 2021-10
• 期刊: Research in the Mathematical Sciences
• 影响因子: 1.2
• 作者: Yulong Lu;Li Wang;Wuzhe Xu