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.

动力学理论已成为研究具有随机运动的多个粒子系统的关键工具，这些粒子系统在等离子体物理学，半导体，动物群，核工程等方面广泛产生。它弥合了微观粒子系统和宏观连续体描述之间的缝隙，因此是多尺度建模的核心。除了其多尺寸的性质外，该项目还旨在推进对动力学理论的理解和计算，这些方面涉及不确定性，非本地性和变异表述。一个平行的教育目标是通过高级课程，主题研讨会和夏季计划为各个级别的学生准备和培训学生，以进行多学科研究。该项目的具体目的包括：（1）利用宏观和动力学方程式的变体配方，以开发可扩展的，结构保存，数学合理的方法，通过先进的优化技术来开发可扩展的结构合理方法； （2）设计非本地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连接； （3）开发具有不确定性双曲线方程的鲁棒算法，尤其是在治疗不连续的溶液时； （4）研究非线性动力学系统的反问题，包括具有不同尺度的稳定性分析，数值正则化和算法。拟议的活动涉及跨学科的主题，并且来自其他领域的计算数学家和科学家的普遍关注。各种方法为克服困难提供了一种新的观点，这些困难在当今大多数部分微分方程（PDE）模型中共享：多个尺度，高维度和保留物理量的必要性。研究结果将对其他学科产生影响，包括计算最佳运输，最佳控制理论，平均现场游戏和机器学习。分数扩散求解器将同样适用于通过宇宙灰尘或大气，电子束剂量计算以及在材料科学，金融和等离子物理学中产生的其他非局部PDE的光子传输。动力学方程中无处不在的不确定性对溶液行为具有深远的影响，必须仔细量化。在前进和反向环境中，通过该项目调查的分析和算法将促进在随机扰动下对系统敏感性的理解，并在很大程度上以最佳的性能推进了现代的设备设计。这一奖项反映了NSF的法定任务，并通过使用基金会的知识优点和广泛的影响来评估NSF的法定任务，并被认为是值得通过评估的。

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

TOWARD UNDERSTANDING THE BOUNDARY PROPAGATION SPEEDS IN TUMOR GROWTH MODELS

理解肿瘤生长模型中的边界传播速度

• DOI: 10.1137/19m1296665
• 发表时间: 2021
• 期刊: SIAM JOURNAL ON APPLIED MATHEMATICS
• 影响因子: 1.9
• 作者: Liu Jian-Guo;Tang Min;Wang Li;Zhou Zhennan
• 通讯作者: Zhou Zhennan