Inverse Problems Arising from Kinetic Theory and Applications

恢复一个隐藏的原因，不能直接观察或测量，仅仅从其影响的测量是一个重要的科学问题。这一挑战出现在广泛的应用中，并且用于重建这些看不见的信息的丰富方法被用于各个领域，包括电子物理学、医学成像、生物学、太阳物理学、遥感和机器学习。该项目的主要目标是促进从可测量数据中重建隐藏属性的理解。该项目的具体目标是研究基本问题并设计重建方法，以揭示动力学方程中的未知系数，包括Boltzmann方程，它描述了二元碰撞的稀释气体动力学，以及Fokker-Planck方程，用于描述等离子体中的动力学。数学工具将被开发，以提供对这些逆问题的新兴主题的理论理解。本项目的主要目的是对非线性动力学方程及其相关的偏微分方程模型在正、逆两种框架下进行基础研究，并对非线性动力学方程及其相关的偏微分方程模型进行研究。这项工作包括三个主要目标。第一个目标集中在分析三个非线性的时间依赖方程，玻尔兹曼方程，Bhatnagar-Gross-Krook方程，和一个传输方程，检索未知的碰撞内核和系数的基础上采取的边界上的测量的目标。第二个目标的目的是唯一和稳定地恢复扩散，吸收和源系数从边界测量使用福克-普朗克方程。第三个目标是分析维格纳方程和薛定谔方程，目的是理解量子力学和经典力学之间的关系。由于各种各样的动力学方程的应用，新的想法和分析技术将被设计，以适应其独特的功能，如运输和相互作用。这项研究将提高逆动力学理论和相关主题的数学理解。该奖项反映了NSF的法定使命，并已被认为是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的支持。

Recovering a hidden cause, which cannot be directly observed or measured, solely from its effected measurements is an important scientific problem. This challenge arises in a wide range of applications, and rich methodologies for reconstructing this unseen information are employed in various fields, including geophysics, medical imaging, biology, solar physics, remote sensing and machine learning. The main goal of this project is to advance the understanding of reconstructing hidden properties from measurable data. This project specifically aims to investigate fundamental questions and devise reconstruction methods to uncover unknown coefficients in kinetic equations, including the Boltzmann equation, which describes the dynamics of dilute gases with binary collisions, and the Fokker-Planck equation, used to describe the dynamics in a plasma. Mathematical tools will be developed to provide a theoretical understanding of emerging topics on these inverse problems. Moreover, the project integrates research and education, providing training and research opportunities for graduate students and postdoctoral researchers.The main objective of this project is to conduct fundamental research on nonlinear kinetic equations and associated partial differential equation models in both the forward and inverse frameworks. The work consists of three primary goals. The first goal centers on analyzing three nonlinear time dependent equations, the Boltzmann equation, the Bhatnagar-Gross-Krook equation, and a transport equation, with the goal of retrieving the unknown collision kernel and coefficients based on measurements taken on the boundaries. The second goal aims to uniquely and stably recover diffusion, absorption, and source coefficients from boundary measurements using the Fokker-Planck equation. The third goal involves an analysis of the Wigner equation and Schrödinger equations, with the purpose of understanding the relationship between quantum and classical mechanics. Due to the wide variety of kinetic equations across applications, novel ideas and analytical techniques will be designed to suit their distinct features, such as transport and interaction. This research will improve the mathematical understanding of inverse kinetic theory and related topics.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.