Inverse Problems Arising from Kinetic Theory and Applications

仅从其影响的测量中恢复无法直接观察或测量的隐藏原因是一个重要的科学问题。这一挑战出现在广泛的应用中，重建这些看不见的信息的丰富方法被用于各个领域，包括地球物理、医学成像、生物学、太阳物理、遥感和机器学习。这个项目的主要目标是提高对从可测量数据中重建隐藏属性的理解。这个项目的具体目的是研究基本问题和设计重建方法，以揭示动力学方程中的未知系数，包括描述具有双碰撞的稀薄气体动力学的Boltzmann方程，以及用于描述等离子体动力学的Fokker-Planck方程。将开发数学工具，以提供对这些逆问题新出现的主题的理论理解。此外，该项目将研究和教育相结合，为研究生和博士后研究人员提供培训和研究机会。该项目的主要目标是在正反向框架下开展非线性动力学方程和相关偏微分方程模型的基础研究。这项工作包括三个主要目标。第一个目标是分析三个与时间有关的非线性方程，即Boltzmann方程、Bhatnagar-Gross-Krook方程和一个传输方程，目的是根据在边界上的测量来恢复未知的碰撞核和系数。第二个目标是利用福克-普朗克方程从边界测量中唯一和稳定地恢复扩散系数、吸收系数和源系数。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。由于各种应用中的动力学方程多种多样，新的想法和分析技术将被设计来适应它们的不同特征，如传输和相互作用。这项研究将提高对逆动力学理论和相关主题的数学理解。这一奖项反映了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。

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.