Mathematical Analysis for Kinetic Equations and Elliptic Equations

该项目解决了通过无损方法重建未知属性的几个基本问题。这些方法允许人们从外部观察中恢复隐藏参数，这些参数通常在无损评估中是看不见的。相关的应用无处不在，从我们日常生活中的医学成像，到太阳系动力学的研究，包括医学成像中肿瘤组织的检测，寻找材料内部的裂缝和界面，以及地球和太阳内部的研究。该项目的一个主要组成部分旨在研究稀带电粒子动力学研究中出现的核心数学问题，以及半导体器件的性能优化。该项目开发的方法将激发无损方法在科学研究中的创新应用。这个项目将把研究部分同研究生的教育训练结合起来，并将特别处理代表性不足的群体的参与问题。这个项目将研究动力学理论和椭圆方程的反问题。主要目标将集中在与这些方程有关的基本问题和重要应用上，目的是发展从给定数据中重建重要信息的数学理论。具体地说，这个项目的第一部分是研究几个正向和逆向的动力学方程，这些方程在等离子体物理、半导体和医学成像中的应用。主题包括确定玻尔兹曼方程中的未知性质，该方程模拟了稀带电粒子的动力学，以及从可测量数据中研究材料参数和复杂碰撞效应。项目的第二部分围绕椭圆算子的反边值问题展开。目标是从边界上的部分或全部数据重构在有界区域的许多物理现象中自然出现的线性和非线性椭圆方程中的未知系数。特别是，研究者将研究重建过程中的独特性和稳定性问题。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。

This project addresses several fundamental questions in reconstructing unknown properties through nondestructive methods. These methods allow one to recover hidden parameters, which are usually unseen in nondestructive evaluation, from external observations. Related applications appear everywhere from the medical imaging in our daily lives, to the study of dynamics of our solar system and beyond, including the detection of tumor tissues in medical imaging, finding cracks and interfaces within materials, and the study of the Earth and solar interior. One primary component of this project aims to study central mathematical questions that arise from the investigation of the dynamics of dilute charged particles, and performance optimization for semiconductor devices. The methodologies developed in this project will excite innovative applications of the nondestructive method in scientific investigations. This project will integrate the research component with the educational training of graduate students, and will particularly address the involvement of underrepresented groups. This project will investigate inverse problems for the kinetic theory and elliptic equations. The major goal will be focused on fundamental questions and important applications related to these equations, with the aim of developing mathematical theories for the reconstruction of significant information from the given data. Specifically, the first part of this project is to study several kinetic equations in both forward and inverse settings with applications in plasma physics, semiconductor, and medical imaging. The topics include the identification of unknown properties in Boltzmann equations, which model the dynamics of dilute charged particles, and the investigation of material parameters and complex collision effects from measurable data. The second part of the project centers around the inverse boundary value problems for elliptic operators. The goal is to reconstruct unknown coefficients in linear and nonlinear elliptic equations that arise naturally in many physical phenomena in a bounded region from partial or full data on the boundary. In particular, the investigator will study uniqueness and stability issues in the reconstruction process.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.