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Mathematical Analysis for Kinetic Equations and Elliptic Equations

动力学方程和椭圆方程的数学分析

基本信息

批准号：
2006731
负责人：
Ru-yu Lai
金额：
$ 21.59万
依托单位：
University of Minnesota-Twin Cities
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-08-15 至 2024-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2006731&HistoricalAwards=false
关键词：
Mathematical Analysis Kinetic Equations Elliptic

项目摘要

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.

这个项目解决了通过非破坏性方法重建未知属性的几个基本问题。这些方法允许一个恢复隐藏的参数，这通常是看不见的无损评价，从外部观察。相关的应用无处不在，从我们日常生活中的医学成像，到我们太阳系及更远的动力学研究，包括医学成像中肿瘤组织的检测，寻找材料中的裂缝和界面，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从稀带电粒子动力学研究中产生的核心数学问题，以及半导体器件的性能优化。在这个项目中开发的方法将激发非破坏性方法在科学调查中的创新应用。该项目将把研究部分与研究生的教育培训结合起来，并将特别解决代表性不足群体的参与问题。本计画将探讨动力学理论与椭圆型方程式之逆问题。主要目标将集中在与这些方程相关的基本问题和重要应用上，目的是发展数学理论，以便从给定数据中重建重要信息。具体来说，这个项目的第一部分是研究几个动力学方程的正向和反向设置与应用在等离子体物理，半导体和医学成像。主题包括在玻尔兹曼方程，模型的动力学稀带电粒子，材料参数和复杂的碰撞效应从可测量的数据的调查未知属性的识别。该项目的第二部分围绕椭圆算子的逆边值问题。我们的目标是重建未知系数的线性和非线性椭圆方程，自然产生的许多物理现象在一个有界区域的部分或全部数据的边界。该奖项反映了NSF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。