Analytical and computational studies of direct and inverse boundary value problems for PDEs

偏微分方程正向和逆边值问题的分析和计算研究

• 批准号: 0307119
• 负责人: Michael Vogelius
• 金额: $ 22.47万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0307119&HistoricalAwards=false
• 关键词: Analytical; computational; studies; direct; inverse

Abstract, Award DMS-0307119Michael Vogelius, Rutgers UniversityTitle: Analytical and computational studies of direct and inverse boundary value problems for PDEsProject Abstract:A main object of this project is to delvelop very detailed information about the response of a system (a PDE boundary value problem) to the presence of internal inhomogeneities (defects), the spacing of these inhomogeneities, or changes in the boundary conditions. One important set of tools to obtainthis information are rigorously verfied asymptotic expansions. In the case of changing, nonlinear boundary conditions another crucial component is the study of mechanisms that govern the development of (near-)singularities. The second main theme is then the application of this information to effectively obtain knowledge about (reconstruct) the parameters of the system (e.g., the location, and the character of the inhomogeneities) from data measured on an accessible part of boundary, or in the ``farfield''. Initially the focus will be on steady state and time harmonic situations, with consideration of scalar equations as well as systems (in particular Maxwell's Equations). Later on fully time-dependent cases will also be studied. The work has an essential computational component. The particular applications that will be investigated concern 1) anti-personnel mine detection, using Ground Penetrating Radar(GPR) data, and 2) corrosion/oxidation imaging (and control) using partial voltage and current data. But many other applications of significant interest come to mind, for instance in the field of medical tomography. It is expected that the combination of analysis, asymptotic analysis and computational work contained in this project will allow for the design of detection (and location) algorithms that are more precise (less sensitive to noise), and much faster than those of a general purpose nature. For reconstructions it is envisaged to work on experimental as well as synthetic data.I intend to organize two small workshops in the second and third year (of the period covered by this grant). The first will be devoted to mathematicaland computational aspects of corrosion/oxidation modeling. The second will be devoted to GPR mine-detection. The participants will be mathematicians and engineers.

摘要：本项目的一个主要目标是开发关于系统（PDE边值问题）对内部非均匀性（缺陷）、这些非均匀性的间距或边界条件变化的响应的非常详细的信息。一组重要的工具来获得这一信息是严格验证渐近展开。在变化的情况下，非线性边界条件的另一个重要组成部分是控制（近）奇点发展的机制的研究。第二个主题是应用这些信息，从边界可达部分或“远场”测量的数据中有效地获取（重建）系统参数的知识（例如，位置和非均匀性的特征）。最初的重点将放在稳态和时间谐波的情况下，考虑标量方程以及系统（特别是麦克斯韦方程）。稍后还将研究完全与时间有关的情况。这项工作有一个基本的计算组成部分。将研究的具体应用包括：1)使用探地雷达（GPR）数据进行杀伤地雷探测；2)使用部分电压和电流数据进行腐蚀/氧化成像（和控制）。但是，我想到了许多其他重要的应用，例如在医学断层扫描领域。预计本项目中包含的分析，渐近分析和计算工作的结合将允许设计更精确（对噪声不太敏感）的检测（和定位）算法，并且比一般用途的算法更快。对于重建，设想在实验和合成数据上工作。我打算在第二年和第三年（资助期内）组织两次小型讲习班。第一个将致力于腐蚀/氧化建模的数学和计算方面。第二个将用于探地雷达地雷探测。参与者将是数学家和工程师。