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

罗格斯大学Michael Vogelius奖题目：PDES正反边值问题的分析和计算研究项目摘要：本项目的一个主要目的是深入研究系统(PDE边值问题)对存在内部不均匀(缺陷)、这些不均匀的间距或边界条件变化的响应的非常详细的信息。获得这些信息的一个重要工具是经过严格验证的渐近展开。在不断变化的非线性边界条件的情况下，另一个关键的组成部分是研究支配(近)奇点发展的机制。第二个主要主题是应用这一信息，从在边界的可接近部分或在“远地”测量的数据中有效地获得关于(重建)系统参数(例如，不均匀的位置和特征)的知识。最初的重点将放在稳态和时间调和的情况上，考虑标量方程和系统(特别是麦克斯韦方程)。稍后还将研究完全依赖于时间的情况。这项工作有一个重要的计算部分。将调查的具体应用涉及1)使用探地雷达(GPR)数据进行杀伤人员地雷探测，以及2)使用部分电压和电流数据进行腐蚀/氧化成像(和控制)。但人们想到了许多其他令人感兴趣的应用，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使检测(和定位)算法的设计更精确(对噪声不那么敏感)，并且比一般用途的算法快得多。我打算在第二年和第三年(这笔赠款所涉期间)组织两个小讲习班。第一部分将致力于腐蚀/氧化模型的数学和计算方面。第二个项目将专门用于探地雷达探雷。参与者将是数学家和工程师。

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.