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

摘要，奖DMS-0307119迈克尔Vogelius，罗格斯大学标题：分析和计算研究的直接和逆边值问题的偏微分方程项目摘要：这个项目的主要目的是delvelop非常详细的信息，系统（偏微分方程边值问题）的响应存在的内部不均匀性（缺陷），这些不均匀性的间距，或边界条件的变化。一套重要的工具来获得这些信息是严格验证的渐近展开。在不断变化的情况下，非线性边界条件的另一个重要组成部分是管理（近）奇点的发展机制的研究。然后，第二个主题是应用这些信息来有效地获得关于（重建）系统参数的知识（例如，位置和不均匀性的特征）从在边界的可接近部分上或在“远场”中测量的数据。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）。稍后还将研究完全时间依赖的情况。这项工作有一个重要的计算组成部分。将研究的特定应用涉及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）。但是，许多其他重要的应用也会浮现在脑海中，例如在医学断层扫描领域。预计该项目中包含的分析、渐近分析和计算工作的组合将允许设计更精确（对噪声不敏感）并且比通用性质的检测（和定位）算法快得多的检测（和定位）算法。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组织两个小型研讨会（该补助金所涵盖的时期）。第一个将致力于腐蚀/氧化建模的计算方面。第二个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.