Analytical and Computational Studies of Direct and Inverse Boundary Value Problems for PDEs

该项目有两个主要目标。第一个目标是揭示关于各种物理系统（由偏微分方程的边值问题描述）对不均匀性（缺陷）、这些不均匀性的间距或边界条件中的参数变化的响应的具体信息。重点放在发展非常精确的渐近公式和先验估计，这可能有助于评估这些公式的准确性。这项工作主要（但不是唯一）集中在稳态和时间谐波情况下，考虑标量方程以及系统（特别是麦克斯韦方程）。在广泛的频率范围内，将投入大量的精力来准确地表征响应。第二个目标是演示如何使用这种关于响应性质的结构知识，结合测量数据（例如，来自边界的可访问部分，或来自“远场”）来有效地推断有关系统参数的信息（例如，位置和非均匀性的特征）。将特别注意有效地利用宽频率范围来改善重建的方法。这个项目研究了无损检测隐藏物体背后的基本数学问题。三种特殊的成像应用涉及探地雷达（例如，用于杀伤地雷探测）涡流成像（例如，用于地雷探测）。用于腐蚀评估)和最佳成像（用于生物医学应用）。这项工作将为设计比目前使用的检测（和定位）算法更精确（对噪声不那么敏感）和更快的算法奠定基础。该项目将开发可供从业人员使用的软件，这些软件将公开提供。这项工作可能会导致理论框架的发展，这也将在许多其他情况下有用。

The project has two major goals. The first goal is to uncover specific information about the response of various physical systems (described by boundary value problems for partial differential equations) to the presence of inhomogeneities (defects), the spacing of these inhomogeneities, or parametric changes in the boundary conditions. Emphasis is placed on developing very precise asymptotic formulas and a priori estimates that may help to assess the accuracy of these formulas. The work largely (but not exclusively) focuses on steady state and time harmonic situations, considering scalar equations as well as systems (in particular Maxwell's equations). Significant effort will be invested in accurate characterization of the response for a wide range of frequencies. The second goal is to demonstrate how to use such structural knowledge about the nature of the response, in combination with measured data (say, from an accessible part of boundary, or from the "far field") to effectively infer information about the parameters of the system (e.g., the location and the character of the inhomogeneities). Special attention will be given to methods that effectively use a wide range of frequencies to improve reconstructions.brbrThis project investigates fundamental mathematical problems that underlie nondestructive inspection of hidden aspects of objects. Three particular imaging applications concern ground penetrating radar (e.g., for anti-personnel mine detection) eddy current imaging (e.g.. for corrosion assessment), and optimal imaging (for biomedical applications). This work will form a basis for the design of detection (and location) algorithms that are more precise (less sensitive to noise) and much faster than those currently in use. The project will develop software of use to practitioners that will be made publicly available. The work may lead to the development of a theoretical framework that will be useful in many other contexts as well.