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

该项目有两个主要目标。第一个目标是揭示各种物理系统（由偏微分方程的边界值问题描述）对非均匀性（缺陷）的存在，这些非均匀性的间距或边界条件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，可能有助于评估这些公式的准确性。 这项工作主要（但不完全）集中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）。 将投入大量的努力，以准确描述各种频率的响应。 第二个目标是演示如何结合测量数据（例如，来自边界的可访问部分或来自“远场”）使用关于响应的性质的这种结构知识来有效地推断关于系统的参数的信息（例如，不均匀性的位置和特征）。将特别注意有效地使用广泛的频率范围，以提高重建的方法。brbr本项目研究的基础数学问题，对象的隐藏方面的无损检测。 三种特定的成像应用涉及探地雷达（例如，对于杀伤人员地雷探测）涡流成像（例如，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更精确（对噪声不敏感）和更快的检测（和定位）算法奠定基础。 该项目将开发可供从业人员使用的软件，并将向公众提供。 这项工作可能会导致一个理论框架的发展，这将是有用的，在许多其他情况下，以及。

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.