Analytical and Numerical Studies of Direct and Inverse Problems for Parabolic Initial-Boundary Value Problems

本计画主要研究抛物型偏微分方程之正问题与反问题。 PI将利用分析和数值方法来解决与域识别中的两个重要问题相关的一些具有挑战性的问题。 第一个问题涉及到检测和识别的不均匀性内，否则已知的域。 第二个问题涉及确定域边界的未知部分。 在每种情况下，以域的边界的一部分上的重叠设定的边界值的形式提供信息。 具体的注意力将集中在发展的现实模型的裂纹和夹杂物内的导热体，理论和数值问题有关的反问题的裂纹和夹杂物检测从热数据，发展稳定和可靠的算法确定一个未知部分的边界域从热数据，在研究物理对象或过程时，许多性质不能直接测量，要么是因为这样做不切实际，要么是因为会对对象造成不必要的损坏。 众所周知的例子包括检测人体内的肿瘤，检测地下深处的储油层或矿藏，以及检测物体（如机翼或机身）内的隐藏损伤。 反问题领域的重点是通过测量其他属性来确定或估计这些不可测量的属性，然后使用数学模型（通常是偏微分方程）将这些测量与所需的属性联系起来。 该项目研究使用热方法来确定物体内部的属性，而不会损坏它们。 加热外表面的一部分，然后在一段时间内测量表面上的温度。 测量的温度和物体内部结构之间的关系可以数学建模。 该项目的重点是分析这一数学模型，并开发一种实用的算法，该算法将允许从这些温度测量中提取有关物体内部的信息。 这种分析将为开发稳定可靠的算法提供理论基础，这些算法可用于各种重要应用，包括分析飞机机翼和机身内部的腐蚀损伤，检查新制造的物体的内部裂缝或空隙（质量控制）以及医疗诊断。

This project focuses on the study of direct and inverse problems for parabolic partial differential equations. The PI will utilize both analytical and numerical methods to address a number of challenging questions related to two important problems in domain identification. The first problem involves the detection and identification of an inhomogeneity inside an otherwise known domain. The second problem involves determination of an unknown portion of the boundary of a domain. In each case, information is provided in the form of overly posed boundary values on a portion of the boundary of the domain. Specific attention will be focused on the development of realistic models for cracks and inclusions inside a thermally conductive body, theoretical and numerical questions related to the inverse problem of crack and inclusion detection from thermal data, the development of stable and reliable algorithms for the determination of an unknown portion of the boundary of a domain from thermal data, and the investigation of a number of issues related to the mathematical modeling of corrosion.When studying physical objects or processes, many properties cannot be measured directly, either because it is impractical to do so, or because unwanted damage to the object would result. Well-known examples include the detection of tumors inside the human body, the detection of oil reservoirs or mineral deposits deep underground, and the detection of hidden damage inside an object such as an aircraft wing or fuselage. The field of inverse problems focuses on determining or estimating these non-measurable properties by measuring other properties, then using a mathematical model (usually a partial differential equation) to connect these measurements to the desired properties. This project investigates the use of thermal methods to determine properties of the inside of objects without damaging them. A portion of the outer surface is heated, then the temperature on the surface is measured over a period of time. The relationship between the measured temperature and the structure of the inside of the object can be modeled mathematically. This project focuses on the analysis of this mathematical model, and on the development of a practical algorithm which will permit the extraction of information about the interior of the object from these temperature measurements. This analysis will provide a theoretical foundation to develop stable and reliable algorithms for use in a wide variety of important applications, including the analysis of corrosion damage inside aircraft wings and fuselages, the inspection of newly manufactured objects for interior cracks or voids (quality control), and in medical diagnostics.