Mathematical Sciences: RUI Inverse Problems in Thermal Imaging

这个研究项目涉及使用热方法对材料进行无损评价时出现的反问题的研究。重点是研究使用成像某些物体的不可接近部分的技术所涉及的数学，例如，边界的不可接近部分，或物体的内部。当人们使用热方法进行腐蚀检测时，就会出现这种相反的问题。该计划包括唯一性和连续依赖性问题的研究，并探讨解对相关模型参数的依赖性的本质。分析结果为设计理论合理、计算高效可靠的反演算法奠定了基础。研究人员还考虑了其他热成像问题，包括内部裂缝、分层或空隙的位置。当对物理系统进行数学建模时，自然会出现反问题。这样的系统几乎总是具有某些不为人知的性质，人们不容易直接测量这些性质。相反，人们会收集更容易获得的信息，并使用数学分析来推断未知的数量。例如，通过测量穿过人体的x射线束的能量，可以创建人体内部的图像，这一过程被称为CAT扫描。反问题的其他重要例子是使用核磁共振成像和PET扫描的医学成像，声学超声图，以及使用电磁方法（例如涡流）对金属部件进行无损评估。这个项目涉及到逆向问题的调查，当人们使用热方法来检测材料物体（如飞机机翼和机身）内部腐蚀的存在和程度时，会出现逆向问题。＊＊＊

959623279 Caudill/Bryan This research project concerns the study of inverse problems which arise in using thermal methods for the nondestructive evaluation of materials. The focus is to investigate the mathematics involved in the use of techniques for imaging inaccessible portions of some object, e.g., an inaccessible part of the boundary, or the interior of the object. Such inverse problems arise when one uses thermal methods for corrosion detection. The program includes the study of uniqueness and continuous dependence questions, and explores the nature of the dependence of solutions on relevant model parameters. The results of this analysis serve as a foundation for designing inversion algorithms which are both theoretically sound and computationally efficient and reliable. The investigators also consider other thermal imaging problems, including the location of interior cracks, delaminations, or voids. %%% Inverse problems arise naturally when one mathematically models physical systems. Such systems almost always have certain unknown properties which one cannot easily measure directly. Instead, one gathers more readily available information and uses mathematical analysis to infer something about the unknown quantities. For example, by measuring the energy of an x-ray beam passed through a human body one can create images of the interior of the body, a process known as a CAT scan. Other important examples of inverse problems are medical imaging using MRI and PET scans, acoustic sonograms, and non-destructive evaluation of metal parts using electromagnetic methods, e.g., eddy currents. This project concerns the investigation of inverse problems which arise when one uses thermal methods to detect the presence and extent of corrosion in the interior of material objects such as aircraft wings and fuselages. ***