New Sampling Algorithms and Inverse Spectral Methods in Scattering

无损检测在工程和医学成像中有许多应用。无损检测的创新已经产生了新的成像方法，例如电阻抗断层扫描（EIT），它可以在医疗应用中作为更昂贵的成像方式的替代品。在许多其他情况下，人们也需要使用声波或电磁波来确定物体的内部结构，而先验信息很少。该项目旨在开发新的算法，用于快速准确的重建，从而能够确定内部特征。研究人员和研究生将研究成像技术的新方法和应用，如EIT。该项目的主要成果将是用于形状恢复的新数学技术，这些技术在很大程度上独立于物体的未知物理特性。该研究有三个部分，连接到逆散射的分析和计算方面。第一个项目是开发新的理论上严格和计算简单的正则化算法的因式分解方法。该项目将侧重于电阻抗断层成像，其中理论将被扩展为一个算子映射到希尔伯特空间的对偶空间。我们的目标是开发一个新的算法形状重建的分析框架。第二个项目是将直接采样方法（DSM）的适用性扩展到近场数据，并减少所需的测量次数。这些是形状重建的简单算法，相对于数据中的噪声稳定，但需要从许多源和接收器的测量。目的是发展新的DSM的分辨率分析。新方法涉及近场数据转换为远场数据和远场算子的因式分解。 第三个项目是研究逆散射中出现的新的透射本征值问题。这些本征值可用于估计或确定散射体中是否存在缺陷。本项目前两部分研究的重建方法对应于形状重建，而这些光谱问题的研究将导致参数识别。该研究的结果预计不仅允许恢复的散射对象，但也提供有关其材料参数的信息。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

Non-destructive testing has many applications in engineering and medical imaging. Innovations in non-destructive testing have given rise to new imaging methods such as electrical impedance tomography (EIT), which can be used in medical applications as an alternative to more expensive imaging modalities. In many other instances as well, one needs to determine the interior structure of an object with little a priori information using acoustic or electromagnetic waves. This project aims to develop new algorithms for fast and accurate reconstructions that will be able to determine interior features. The investigator and graduate students will study new methods and applications for imaging techniques such as EIT. The primary result of the project will be new mathematical techniques for shape recovery that are largely independent of unknown physical properties of an object. The research has three parts that are connected to the analytical and computational aspects of inverse scattering. The first project is to develop new theoretically rigorous and computationally simple regularization algorithms for the factorization method. The project will focus on electrical impedance tomography, where the theory will be extended for an operator mapping a Hilbert space into the dual space. The goal is to develop the analytical framework for a new algorithm for shape reconstruction. The second project is to extend the applicability of the direct sampling method (DSM) to near-field data as well as reduce the number of measurements needed. These are simple algorithms for shape reconstruction stable with respect to noise in the data but requiring measurements from many sources and receivers. The aim is to develop the resolution analysis for new DSMs. The new methods involve transforming near-field data into far-field data and applying factorization of the far-field operator. The third project is to study new transmission eigenvalue problems arising in inverse scattering. These eigenvalues can be used to estimate or determine if there are defects in a scatterer. The reconstruction methods studied in the first two parts of the project correspond to shape reconstruction, whereas the study of these spectral problems will lead to parameter identification. Results of the research are expected not only to allow recovery of a scattering object but also to provide information about its material parameters.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.