Multilinear Operators and Microlocal Analysis of Electrical Impedance Tomography, Radar, and Seismology

该研究项目使用数学模型来描述如何测量物体表面的静电波或声波，或远程进行的雷达观测，以确定物体的物理特性。这种非破坏性成像在现代世界中很普遍，对更高精度和更广泛适用性的需求推动了技术的不断改进。本项目将研究的模式出现在医疗和工业成像、遥感环境和勘探地震学中。一个是与行程诊断或检测制造零件中的缺陷相关的。另一个将支持低功率雷达成像。第三个将更好地证明从勘探地震数据中提取信息的算法。通过进一步改进和完善这些成像方法，并提供所需技术方面的培训，该项目将有助于发展非侵入性成像和遥感技术，并加强科学工作队伍。前三个项目是从偏微分方程的反问题，特别是在（A）电阻抗层析成像，（B）多普勒合成孔径雷达，和（C）线性地震成像。将开发技术来检测可能存在于成像的物理参数中的跳跃和其他尖锐特征。第四个项目将在（D）谐波和微局部分析，其中涉及类似于前三个项目中遇到的多线性算子。项目D的工作将把点云的大小与云中某些点配置集的大小联系起来。项目A将使用复杂主型算子的奇点传播来改进从电压和电流测量中获得的图像。在项目B中，研究人员将改进早期的模型，例如通过加入双重反射来改善墙壁和角落的成像。项目C中线性化全波反演的合理性将提高声学、地震成像算法准确性的置信度，并可能带来其他改进。项目D将开发新的方法来显示配置集有非空的内部。用于反演和配置问题的多线性算子具有很强的相似性，项目D的进展将应用于项目A、B和C。该奖项反映了NSF的法定使命，并被认为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来支持。

This research project uses mathematical models to describe how measurements of waves, either electrostatic or acoustic at the surface of an object, or radar observations made remotely, can be used to determine physical properties of the object. Such non-destructive imaging is widespread in the modern world, and the need for higher precision and broader applicability drives continuous improvement in the technology. The modalities to be studied in this project occur in medical and industrial imaging, remote environmental sensing, and exploration seismology. One is relevant for stroke diagnosis or detecting defects in manufactured parts. Another would support low-power radar imaging. A third will better justify algorithms for extracting information from exploration seismology data. By further improving and refining these imaging methods and providing training in the techniques needed, the project will contribute to the development of noninvasive imaging and remote sensing and strengthen the scientific workforce.The investigator will work on four projects. The first three projects are from inverse problems for partial differential equations, specifically in (A) electrical impedance tomography, (B) Doppler synthetic aperture radar, and (C) linearized seismic imaging. Techniques will be developed to detect jumps and other sharp features that might be present in the physical parameters being imaged. The fourth project will be in (D) harmonic and microlocal analysis, which involves multilinear operators similar to those encountered in the first three projects. Work on Project D will relate the size of a point cloud to the size of certain sets of point configurations in the cloud. Project A will use propagation of singularities for complex principal type operators to improve images obtained from electrical voltage and current measurements. In Project B, the investigator will refine an earlier model, for example by incorporating double reflections to improve imaging of walls and corners. The justification of linearized full wave inversion in Project C will lead to improved confidence in the accuracy of acoustic, seismic imaging algorithms, and possibly other improvements. Project D will develop new methods for showing that configuration sets have nonempty interior. Multilinear operators used in inverse and those in configuration problems have strong similarities, and progress on Project D will be applied to Projects A, B, and C.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.