Microlocal Analysis of Inverse Problems in Electrical Impedance Tomography, Radar, and Seismics

该项目涉及反问题，使用数学模型来描述如何在近场（物体表面）或远场（远程成像）测量静电、电磁或声学波，以无损地确定物体内部或物体上感兴趣的特征或物理特性。这些问题在现代技术世界中无处不在；在这个建议发生在医学和工业成像，雷达和勘探地震学使用声学。该奖项将支持由椭圆型和双曲型偏微分方程建模的逆问题领域的三个项目。通过非破坏性成像或遥感技术的进步，将改进对材料参数（如电导率、雷达反射率系数和声速）中的跳跃和其他奇异点的重建。研究生将接受这种分析所需的数学技术培训，为STEM劳动力提供高级培训。这三个项目的目标是更好地理解这些逆问题背后的几何结构与分析之间的相互作用，特别是改进重建所需的微局部分析。这些改进将从更好地理解微局部几何底层滤波反投影方法。第一个项目将扩展已经在进行的工作，即使用复杂主型奇点传播来提高电阻抗断层扫描成像内含物中的内含物的能力。一个可能的应用是中风诊断；另一个是检测制造零件的缺陷。研究者将尝试完善他在二维模型上的早期工作，以产生更清晰的图像，并将方法扩展到物理上重要的三维情况。第二个项目将开发控制退化傅立叶积分算子组成的技术，这是研究多普勒合成孔径雷达所需要的，改进和扩展目前的技术水平。最后一个项目的线性化全波反演分析将需要分解成像算子，Littlewood-Paley类型，但适应速度模型的射线几何形状，从而改进地震成像算法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project concerns inverse problems, which use mathematical models to describe how measurements of waves, whether electrostatic, electromagnetic or acoustic, in the near field (at the surface of an object) or in the far field (imaging remotely) can be used to nondestructively determine features or physical properties of interest inside or on the object. Such problems are ubiquitous in the modern technological world; the ones in this proposal occur in medical and industrial imaging, radar, and exploration seismology using acoustics. This award will support work on three projects in the area of inverse problems modeled by elliptic and hyperbolic partial differential equations. Progress will lead to improved reconstruction, via nondestructive imaging or remote sensing, of jumps and other singularities in material parameters of interest, such as electrical conductivity, radar reflectivity coefficients, and acoustic sound speeds. Graduate students will be trained in the mathematical techniques needed for this analysis, adding to the STEM workforce with advanced training.The goal of all three projects is to better understand the interaction between the geometry underlying these inverse problems and the analysis, particularly microlocal analysis, needed to improve reconstructions. These improvements will follow from a better understanding of the microlocal geometry underlying filtered back-projection methods. The first project will extend work already under way on using complex principal type propagation of singularities to improve the ability of Electrical Impedance Tomography to image inclusions within inclusions. One possible application is to stroke diagnosis; another is detecting defects in manufactured parts. The investigator will both try to refine his earlier work on a two-dimensional model to produce sharper images and extend the approach to the physically significant three-dimensional situation. The second project will develop techniques for controlling the composition of degenerate Fourier integral operators, needed in the study of Doppler Synthetic Aperture Radar, refining and extending the current state of the art. The analysis of linearized full wave inversion in the last project will require decompositions of imaging operators, of Littlewood-Paley type but adapted to the ray geometry of the velocity model, leading to improved seismic imaging algorithms.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.