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

该项目涉及反问题，它使用数学模型来描述如何使用近场(物体表面)或远场(远程成像)中的波的测量结果来非破坏性地确定物体内部或物体上感兴趣的特征或物理特性。这样的问题在现代技术世界中无处不在；本提案中的问题出现在医学和工业成像、雷达和使用声学的勘探地震学中。该奖项将支持椭圆和双曲型偏微分方程反问题领域的三个项目的工作。这一进展将导致通过非破坏性成像或遥感改进对跳跃和其他感兴趣的材料参数奇异性的重建，如电导率、雷达反射率系数和声速。研究生将接受这种分析所需的数学技术培训，并通过高级培训增加STEM工作人员。所有三个项目的目标都是更好地了解这些逆问题背后的几何图形与改进重建所需的分析之间的相互作用，特别是微局部分析。这些改进将源于对过滤反投影法背后的微局部几何的更好理解。第一个项目将扩展已经开展的利用奇点的复主类型传播来提高电阻抗断层成像对包裹体内包裹体成像的能力的工作。一种可能的应用是中风诊断；另一种可能是检测制造部件中的缺陷。这位研究人员将努力改进他早期在二维模型上的工作，以产生更清晰的图像，并将方法扩展到具有物理意义的三维情况。第二个项目将开发控制简并傅立叶积分算子的组成的技术，这是研究多普勒合成孔径雷达所需的，完善和扩展了当前的技术水平。最后一个项目中的线性化全波反演分析将需要对Littlewood-Paley类型的成像算子进行分解，但需要适应速度模型的射线几何，从而导致改进的地震成像算法。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.