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

电阻抗断层扫描、雷达和地震反演问题的微局域分析

• 批准号: 1906186
• 负责人: Allan Greenleaf
• 金额: $ 28.26万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-15 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906186&HistoricalAwards=false
• 关键词: Microlocal; Analysis; Inverse; Problems; Electrical

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.

该项目涉及逆问题，即利用数学模型来描述如何利用近场（物体表面）或远场（远程成像）的波测量（无论是静电波、电磁波还是声波）来无损地确定物体内部或表面的感兴趣特征或物理特性。这些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。该奖项将支持在椭圆和双曲偏微分方程建模的反问题领域的三个项目的工作。通过非破坏性成像或遥感技术，这方面的进展将有助于改进材料参数（如电导率、雷达反射系数和声学声速）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，通过高级培训增加STEM劳动力。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形状与分析（特别是微局部分析）之间的相互作用，需要改进重建。这些改进将遵循从一个更好地理解的微局部几何基础过滤反投影方法。第一个项目将扩展已经在进行的工作，使用复杂的主要类型传播的奇异性，以提高电阻抗断层成像的能力，以图像夹杂物内的夹杂物。一个可能的应用是中风诊断;另一个是检测制造零件中的缺陷。研究人员将尝试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的合成的技术，这是研究多普勒合成孔径雷达所需要的，改进和扩展了现有技术水平。最后一个项目中的线性化全波反演分析将需要分解成像算子，Littlewood-Paley类型，但适应于速度模型的射线几何形状，该奖项反映了NSF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。

项目成果

On k ‐point configuration sets with nonempty interior

内部非空的 k 点配置集

• DOI: 10.1112/mtk.12114
• 发表时间: 2022
• 期刊: Mathematika
• 影响因子: 0.8
• 作者: Greenleaf, Allan;Iosevich, Alex;Taylor, Krystal
• 通讯作者: Taylor, Krystal

Microlocal analysis of borehole seismic data

钻孔地震数据的微局域分析

• DOI: 10.3934/ipi.2022026
• 发表时间: 2022
• 期刊: Inverse Problems and Imaging
• 影响因子: 1.3
• 作者: Felea, Raluca;Gaburro, Romina;Greenleaf, Allan;Nolan, Clifford
• 通讯作者: Nolan, Clifford

Configuration Sets with Nonempty Interior

具有非空内部的配置集

• DOI: 10.1007/s12220-019-00288-y
• 发表时间: 2021
• 期刊: The Journal of Geometric Analysis
• 作者: Greenleaf, Allan;Iosevich, Alex;Taylor, Krystal
• 通讯作者: Taylor, Krystal

Existence of similar point configurations in thin subsets of $${\mathbb {R}}^d$$

$${mathbb {R}}^d$$ 的薄子集中存在相似的点配置

• DOI: 10.1007/s00209-020-02537-1
• 发表时间: 2021
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Greenleaf, Allan;Iosevich, Alex;Mkrtchyan, Sevak
• 通讯作者: Mkrtchyan, Sevak