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

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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.