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

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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.