Inverse Problems in Partial Differential Equations and Geometry

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The primary focus of this project is on inverse problems of determining parameters of media from remote measurements, in particular those arising in Earth exploration, in cosmology and imaging of moving and changing media, in medical imaging, and the sampling theory of linear and non-linear inverse problems. The principal investigator will study the elastic earth model with coefficients jumping across smooth surfaces and propagation and mode conversion of pressure and shear waves in it. The goal is to show that one can recover those coefficients stably from travel times of seismic waves. The principal investigator will study problems of recovery of moving media and Lorentzian metrics arising in relativity from remote observations. Tomography problems arising in medical imaging will be studied as well. Finally, the principal investigator will study the problem of finding the optimal sampling rate of linear and non-linear operators with applications to Inverse Problems. The interest to those is motivated by the fact that real life measurements are discrete, they typically average over small detectors, and numerical simulations are done on discrete grids as well. The principal investigator plans to do a full analysis of propagation, reflection, transmission and mode conversion of elastic pressure and shear waves (singularities) in isotropic elasticity with variable coefficients jumping across smooth surfaces modeling the boundaries between the Mantle, etc. Some of this analysis has been done in the flat constant coefficient case. The principal investigator will study Rayleigh and Stoneley surface waves as well. Next, the principal investigator will analyze the tensor tomography problem in Lorentzian geometry and the non-linear problem of recovery of a Lorentzian metric from remote observations. This problem is harder than its Riemannian counterpart because signals moving faster than light are unrecoverable in a stable way. The tomography problems arising in medical imaging, including Compton camera imaging, will be treated as Fourier Integral Operators and its analysis would require specific tools from that calculus. Finally, the principal investigator plans to study sampling of images of linear and non-linear operators motivated by practical considerations of ability to measure discrete data only. The principal investigator will connect sampling theory with semiclassical analysis and instead of looking into the "band limit" (the support of the Fourier transform) he will relate the sampling requirements to the semi-classical wave front set.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.