Inverse Problems for the Wave Equation

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Many measurements in science, engineering, medicine, and business call for determining properties of the interior of an object without physical probing or other destructive testing. Areas in which such noninvasive testing plays a major role range from medical imaging to oil and gas prospecting. In these situations, objects are probed by noninvasive methods utilizing sound waves, electromagnetic radiation, electrical currents, or other means. The expectation is that the interior of the object will affect the waves or currents, and the body response will provide a window into its properties so that one can construct an image of the interior of the object. The goal of this research project is to study the mathematics behind such imaging techniques and advance the rigorous foundations on which this field is based. Results have the potential for important advances in imaging technology.This project will encompass a study of three inverse problems for hyperbolic partial differential equations. The first two problems are associated with a hyperbolic operator in three space (and one time) dimensions, with the operator consisting of the wave operator plus a compactly-supported potential, where the potential is supported, say, in a ball of radius one. For the inverse back-scattering problem, one considers the solution of the hyperbolic initial value problem corresponding to a point source located on the unit sphere. The goal is the recovery of the potential from the back-scattering data consisting of the value of the solution measured at the source location, for all time. To do this, a study of the injectivity of the map from the potential to the backscattering data will be performed. For the second problem, one considers the solution of the hyperbolic initial value problem with the source placed at the origin, and the goal is the determination of the potential from the trace of the solution on the unit sphere, for all time. For the third problem, one considers the solution of the standard wave equation with zero initial position function and arbitrary initial velocity function. The goal of this part of the project is to recover the velocity function from the trace of the solution on the light cone, the cone where time equals the magnitude of the position.