Research

Background

I am currently interested in formally determined inverse problems for hyperbolic PDE, the applications of which lie in disciplines such as geophysics and medical imaging.

Consider the following experiment of searching for oil underneath the surface: We set off explosions at points around the surface of some chunk of the Earth. We then record the pressure variation of the sound waves that reflect back to the surface. The question is, can we recover the properties of the material underneath the surface (i.e. location of oil) given the data we record on the surface?

I study mathematical problems that help us better understand questions like this.

An Open Problem

Here is a type of problem often referred to as a "backscattering problem". Let 'a' be any point on the unit sphere in three dimensions, q(x) a smooth function supported in the open unit ball (away from the boundary), and U(x,t) the solution of the initial value problem:

The inverse problem is as follows: Given data U(a,t) on the cylinder |a| = 1 & 0 < t < T, can we uniquely recover the coefficient q(x)? In other words is the map F: q(x) -> U(a,t) injective? Can we find a method to reconstruct q(x) given U(a,t)? Is the reconstruction stable under small perturbations? Rakesh & Uhlmann proved this to be injective for a certain class of functions q, but problem in general is still open.

Much of my Ph.D. thesis concerns this problem and similar ones though I do not attempt to solve the open problem.