The recovery of boundaries and boundary conditions in elliptic partial differential equations
William Rundell, Texas A&M University

Radar and sonar, satellite observations CAT scans, indeed most of medical imaging, are ubiquitous examples of the recovery of a hidden or remote object by making measurements from a distance. They are wonderful case studies of the application of mathematics. In all such problems there is a partial differential equation lurking behind the scenes. If we knew the shape, location and material properties of the object then mathematics developed in the latter half of the nineteen century and first half of the twentieth would let us predict, in theory, exactly the kind of measurements we would obtain.

The much harder, but interesting part, is the converse; given the measurements, where, and what, is the object? This talk will explore this inverse problem, specifically the issues of uniqueness and determination. What is the least amount of measurements one can make in order to obtain a unique recovery? Are there algorithms that would let us reconstruct the object and what are their limitations? And, of course, what are the pdes and what makes these problems so challenging from an analysis viewpoint?

Colloquium, Department of Mathematics, Rice University
February 7, 2008, 4:00 - 5:00 PM, Herman Brown 227

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