Kyle Rosa

Kyle RosaKyle Rosa

The University of Western Australia

Biography

Kyle is a third year Mathematics and Statistics major at the University of Western Australia. His honours year will be specialised in finite group theory.

Inverse Scattering by Finitely Parameterised Planar Obstacles

Given a closed strictly convex plane curve C which bounds a finite number of unknown strictly convex obstacles, we aim to reconstruct the obstaccles from the travelling-time spectrum, defined as follows:A generalized geodesic is a track-sum of line segments which reflect at the obstacles according to Snell’s Law. The travelling-time spectrum is then the set of all (x,y,t) where t is the length of a generalised geodesic from x to y, with x and y laying on the curve C. The travelling-time spectrum is the subject of the recent papers [1], [2], where nonconstructive proofs are given that a finite number of strictly convex objects can be recovered from travelling-time data.Constructive proofs are known in simple cases, such as for one or two objects, but the general case is quite complicated and the subject of ongoing work by Lyle Noakes and Luchezar Stoyanov.Infinitely many parameters are needed to describe arbitrary families of convex objects. The aim of the present project is to study constructive algorithms for the more restricted case where planar obstacles are bounded by convex closed splines of low order, or possibly even by convex polyhedra. This more limited setting offers a number of attractive features not found in the general case. First, the obstacles are finitely parameterised and therefore, in principle, easier to determine. In practice there will be more equations than parameters, raising interesting questions about how to best use the additional information to approximate boundaries that are not exactly splines. So some numerical optimisation is needed, in keeping with the geometry of the inverse problem.We are also interested in the limiting process whereby arbitrary convex closed planar curves are approximated by splines with increasing numbers of polynomial segments. It would be very interesting to examine relationships between inverse problems for the finitely parameterised objects and for the general case. For instance it might be possible to say something more definitive about the internal structure of the travelling-time spectrum for finitely parameterised obstacles than is presently possible for the general case.

References:

[1] “Rigidity of Scattering Lengths and Travelling Times for Disjoint Unions of Convex Bodies,” L. Noakes and L. Stoyanov, Proc Amer Math. Soc, to appear (April 29, 2014).

[2] http://arxiv.org/pdf/1404.4147v1.pdf (in preparation)

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