Determining the Profile of a Nonmonotonic Hill

摘要

In a well-known generalization of the tautochrone problem, the shape of a monotone hill is determined by observing the travel time of a particle on the surface of the hill. In this paper we present a technique for uniquely determining the shape of any smooth hill, whether monotone or not, by observing collisions of particles on the hill. It is shown that this inverse problem is well posed. Error bounds for the solution are given for the case in which there are incomplete data. The efficiency of these bounds is shown by means of an example.