Reconstruction of an unknown boundary portion from Cauchy data in n dimensions

Abstract

We consider the inverse problem of determining the shape of some inaccessible portion of the boundary of a region in n dimensions from Cauchy data for the heat equation on an accessible portion of the boundary. The inverse problem is quite ill-posed and nonlinear. We develop a Newton-like algorithm for solving the problem, with a simple and efficient means for computing the required derivatives, develop methods for regularizing the process, and provide computational examples.