Stekloff Eigenvalues in Inverse Scattering
Wednesday, February 15, 2017 - 9:00am - 9:50am
In many problems in nondestructive testing, imaging is not a practical method for detecting flaws. An example of this is the nondestructive testing of airplane canopies in which the absorbing anisotropic material is degraded through prolonged exposure to ultraviolet radiation. In cases such as this, the use of target signatures (if they exist!) may present an alternative to imaging. In this talk we consider the situation when small changes in the (possibly complex valued) index of refraction are to be determined from changes in the measured far field (or near field) data. This problem is studied by considering a modified far field operator F whose kernel is the difference of the measured far field pattern due to the scattering object and the far field pattern of an auxiliary scattering problem with the Stekloff boundary condition imposed on the boundary of a domain B, where B is either the support of the scattering object D or a ball containing D in its interior. It is shown that F can be used to determine the Stekloff eigenvalues ( i.e. the eigenvalues of the Neumann to Dirichlet mapping) corresponding to B where if B does not equal D the refractive index is set equal to one in B\D. Numerical examples are given showing the effectiveness of determining changes in the refractive index in this way. This is joint work with F. Cakoni, S. Meng and P. Monk and is supported in part by a grant from the Air Force Office of Scientific Research.