New eigenvalue problems in inverse scattering
Date
2019
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
University of Delaware
Abstract
The ability to detect changes in the constitutive parameters of a material from
measured scattering data is an important tool in the field of nondestructive testing of
materials. Previously considered target signature candidates such as scattering resonances
and transmission eigenvalues have fallen short in this effort due to their intimate
connection with the interrogating frequency used during evaluation. The main theme
of this thesis is to develop and study new eigenvalue problems in inverse scattering
theory which rely on artificially introduced parameters and consequently provide more
practical target signatures for nondestructive testing of materials. We generate these
eigenvalue problems from a modification of the physical scattering data by subtracting
scattering data from an artificial problem depending on one or more parameters. With
the exception of our discussion of an inverse spectral theorem, we fix the interrogating
frequency and consider one of the parameters appearing in the artificial scattering
problem as the eigenparameter and target signature. We investigate three different
acoustic scattering scenarios: scattering by an inhomogeneous medium with far field
data, scattering by an anisotropic and inhomogeneous medium with near field data
measured inside a cavity in the medium, and scattering by a partially coated crack
with far field data. In each of these cases we establish that the eigenvalues are discrete
and may be computed from measured scattering data using the classic or generalized
linear sampling method. In the first scenario we also establish existence of eigenvalues
for an absorbing medium, which is made difficult by the fact that the eigenvalue problem
is not self-adjoint in this case. We present numerical examples which suggest that
the associated eigenvalues may potentially serve as effective target signatures for the
detection of changes in the relevant constitutive parameters.