Inverse Problems for Anisotropic Scatterers with Conductive Boundary Conditions

In this thesis, we consider the inverse problem of detecting the shape, size, and some information about the material properties of an anisotropic object with a conductive boundary condition. First, we consider the transmission eigenvalue problem for this type of material, where the conductive boundary condition is modeled by a Robin condition that states the normal/co-normal derivative of the total field has a jump across the boundary of the scatterer that is proportional to the total field. Transmission eigenvalues are connected to non-scattering frequencies and are a relevant area of research since they can be used to retrieve information about the material properties of the scattering object. We prove that these transmission eigenvalues exist for this material and are at most a discrete set. We also study the dependence of the transmission eigenvalues on the physical parameters of the material, we show that the first transmission eigenvalue is monotonic with respect to the physical parameters. We also study the limiting behavior of the transmission eigenvalues as the conductivity parameter vanishes or goes to infinity in magnitude. We then provide some numerical examples of the dependence of the transmission eigenvalues on the physical parameters to validate our theoretical results.

Developing non-destructive testing in complex media continues to be a popular and growing field of study. This type of testing has applications in medical imaging, structural mechanics, and other engineering disciplines. Because this field continues to grow, there is a need for the development of new mathematical theories and numerical techniques for inverse problems in partial differential equations. We next study the inverse shape problem of recovering the same anisotropic scatterer as well as the inverse boundary parameter problem given the far--field data. We develop the monotonicity method for recovering an anisotropic scatterer with a conductive boundary. This is a qualitative method that uses the known/measured far--field data to derive an imaging function that relies on a symmetric factorization of the far--field operator. First, we show that the far--field data uniquely determines the boundary coefficient and then we turn our attention to the inverse shape problem. Our main contribution will be to study the application of the monotonicity method for this anisotropic material on an unbounded domain. This problem has not been studied before and will require original analysis to complete. We also provide numerical examples of the application of this method to circular domains with the physical parameters being constant.

Finally, we consider the inverse shape problem for a new anisotropic material and develop two direct sampling methods to reconstruct our scatterer. Here, the conductive boundary condition is modeled by a Robin condition that states the total field has a jump across the boundary of the scatterer that is proportional to the co-normal derivative of the total field. We will assume that the corresponding far–field pattern and Cauchy data are known/measured and we will develop two qualitative methods (called Direct Sampling Methods) to recover the scatterer. These direct sampling methods (DSMs) have been applied to inverse problems for other materials, but this is the first time they will be applied to an anisotropic scatterer with a conductive boundary condition. One direct sampling method allows us to derive an imaging function using the far--field data; to prove it is viable, we need to study the operators in the symmetric factorization of the far--field operator. The other direct sampling method allows us to reconstruct via the Cauchy data by considering the reciprocity gap integral on the boundary of a region that completely encloses the scatterer. We then provide numerical examples of each direct sampling method for circular and non-circular domains with constant physical parameters.