A hybrid finite element/finite difference method for reconstruction of dielectric properties of conductive objects

The aim of this article is to present a hybrid finite element/finite difference method which is used for reconstructions of electromagnetic properties within a realistic breast phantom. This is done by studying the mentioned properties' (electric permittivity and conductivity in this case) representing coefficients in a constellation of Maxwell's equations. This information is valuable since these coefficient can reveal types of tissues within the breast, and in applications could be used to detect shapes and locations of tumours. Because of the ill-posed nature of this coefficient inverse problem, we approach it as an optimization problem by introducing the corresponding Tikhonov functional and in turn Lagrangian. These are then minimized by utilizing an interplay between finite element and finite difference methods for solutions of direct and adjoint problems, and thereafter by applying a conjugate gradient method to an adaptively refined mesh.