The Factorization Method for Electrical Impedance Tomography in the Half-Space

We consider the inverse problem of electrical impedance tomography in a conducting half space, given electrostatic measurements on its boundary, i.e., a hyperplane. We first provide a rigorous weak analysis of the corresponding forward problem, and then develop a numerical algorithm to solve an associated inverse problem. This inverse problem consists in the reconstruction of certain inclusions within the half space, which have a different conductivity than the background. To solve the inverse problem we employ the so-called factorization method of Kirsch, which so far has only been considered for the impedance tomography problem in bounded domains. Our analysis of the forward problem makes use of a Liouville type argument which says that a harmonic function in the entire two dimensional plane must be a constant if some weighted L norm of this function is bounded.