Magnetic Resonance Electrical Impedance Tomography (MREIT): Convergence and Reduced Basis Approach

Abstract

This article considers the inverse problem of magnetic resonance electrical impedance tomography (MREIT) in two dimensions. A rigorous mathematical framework for this inverse problem as well as the existing Harmonic $B_z$ Algorithm as a solution algorithm are presented. The convergence theory of this algorithm is extended, such that the usage of an approximative forward solution of the underlying partial differential equation (PDE) in the algorithm is sufficient for convergence. Motivated by this result, a novel algorithm is developed, the aim of which is to speed up the existing Harmonic $B_z$ Algorithm. This is achieved by combining it with an adaptive variant of the reduced basis method, a model order reduction technique. In a numerical experiment, a high-resolution image of the Shepp--Logan phantom is reconstructed and both algorithms are compared.