  This article considers the inverse problem of Magnet 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 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 where it is the aim 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.
