  We consider the inverse problem of recovering an isotropic electrical
conductivity from interior knowledge of the magnitude of one current density
field generated by applying current on a set of electrodes. The required
interior data can be obtained by means of MRI measurements. On the boundary we
only require knowledge of the electrodes, their impedances, and the
corresponding average input currents. From the mathematical point of view, this
practical question leads us to consider a new weighted minimum gradient problem
for functions satisfying the boundary conditions coming from the Complete
Electrode Model of Somersalo, Cheney and Isaacson. This variational problem has
non-unique solutions. The surprising discovery is that the physical data is
still sufficient to determine the geometry of the level sets of the minimizers.
In particular, we obtain an interesting phase retrieval result: knowledge of
the input current at the boundary allows determination of the full current
vector field from its magnitude. We characterize the non-uniqueness in the
variational problem. We also show that additional measurements of the voltage
potential along one curve joining the electrodes yield unique determination of
the conductivity. A nonlinear algorithm is proposed and implemented to
illustrate the theoretical results.
