  We study the numerical reconstruction problem in acousto-electric tomography
of recovering the conductivity distribution in a bounded domain from interior
power density data. We propose a numerical method for recovering discontinuous
conductivity distributions, by reformulating it as an optimization problem with
$ L^1 $ fitting and total variation penalty subject to PDE constraints. We
establish continuity and differentiability results for the forward map, the
well-posedness of the optimization problem, and present an easy-to-implement
and robust numerical method based on successive linearization, smoothing and
iterative reweighing. Extensive numerical experiments are presented to
illustrate the feasibility of the proposed approach.
