  The inverse problem in Acousto-Electric tomography concerns the
reconstruction of the electric conductivity in a domain from knowledge of the
power density function in the interior of the body. This interior power density
results from currents prescribed at boundary electrodes (and can be obtained
through electro-static boundary measurements together with auxiliary acoustic
measurement. In Electrical Impedance Tomography, the complete electrode model
is known to be the most accurate model for the forward modelling. In this
paper, the reconstruction problem of Acousto-Electric tomography is posed using
the (smooth) complete electrode model, and a Levenberg-Marquardt iteration is
formulated in appropriate function spaces. This results in a system of partial
differential equations to be solved in each iteration. To increase the
computational efficiency and stability, a strategy based on both the complete
electrode model and the continuum model with Dirichlet boundary condition is
proposed. The system of equations is implemented numerically for a two
dimensional scenario and the algorithm is tested on two different numerical
phantoms, a heart and lung model and a human brain model. Several numerical
experiments are carried out confirming the feasibility, accuracy and stability
of the methods.
