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.
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