Levenberg-Marquardt Algorithm for Acousto-Electric Tomography based on the Complete Electrode Model

The inverse problem in acousto-electric tomography concerns the reconstruction of the electric conductivity in a body 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 it can be obtained through electro-static boundary measurements together with auxiliary acoustic probing. Previous works on acousto-electric tomography used the continuum model for the electrostatic boundary conditions; however, from Electrical Impedance Tomography, it is known that the complete electrode model is much more realistic and accurate. In this paper, the inverse problem of acousto-electric tomography is posed using the (smoothened) complete electrode model, and a reconstruction method based on the Levenberg-Marquardt iteration is formulated in appropriate function spaces. This results in a system of partial differential equations to be solved in each step. To increase the computational efficiency and stability, a strategy based on both the complete electrode model and the continuum model is proposed. The method 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 developed method.