# The Levenberg-Marquardt Method for Acousto-Electric Tomography on Different Conductivity Contrast
## 1 Introduction  
Electrical Impedance Tomography (EIT) is a noninvasive technique for reconstructing the conductivity distribution from the measured potential on the electrodes that are attached to the surface of the domain of interest (DOI), and current is injected through the same electrodes into DOI [1]. As the total number of measurements is much smaller than the total number of points to be reconstructed, the EIT problem is severely ill-posed. Acousto-electric tomography [2; 3; 4; 5] is developed to improve the ill-posedness and spatial imaging resolution of electrical impedance tomography (EIT) while conserving the high contrast property of EIT. Other methods, e.g., learning structural sparsity via a Bayesian approach [6; 7], and some based on multi-physics models [8; 9], exist as well for the same purpose. AET is applied to determine the internal conductivity of a physical body with better stability and resolution compared to the ill-posed EIT. The main idea is to carry out a classical EIT measurement while a known focused ultrasonic wave travels through the DOI. The high intensity of the pressure produced by the acoustic wave creates a small local deformation within the DOI, and this deformation changes the electrical conductivity locally, and therefore the electrical current distribution, which further changes the electric voltages measured on the electrodes. The higher the electrical conductivity in the acoustic wave focal zone, the greater the change. This deformation also influences the resolution of the imaging [2]. The resulting changes of the boundary potential can be recorded with EIT measurements [10], which are then used to compute the power density [2; 11] for reconstructingthe distribution of conductivity in the DOI. This method also has the potential applications in the materials damage detection [12; 13]. The conductivity reconstruction from the power density is an ill-posed nonlinear inverse problem. Many works have been reported for solving it in the existing literature, and many for improving the performance of these algorithms [14; 15]. The Levenberg-Marquardt method is a nonlinear optimization method well known for its convergence and stability, and it has been applied to solve the nonlinear problem of AET [14; 16; 17]. A Tikhonov regularization is used in the computation for better computation performance on stability, noise tolerance and accuracy. The adjoint operators required in the method are computed via constructing several PDEs for reconstructing the conductivity distribution iteratively. Numerical results presented in [14; 16; 17] show a good performance of the Levenberg-Marquardt method to different numerical phantoms, but the algorithm has several theoretical requirements to be meet for a stable computational performance, these include a well-chosen regularization parameter, the continuity requirements on the power density, and the existence of the second-order derivative of the conductivity distribution for the computation of the adjoint operator of power density. However, all the above requirements can be coupled together to give a synthesized influence on the performance of the algorithm, and neither the influence of each parameter nor the synthesized influence of all the parameters are discussed in the literature. Especially the contrast of the conductivity between regions of the whole DOI, which can render both conductivity and power density discontinue to violate the continuity requirements on the algorithm, which in turn requires the regularization parameter to be properly chosen. The focus of the present contribution is on the stability of the Levenberg-Marquardt method with respect to the above factors when it is applied to solve the nonlinear AET problem with multi-domain and different levels of conductivity contrasts.