# The Levenberg-Marquardt Method for Acousto-Electric Tomography on Different Conductivity Contrast
## 2 The Reconstruction Algorithm Based on Levenberg-Marquardt Method
### The Reconstruction Algorithm
The reconstructed conductivity map and the power density of the 7 th step are given in Figure 9a,b, and a larger change on the boundary of the interior domain is still observed, but the influence of the noise is not obvious. \[\
abla\cdot\sigma\
abla\xi+\
abla\cdot(\tau\
abla u(\sigma)) =0, \text{in }\Omega, \tag{8a}\]9 When SNR = 40 dB, \(\sigma=4.0\,\mathrm{S/m}\) and the step is 11: (**a**) the reconstructed image and (**b**) the power density. \[\xi =0, \text{on }e_{l}. \tag{8b}\]0 The relative error of each conductivity when SNR = 40 dB. However, when the noise level is further increased to SNR = 20 dB, the performance of the algorithm becomes more sensitive to the contrast. The variation of the relative errors corresponding to different conductivities is shown in \[\xi =0, \text{on }e_{l}. \tag{8b}\]1 The convergence of the computation is poor even for \(\sigma_{i}=8\,\mathrm{S/m}\). For higher contrast, the computation converges well in the first several steps, but it starts to diverge afterwards. The minimum relative error \(\eta=0.050\,\mathrm{S/m}\) is achieved at the 5 th step when the conductivity of interior domain is 4.0 S/m. The reconstructed conductivity map is shown in Figure 10(a), and the corresponding power density is shown in Figure 10(b). The results are obviously contaminated by noise, and fluctuation is observed in the DOI. This is caused by the large noise introduced into the power density as shown in Figure 10(b). To conclude, the performance of the introduced Levenberg-Marquardt method for acousto-electric tomography has a close relation with the contrast of the conductivity. It works adequately for a problem with low contrast and low noise level. This is observed when the contrast is smaller than 5. Then, the performance becomes worse with the increase of the contrast, and the minimum relative error also increases. However, higher noise will increase the sensitivity of the method to the contrast, and therefore decrease the stability of the algorithm and the quality of the reconstruction. These performances are also closely related to the smoothness of the distribution of power density in the domain of interest. However, it should not be neglected that the parameters \(\alpha_{k}\) and \(\beta_{k}\) introduced in (20) also play an important role. \[\xi =0, \text{on }e_{l}. \tag{8b}\]2 The relative error of each conductivity when SNR = 20 dB. \[\xi =0, \text{on }e_{l}. \tag{8b}\]3 When SNR = 40 dB, \(\sigma=12.0\,\mathrm{S/m}\) and the step is 7: (**a**) the reconstructed image and (**b**) the power density. ### Heart Lung Model

To further investigate the performance of the algorithm, a heart-lung model is created and used here, which is shown in \[\xi =0, \text{on }e_{l}. \tag{8b}\]4 The considered tissues are heart (red, \(\sigma=0.7\,\mathrm{S/m}\)), lung (cyan, \(\sigma=0.26\,\mathrm{S/m}\)), and soft tissues (blue, \(\sigma=0.33\,\mathrm{S/m}\)). Refer to the work in [16] for conductivities of different tissues. An additional circular domain is added to the left lung to simulate a lung cancer, and the tissue is in purple color. The conductivity of lung cancer tissue is generally 1.6 to 3.3 times larger than the normal tissues; here, it is set to \(\sigma=0.50\,\mathrm{S/m}\), which is \(\sim\)1.92 times of the lung tissue. The model is placed into a circular region with a background material \(\sigma=0.22\,\mathrm{S/m}\) and the radius is 0.25 m. The true conductivity distributions are shown in \[\xi =0, \text{on }e_{l}. \tag{8b}\]5 The inverse mesh consists of 12,539 vertices and 24,730 cells.The initial guess for the interaction is set to 0.22 S/m for the whole DOI. Here, \(\alpha_{k}\) for the \(k\)-th iteration is set as \(\alpha_{k}=\alpha_{0}/a^{k}\) with \(\alpha_{0}=1.0\), \(a=2.0\), and \(\beta_{k}\) is set as constant \(\beta_{0}=1\times 10^{-3}\). The conductivity is reconstructed with the method introduced in this paper.