# 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 minimum relative error becomes larger as the contrast increases, as obviously seen in \[\
abla\cdot\sigma\
abla\xi+\
abla\cdot(\tau\
abla u(\sigma)) =0, \text{in }\Omega, \tag{8a}\]3 This phenomenon is also observed in Figure 3a for high contrast cases and Figure 3b for almost all the noisy cases. Theoretically, the conductivity distribution should be in the \(H^{2}(\Omega)\) space, and the value of \(\beta_{k}\) is closely related to the second order derivative of the conductivity distribution as shown in Equation (15). When the contrast is too high, this continuity requirement can not be satisfied, this \"V-shape\" curve of the relative error appears. When the conductivity of the inclusion is \(\sigma_{i}=4.0\,\mathrm{S/m}\), the minimum value of the relative error \(\eta=0.8\%\), and this value is achieved at the 12 th iteration step. The reconstructed conductivity and the power density are shown in Figure 5a,b, respectively. A very good level of reconstruction is achieved. The change of the power density across the boundary between the inclusion and the background is not large. When \(\sigma_{i}\) is further increased to \(\sigma_{i}=12.0\,\mathrm{S/m}\), the minimum relative error is achieved at the 7 th iteration step and \(\eta=0.120\). The reconstructed conductivity is given in Figure 6a. However, \(\eta\) starts to

\[\
abla\cdot\sigma\
abla\xi+\
abla\cdot(\tau\
abla u(\sigma)) =0, \text{in }\Omega, \tag{8a}\]4 The relative error of different \(\beta_{0}\) with different conductivities: (**a**) when SNR = 40 dB and (**b**) when SNR = 20 dB. increase as the iteration proceeding. The reason of the increased error can be explored from power density, and the one at the 7 th step is given in Figure 6b. Comparing to Figure 5b, there is a larger jump of the power density across the boundary between the inclusion and the background. This may cause the relative error increase and make the algorithm become unstable and poorly convergent. \[\
abla\cdot\sigma\
abla\xi+\
abla\cdot(\tau\
abla u(\sigma)) =0, \text{in }\Omega, \tag{8a}\]5 The relative error of each conductivity when \(\text{SNR}=\infty\). \[\
abla\cdot\sigma\
abla\xi+\
abla\cdot(\tau\
abla u(\sigma)) =0, \text{in }\Omega, \tag{8a}\]6 When \(\text{SNR}=\infty\), \(\sigma=4.0\,\text{S}/\text{m}\) and the step is 12: (**a**) the reconstructed image and (**b**) the power density. \[\
abla\cdot\sigma\
abla\xi+\
abla\cdot(\tau\
abla u(\sigma)) =0, \text{in }\Omega, \tag{8a}\]7 When \(\text{SNR}=\infty\), \(\sigma=12.0\,\text{S}/\text{m}\) and the step is 7: (**a**) the reconstructed image and (**b**) the power density. #### 3.2.2 Numerical Results with Noise

When SNR = 40 dB, 1% of noise is added to the computed power density for the reconstruction. Here, \(\alpha_{k}\) for the \(k\)-th iteration is set as \(\alpha_{k}=\alpha_{0}/a^{k}\) with \(\alpha_{0}=1.0\) and \(a=2.0\), and \(\beta_{k}\) is set as constant \(\beta_{0}=1\times 10^{-3}\). The variation of the relative errors of reconstruction corresponding to different conductivities is shown in \[\
abla\cdot\sigma\
abla\xi+\
abla\cdot(\tau\
abla u(\sigma)) =0, \text{in }\Omega, \tag{8a}\]8 The algorithm works fine with \(\sigma_{i}<10\,\mathrm{S/m}\) though the relative error increases even with such a low level noise. If the contrast is continuously increased, the instability of the algorithm becomes severe, and the minimum relative error also becomes larger. The minimum relative error \(\eta=0.011\) is achieved at the 11 th step when the conductivity of the interior circular domain is \(\sigma_{i}=4.0\,\mathrm{S/m}\). The reconstructed conductivity distribution is shown in Figure 8a, and the corresponding power density is shown in Figure 8b. The reconstructed conductivity map is attractive although noise exists in the DOI, and the change of the power density across the boundary is not much. When the conductivity is further increased to \(\sigma_{i}=12.0\,\mathrm{S/m}\), the computation starts to diverge after the 7 th step, and the relative error begins to increase until to \(\eta=0.142\).