# The Levenberg-Marquardt Method for Acousto-Electric Tomography on Different Conductivity Contrast
## 2 The Reconstruction Algorithm Based on Levenberg-Marquardt Method
### The Reconstruction Algorithm
This divergence becomes severe when the noise level is increased, as shown in Figure 2b. When the value of \(\alpha_{0}\) is increased, the divergence phenomenon disappear for both lower and higher noisy cases, but the speed of convergence slows down, and it takes much more iteration steps for the computations to achieve a similar level of relative error as \(\alpha_{0}=0.1\). A direct comparison of the results is given in Figure 2a,b. A larger value of \(\alpha_{0}\) brings higher stability even with higher noise. Therefore, a smaller \(\alpha_{0}\) will accelerate the convergence of the computation, and a larger \(\alpha_{0}\) will improve the noise tolerance and the performance of the method for high contrast conductivity distribution. To accelerate the convergence

\[\
abla\cdot\sigma\
abla\xi+\
abla\cdot(\tau\
abla u(\sigma)) =0, \text{in }\Omega, \tag{8a}\]0 The mesh used in numerical examples: (a) the mesh of forward model and (b) the mesh of inverse problem. \[\
abla\cdot\sigma\
abla\xi+\
abla\cdot(\tau\
abla u(\sigma)) =0, \text{in }\Omega, \tag{8a}\]1 The variation of relative error of different \(\alpha_{0}\) with various conductivities: (a) when SNR = 40 dB and (b) when SNR = 20 dB. without losing the stability of the computation, \(\alpha_{k}\) for the \(k\)-th iteration is set as \(\alpha_{k}=\alpha_{0}/a^{k}\) with \(\alpha_{0}=1.0\), \(a=2.0\). The values of \(\beta_{k}=\beta_{0}\) are fixed in the following computations. However, a different value of \(\beta_{0}\) is possible to provide different performance of the Levenberg-Marquardt method for high contrast cases. The model with two different levels of noise (SNR = 40 dB, 20 dB) will be considered when conductivity is \(10.0\,\mathrm{S/m}\), \(12.0\,\mathrm{S/m}\), and \(14.0\,\mathrm{S/m}\). The \(\beta_{0}\) is set as 0.0001, 0.001, and 0.01. The results are shown in Figure 3a,b. When the value of \(\beta\) is increased from 0.0001 to 0.001, the speed of convergence is similar, this indicates the convergence is mainly dominated by the value of \(\alpha_{k}\). However, the minimum relative error achieved during the iteration process is decreased. For the high contrast case, the instability is also influenced by the value of the \(\beta_{k}\), a larger \(\beta_{k}\) with a better stability, and this becomes more obvious when more noise is introduced into the measured data, as shown in Figure 3b. Therefore, a smaller \(\beta_{0}\) will make the minimum relative error smaller during computation, however it decreases the noise tolerance. Therefore, the value of \(\beta_{k}\) is set to \(\beta_{0}=1e-3\) in the following computations. The values of \(\beta_{0}\) and \(\alpha_{0}\) could be greater or smaller whether needed. Another phenomenon observed in Figure 3a,b is that when \(\alpha_{k}\) becomes very small, the relative error for the high contrast cases starts to increase, and it stops at certain level. This is the instability caused by the high contrast of the conductivity. Therefore, the value of \(\alpha_{k}\) can not be too small. In what follows, the variation of \(\alpha_{k}\) follows \(\alpha_{k}=\alpha_{0}/a^{k}\) at the beginning of the iteration, but it is fixed to the value where the minimum relative error is achieved. #### 3.2.1 Numerical results with SNR = \(\infty\)

When SNR = \(\infty\), the relative error corresponding to each conductivity is shown in \[\
abla\cdot\sigma\
abla\xi+\
abla\cdot(\tau\
abla u(\sigma)) =0, \text{in }\Omega, \tag{8a}\]2 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 \(\beta_{0}=1e-3\). The figure shows that the algorithm works stably with good convergence when the conductivity \(\sigma_{i}\) is less than \(10.0\,\mathrm{S/m}\), where the contrast is 5. When the contrast is more than 5, the algorithm converges to certain level, and then it starts to diverge. The curve of the relative error varying with iteration steps becomes a \"V-shape\".