# The Levenberg-Marquardt Method for Acousto-Electric Tomography on Different Conductivity Contrast
## 2 The Reconstruction Algorithm Based on Levenberg-Marquardt Method
### The Reconstruction Algorithm
\[\mathcal{E}^{\prime}(\sigma)\tau\approx\mathcal{E}(\sigma+\tau)-\mathcal{E}( \sigma)\approx\tau|\
abla u(\sigma)|^{2}+2\sigma\
abla u(\sigma)\
abla\xi \tag{9}\]3 The true conductivity distributions of smooth–heart–lung model. \[\mathcal{E}^{\prime}(\sigma)\tau\approx\mathcal{E}(\sigma+\tau)-\mathcal{E}( \sigma)\approx\tau|\
abla u(\sigma)|^{2}+2\sigma\
abla u(\sigma)\
abla\xi \tag{9}\]4 The relative error and reconstructed conductivity map of smooth-heart–lung model with (**a**,**d**) for SNR = \(\infty\), (**b**,**e**) for SNR = 40 dB, (**c**,**f**) for SNR = 20 dB. ## 4 Conclusions

The Levenberg-Marquardt method for the acousto-electric tomography is applied to reconstruct the conductivity distribution in the DOI from voltages measured on its boundary. The main focus has been on the influence of the contrast on the stability and convergence performance of the Levenberg-Marquardt method. It also discussed the influences of the regularization parameters. Several numerical investigations have been carried out for the illustration. The work herein, on the one hand, focuses on the higher contrast potential reconstruction method, and on the other hand, on validations of its practicability on detecting lung cancer. It has been discovered that the algorithm becomes unstable when the contrast is approximately beyond 5. A contrast beyond this level will bring larger minimum relative error and a divergence after several iterations. The regularization parameter \(\alpha_{k}\) plays an important role in the computation. A larger value of the \(\alpha_{k}\) can make the computation converge better, but it will also render the convergence slower. A smaller \(\alpha_{k}\) will bring a faster convergence as long as the solution is not too far from the true value, otherwise the algorithm will be unstable. Therefore, a strategy of decreasing \(\alpha_{k}\) with the progress of the iteration can accelerate the convergence and keep the computation stable. 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. When the contrast is too high, this continuity requirement can not be satisfied, a \"V-shape\" curve of the relative error appears. Therefore, the value of \(\beta_{k}\) is also important for the stability and accuracy of the reconstruction. A large \(\beta_{k}\) will enforce strong continuity requirement on the conductivity distribution, which will render the computation unstable. A small \(\beta_{k}\) releases the continuity requirement, which will produce a poor reconstructed results. Therefore, the value of \(\beta_{k}\) needs to be properly chosen. It is well known that the Tikhonov regularization can smooth the reconstructed conductivity, therefore a large \(\alpha_{k}\) will balance the influence of a small \(\beta_{k}\). Besides the above parameters, a high contrast between the conductivity of different regions will cause a discontinuity of power density across the boundaries; this will not meet the continuity

\[\mathcal{E}^{\prime}(\sigma)\tau\approx\mathcal{E}(\sigma+\tau)-\mathcal{E}( \sigma)\approx\tau|\
abla u(\sigma)|^{2}+2\sigma\
abla u(\sigma)\
abla\xi \tag{9}\]5 The power density corresponding to heart–lung model (top row) and smooth–heart–lung model (bottom row) at the iteration step where the minimum relative error achieved. (**a**,**d**) for SNR = \(\infty\), (**b**,**e**) for SNR = 40 dB, (**c**,**f**) for SNR = 20 dB. requirement in the reconstruction algorithm, which will also bring the instability and poor convergence, as demonstrated by the numerical results of the high noisy and high contrast cases. The reconstructed results of a heart-lung model and a smooth-heart-lung model are compared to support the above conclusions, and a much better convergence and stability are observed with this smoothed heart-lung model. However, it has been shown that the Levenberg-Marquardt method works quite well in general, and performs much better in low contrast case.