# 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 relative error and reconstructed conductivity map with SNR = \(\infty\), 40 dB, and 20 dB are shown in \[\xi =0, \text{on }e_{l}. \tag{8b}\]6 The computation, each iteration takes about 16 s and the total 50

\[\xi =0, \text{on }e_{l}. \tag{8b}\]7 When SNR = 20 dB, \(\sigma=4.0\,\mathrm{S/m}\) and the step is 5: (a) the reconstructed image and (b) the power density. \[\xi =0, \text{on }e_{l}. \tag{8b}\]8 The heart–lung model: (a) the region of different organs and (b) true conductivity distributions. 13 min. For the noise free case and the case with very low noise level as SNR = 40 dB, the minimum relative error is \(\sim\)0.074. The convergence of the computation is fast, but the reconstructed results are not good, as can be seen in Figure 13a,b. When higher noise level is considered, the convergence of the computation becomes worse, and the reconstructed results are also not good. As seen in Figure 13c for SNR = 20 dB, although the general conductivity contribution is observed, and a minimum relative error 0.094 is achieved, a lot of noise-like areas appear in the whole DOI. However, different tissues can still be clearly located in the reconstructed conductivity distribution, although their shapes are not so clear. To investigate the influences of the conductivity jump on the boundaries of two subdomains in the DOI, a smooth-heart-lung model is obtained with a mollification operation. The piece-wised conductivity distribution is mollified with

\[\xi =0, \text{on }e_{l}. \tag{8b}\]9

to produce \(\sigma\in\mathcal{H}^{s}(\Omega)\) with \(s=2\) for 2D problem. The constant \(C>0\) is selected so that \(\int_{R^{2}}\eta_{e}=1\). The value of \(\epsilon\) is 1 cm for heart-lung model. The true smoothed distribution of \(\sigma\) for the heart-lung model is shown in \[\mathcal{E}^{\prime}(\sigma)\tau\approx\mathcal{E}(\sigma+\tau)-\mathcal{E}( \sigma)\approx\tau|\
abla u(\sigma)|^{2}+2\sigma\
abla u(\sigma)\
abla\xi \tag{9}\]0 The numerical setup for the reconstruction of smoothed heart-lung model is kept. The relative error and reconstructed conductivity map of smooth-heart-lung model with SNR = \(\infty\), 40 dB, 20 dB are shown in \[\mathcal{E}^{\prime}(\sigma)\tau\approx\mathcal{E}(\sigma+\tau)-\mathcal{E}( \sigma)\approx\tau|\
abla u(\sigma)|^{2}+2\sigma\
abla u(\sigma)\
abla\xi \tag{9}\]1 The minimum relative errors are \(\sim\)0.002 when SNR = \(\infty\) and 0.007 when SNR = 40 dB, which is much smaller than the case with a piece-wised conductivity distribution. Even for the more noisy case with SNR = 20 dB, the minimum relative error is about 0.037, which is also much smaller. The convergence of the computation also performs much better. This significant decrease of relative error and increase of convergence speed are mainly caused by the smoothness of the power density which is further resulted from the smoothed conductivity distribution. The power density of corresponding to the minimum relative errors for SNR = \(\infty\), 40 dB, 20 dB are given in Figure 16 for both piece-wised and smoothed conductivity distributions of heart-lung model, respectively. A comparison shows that the power density for the smoothed conductivity is smoother,

\[\mathcal{E}^{\prime}(\sigma)\tau\approx\mathcal{E}(\sigma+\tau)-\mathcal{E}( \sigma)\approx\tau|\
abla u(\sigma)|^{2}+2\sigma\
abla u(\sigma)\
abla\xi \tag{9}\]2 The variation of the relative error for (**a**) SNR = \(\infty\), (**b**) SNR = 40 dB, and (**c**) SNR = 20 dB. The reconstructed conductivity distribution for (**d**) SNR = \(\infty\), (**e**) SNR = 40 dB, (**f**) SNR = 20 dB. and there is no obvious discontinuity across the boundary of different subdomains. Therefore, a rather smooth distribution of the conductivity can lead to a smoothed power density distribution, and this can well improve the stability of the algorithm and accelerate the convergence as well.