# 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 latter cannot be assumed to be differentiable, then \(\mathcal{E}(\sigma):\mathcal{H}^{2}(\Omega)\to\mathcal{L}^{2}(\Omega)\). As \(\mathcal{E}^{\prime}(\sigma):\mathcal{H}^{2}(\Omega)\to\mathcal{L}^{2}(\Omega)\) is a bounded operator, its Hilbert adjoint \(\mathcal{E}^{\prime}(\sigma)^{*}:\,\mathcal{L}^{2}(\Omega)\,\to\,\mathcal{H}^{ 2}(\Omega)\) is bounded as well [30]. With embedding operator \(B:\mathcal{H}^{2}(\Omega)\to\mathcal{L}^{2}(\Omega)\), \(\mathcal{E}^{\prime}(\sigma)^{*}\) can be written as  
\[\mathcal{E}^{\prime}(\sigma)^{*}:\mathcal{L}^{2}(\Omega)\to \mathcal{H}^{2}(\Omega), \tag{10}\] \[\mathcal{E}^{\prime}(\sigma)^{*}z=B^{*}\tilde{\mathcal{E}}^{ \prime}(\sigma)^{*}z. \tag{11}\]  
\(\tilde{\mathcal{E}}^{\prime}(\sigma)^{*}\) is the \(\mathcal{L}^{2}(\Omega)\)-adjoint of \(\mathcal{E}^{\prime}(\sigma)\). \(z\in\mathcal{L}^{2}(\Omega)\), and \(\tilde{\mathcal{E}}^{\prime}(\sigma)^{*}z=|\
abla u(\sigma)|^{2}z+2\
abla u \
abla Vz\). The map \(V:\mathcal{H}^{2}(\Omega)\to\mathcal{H}^{2}(\Omega)\) is linear and defined by  
\[L_{\sigma}Vz =-\
abla\cdot(z\sigma\
abla u(\sigma)) \text{in}\;\Omega, \tag{12a}\] \[\sigma_{k+1}=\sigma_{k}+(F^{\prime}(\sigma_{k})^{*}F^{\prime}(\sigma_{k})+ \alpha_{k})^{-1}F^{\prime}(\sigma_{k})^{*}(\mathcal{E}(\sigma)^{\delta}-F( \sigma_{k})), \tag{5}\]0  
If multiple measurements are considered, \(\tilde{\mathcal{E}}^{\prime}_{H}(\sigma)^{*}z\) in Equation (11) is replaced with \(\tilde{\mathcal{E}}^{\prime}(\sigma)\) defined as  
\[\sigma_{k+1}=\sigma_{k}+(F^{\prime}(\sigma_{k})^{*}F^{\prime}(\sigma_{k})+ \alpha_{k})^{-1}F^{\prime}(\sigma_{k})^{*}(\mathcal{E}(\sigma)^{\delta}-F( \sigma_{k})), \tag{5}\]1  
To calculate \(B^{*}:\mathcal{L}^{2}(\Omega)\to\mathcal{H}^{2}(\Omega)\), we allow  
\[\sigma_{k+1}=\sigma_{k}+(F^{\prime}(\sigma_{k})^{*}F^{\prime}(\sigma_{k})+ \alpha_{k})^{-1}F^{\prime}(\sigma_{k})^{*}(\mathcal{E}(\sigma)^{\delta}-F( \sigma_{k})), \tag{5}\]2  
with its inner product given as  
\[\sigma_{k+1}=\sigma_{k}+(F^{\prime}(\sigma_{k})^{*}F^{\prime}(\sigma_{k})+ \alpha_{k})^{-1}F^{\prime}(\sigma_{k})^{*}(\mathcal{E}(\sigma)^{\delta}-F( \sigma_{k})), \tag{5}\]3  
\(\beta\) is a regularization parameter, which enforces a second-order weak derivative. Upon integration by parts, \(B^{*}\zeta\) is the solution of the Neumann problem  
\[\sigma_{k+1}=\sigma_{k}+(F^{\prime}(\sigma_{k})^{*}F^{\prime}(\sigma_{k})+ \alpha_{k})^{-1}F^{\prime}(\sigma_{k})^{*}(\mathcal{E}(\sigma)^{\delta}-F( \sigma_{k})), \tag{5}\]4 \[\sigma_{k+1}=\sigma_{k}+(F^{\prime}(\sigma_{k})^{*}F^{\prime}(\sigma_{k})+ \alpha_{k})^{-1}F^{\prime}(\sigma_{k})^{*}(\mathcal{E}(\sigma)^{\delta}-F( \sigma_{k})), \tag{5}\]5 (16b) \[\sigma_{k+1}=\sigma_{k}+(F^{\prime}(\sigma_{k})^{*}F^{\prime}(\sigma_{k})+ \alpha_{k})^{-1}F^{\prime}(\sigma_{k})^{*}(\mathcal{E}(\sigma)^{\delta}-F( \sigma_{k})), \tag{5}\]6  
with \(\zeta\in\mathcal{L}^{2}(\Omega)\). Refer to the work in [17] for more detail. This fourth-order partial differential equation is not practical, but it can be written into an equivalent form  
\[\sigma_{k+1}=\sigma_{k}+(F^{\prime}(\sigma_{k})^{*}F^{\prime}(\sigma_{k})+ \alpha_{k})^{-1}F^{\prime}(\sigma_{k})^{*}(\mathcal{E}(\sigma)^{\delta}-F( \sigma_{k})), \tag{5}\]7 \[\sigma_{k+1}=\sigma_{k}+(F^{\prime}(\sigma_{k})^{*}F^{\prime}(\sigma_{k})+ \alpha_{k})^{-1}F^{\prime}(\sigma_{k})^{*}(\mathcal{E}(\sigma)^{\delta}-F( \sigma_{k})), \tag{5}\]8 (17b) \[\sigma_{k+1}=\sigma_{k}+(F^{\prime}(\sigma_{k})^{*}F^{\prime}(\sigma_{k})+ \alpha_{k})^{-1}F^{\prime}(\sigma_{k})^{*}(\mathcal{E}(\sigma)^{\delta}-F( \sigma_{k})), \tag{5}\]9 (17c) \[u(\sigma+\tau)=u(\sigma)+u^{\prime}(\sigma)\tau+o(\tau) \tag{6}\]0  
According to the Levenberg-Marquardt iteration in Equation (5), the formula for calculating the \(k\)-th updating step \(\tau_{k}\) for the problem at hand is given as  
\[u(\sigma+\tau)=u(\sigma)+u^{\prime}(\sigma)\tau+o(\tau) \tag{6}\]1where \(\mathcal{E}^{\delta}\) represents the measured power density derived from \(u_{b}\) and \(u_{b,\epsilon}\) in the last section.