# The Levenberg-Marquardt Method for Acousto-Electric Tomography on Different Conductivity Contrast
## 2 The Reconstruction Algorithm Based on Levenberg-Marquardt Method
### The Reconstruction Algorithm
For the left hand side of (18), \(\mathcal{E}^{\prime}(\sigma_{k})^{*}\mathcal{E}^{\prime}(\sigma_{k})\tau=B^{*}M\tau\) with  
\[u(\sigma+\tau)=u(\sigma)+u^{\prime}(\sigma)\tau+o(\tau) \tag{6}\]2 \[u(\sigma+\tau)=u(\sigma)+u^{\prime}(\sigma)\tau+o(\tau) \tag{6}\]3  
Here, \(M\tau\) is easily obtained with Equations (9) and (11). After computing \(\tau_{k}\) from (18), the conductivity map \(\sigma_{k}\) obtained at the \(k\)-th iteration is updated by \(\sigma_{k+1}=\sigma_{k}+\tau_{k}\) for a new iteration. To calculate \(B^{*}M\tau\), Equation (17) needs to be solved with \(\varsigma=M\tau\), which requires first solving Equations (8) and (12). As all these PDE systems are coupled to each other via \(\tau\), they need to be collected into one PDE system and solved in one variational form. It yields the following system,  
\[u(\sigma+\tau)=u(\sigma)+u^{\prime}(\sigma)\tau+o(\tau) \tag{6}\]4 \[u(\sigma+\tau)=u(\sigma)+u^{\prime}(\sigma)\tau+o(\tau) \tag{6}\]5 (20b) \[u(\sigma+\tau)=u(\sigma)+u^{\prime}(\sigma)\tau+o(\tau) \tag{6}\]6 (20c) \[u(\sigma+\tau)=u(\sigma)+u^{\prime}(\sigma)\tau+o(\tau) \tag{6}\]7 (20d) \[u(\sigma+\tau)=u(\sigma)+u^{\prime}(\sigma)\tau+o(\tau) \tag{6}\]8 (20e) \[u(\sigma+\tau)=u(\sigma)+u^{\prime}(\sigma)\tau+o(\tau) \tag{6}\]9 (20f) \[L_{\sigma}u^{\prime}(\sigma)\tau+\
abla\cdot(\tau\
abla u(\sigma))=0. \tag{7}\]0 (20g) \[L_{\sigma}u^{\prime}(\sigma)\tau+\
abla\cdot(\tau\
abla u(\sigma))=0. \tag{7}\]1 (20h) \[L_{\sigma}u^{\prime}(\sigma)\tau+\
abla\cdot(\tau\
abla u(\sigma))=0. \tag{7}\]2 (20i) \[L_{\sigma}u^{\prime}(\sigma)\tau+\
abla\cdot(\tau\
abla u(\sigma))=0. \tag{7}\]3 (20j) \[L_{\sigma}u^{\prime}(\sigma)\tau+\
abla\cdot(\tau\
abla u(\sigma))=0. \tag{7}\]4 (20k) \[L_{\sigma}u^{\prime}(\sigma)\tau+\
abla\cdot(\tau\
abla u(\sigma))=0. \tag{7}\]5  
with \(y=\mathcal{E}^{\prime}(\sigma_{k})^{*}(\mathcal{E}^{\delta}-\mathcal{E}( \sigma_{k}))\), \(\phi=B^{*}M\tau\), \(\rho=V(\tau|\
abla u|^{2})\), and \(\kappa=V(\sigma\
abla u\
abla u^{\prime}(\sigma)\tau)\). If the number of measurements \(N>1\), \(y=\mathcal{E}(\sigma_{k})^{*}(\mathcal{E}^{\delta}-\mathcal{E}(\sigma_{k}))\) and \(\gamma=\sum_{n}^{N}M_{n}\tau\) with  
\[L_{\sigma}u^{\prime}(\sigma)\tau+\
abla\cdot(\tau\
abla u(\sigma))=0. \tag{7}\]6 \[L_{\sigma}u^{\prime}(\sigma)\tau+\
abla\cdot(\tau\
abla u(\sigma))=0. \tag{7}\]7  
Equations (20g)-(20l) must be solved for each measurement, so one additional measurement will require to solve in addition three partial differential equations. The smoothness requirement on \(\sigma\) can be achieved by a so-called mollification operation. To solve the above system, \(\mathcal{E}^{\prime}(\sigma_{k})^{*}(\mathcal{E}^{\delta}-\mathcal{E}(\sigma _{k}))\) is needed first. Let \(g=\mathcal{E}^{\delta}-\mathcal{E}(\sigma_{k})\), then \(\mathcal{E}^{\prime}(\sigma_{k})^{*}g=B^{*}z\) with \(z=|\
abla u|^{2}\Phi_{\epsilon}g+2\
abla u\
abla w\) and \(h=V\Phi_{\epsilon}g\). Thus, \(w\) can be calculated with (12), and \(x=B^{*}z\) is ```
Data: Measured power density \(\mathcal{E}^{\delta}\) and initial guess \(\sigma_{0}\) Result: Retrieved conductivity map \(\sigma_{r}\) with relative error \(\eta\)
1\(N\): maximum number of iterations;
2\(\sigma_{k}\leftarrow\sigma_{0}\);
3\(\alpha_{k}\leftarrow\alpha_{0}\);
4\(num\gets 1\);
5\(norm\gets 1\);
6 whilenorm \(>\delta\) and \(num<N\)do
7 Update \(u_{k}\) from \(\sigma_{k}\) with the forward model;
8 Compute \(\mathcal{E}(\sigma_{k})\) from \(u_{k}\);
9 Compute \(y=\mathcal{E}^{\prime}(\sigma_{k})^{*}(\mathcal{E}^{\delta}-\mathcal{E}( \sigma_{k}))\) with (12) and (17);
10 Compute \(\tau_{k}\) with the linear system defined by (20);
11\(\sigma_{k}\leftarrow\sigma_{k}+\tau_{k}\);
12\(norm\leftarrow\|\tau_{k}\|_{L_{2}}\);
13\(num\gets num+1\);
14 Update \(\alpha_{k}\);
15
16 end while
```

**Algorithm 1**LM-CM algorithm for retrieving the conductivity map of a domain \(\Omega\) from a single measurement of power density.