# 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 Levenberg-Marquardt algorithm (LMA) locates the minimum of a function expressed as the sum of squares of errors between function and measured data through iterative updating. For a general operator equation \(F(\sigma)=\mathcal{E}(\sigma)^{\delta}\), where \(F:X\to Y\) and \(X\), \(Y\) are Hilbert spaces [23]. \(\mathcal{E}(\sigma)^{\delta}\) is the measured noisy data of \(\mathcal{E}(\sigma)\). More practically, one could instead minimize [24; 25]  
\[\sigma_{k+1}=\min_{\sigma}\left\|\mathcal{E}(\sigma)^{\delta}-F(\sigma_{k})-F ^{\prime}(\sigma_{k})\delta\sigma\right\|_{Y}^{2}+\alpha_{k}\left\|\delta \sigma\right\|_{X}^{2}, \tag{4}\]  
letting \(\alpha_{k}\) the Tikhonov regularization parameter, and \(\delta\sigma=\sigma-\sigma_{k}\). This minimization problem regarding to (4) is equivalent to the linearized one  
\[\sigma_{k+1}=\sigma_{k}+(F^{\prime}(\sigma_{k})^{*}F^{\prime}(\sigma_{k})+ \alpha_{k})^{-1}F^{\prime}(\sigma_{k})^{*}(\mathcal{E}(\sigma)^{\delta}-F( \sigma_{k})), \tag{5}\]  
letting \(F^{\prime}(\sigma_{k})^{*}\) the adjoint of \(F^{\prime}(\sigma_{k})\). It has been proven in [17; 26; 27; 28; 29], with a properly chosen parameter \(\alpha_{k}\) and an initial guess \(\sigma_{0}\) sufficiently close to the desired solution, that the LMA converges to a solution \(\sigma^{\delta}\) of \(F(\sigma^{\delta})=\mathcal{E}(\sigma)^{\delta}\). The LMA based on Equation (5) can be thought of as a combination of steepest descent and Gauss-Newton method. When the current solution is far from the correct one, a large value is given to \(\alpha_{k}\), and LMA behaves like a steepest-descent method, which converges slowly but well. When the current solution is close to the correct solution, a small \(\alpha_{k}\) is used, and LMA behaves like a Gauss-Newton method which is known as a faster iterative method than the steepest descent. The Frechet derivative and adjoint operator needed in (5) are computed with a method proposed in [17], as follows. It is assumed that \(\sigma\) is positive, and \(\sigma\in\mathcal{H}^{l}(\Omega)\) for \(l>n/2\). For \(f\in\mathcal{H}^{l+\frac{1}{2}}(\Omega)\), then the forward model has a unique solution \(u(\sigma)\in\mathcal{H}^{l+1}(\Omega)\), which indicates the operators \(\mathcal{E}(\sigma)\), \(\mathcal{E}^{\prime}(\sigma)\), and \(\mathcal{E}^{\prime}(\sigma)^{*}\) are all maps from \(\mathcal{H}^{s}(\Omega)\) to \(\mathcal{H}^{s}(\Omega)\)[17]. If \(u^{\prime}(\sigma)\) is the Frechet derivative of \(u(\sigma)\) in \(\Omega\),  
\[u(\sigma+\tau)=u(\sigma)+u^{\prime}(\sigma)\tau+o(\tau) \tag{6}\]  
is established for \(\tau\to 0\). \(o(\tau)\) is the remainder. Define \(L_{\sigma}(u)=u\to\
abla\cdot(\sigma\
abla u)\). According to Equations (3a) and (6),  
\[L_{\sigma}u^{\prime}(\sigma)\tau+\
abla\cdot(\tau\
abla u(\sigma))=0. \tag{7}\]  
The boundary conditions for \(u^{\prime}(\sigma)\tau\) can also be derived from Equation (6). If \(\xi=u^{\prime}(\sigma)\tau\), then \(\xi\) satisfies  
\[\
abla\cdot\sigma\
abla\xi+\
abla\cdot(\tau\
abla u(\sigma)) =0, \text{in }\Omega, \tag{8a}\] \[\xi =0, \text{on }e_{l}. \tag{8b}\]  
Here, it is assumed that \(\tau\) vanishes in a domain \(\Omega_{\epsilon}=\{x\in\bar{\Omega}|d(x,\partial\Omega)<\epsilon\}\) for a sufficiently small \(\epsilon\). Therefore, all terms regarding to \(\tau\) in the boundary conditions disappear. It is well known that the solution of Equations (8) uniquely exists in a weak sense. \(u^{\prime}(\sigma)\) as defined in (8) is the Frechet derivative of \(u(\sigma)\). The directional derivative of \(\mathcal{E}(\sigma)\) is given as  
\[\mathcal{E}^{\prime}(\sigma)\tau\approx\mathcal{E}(\sigma+\tau)-\mathcal{E}( \sigma)\approx\tau|\
abla u(\sigma)|^{2}+2\sigma\
abla u(\sigma)\
abla\xi \tag{9}\]  
with terms of the order of \(\|\tau\|^{2}\) neglected. \(\mathcal{E}^{\prime}(\sigma)\tau\) is actually the Frechet derivative of \(\mathcal{E}(\sigma)\). Let us assume \(\Omega\subset\mathbb{R}^{2}\) for simplicity, and consider the effect of noise.