# The Levenberg-Marquardt Method for Acousto-Electric Tomography on Different Conductivity Contrast
## 2 The Reconstruction Algorithm Based on Levenberg-Marquardt Method
### The Reconstruction Algorithm
## 3 Numerical Investigations

The conductivity distribution can be obtained from the power density \(\mathcal{E}(\sigma)\) with Algorithm 1 based on the Levenberg-Marquardt method. In the following computation, \(\mathcal{E}(\sigma)\) is computed directly from its definition \(\sigma|\
abla u|^{2}\), where \(u\) is obtained with the forward model introduced in Section 2. Different phantoms with different contrast between the inclusion and the background are studied here. The weak forms of the equations in (20) can be easily obtained by applying integration by parts, and the equations are solved with finite element method implemented by using the open-source finite element library FEniCS [31]. Two boundary conditions with \(f_{1}(x,y)=x\) and \(f_{2}(x,y)=y\) are given to Equation (3a) for a reconstruction with multiple measurements. A relative error \(\eta\) is defined as \(\eta=\|\sigma_{t}-\sigma_{r}\|_{L_{2}}/\|\sigma_{t}\|_{L_{2}}\) to indicate the convergence of the Levenberg-Marquardt iteration. Index \(t\) indicates the true value, and index \(r\) means retrieved value. Practical measurements are always affected by various noises introduced by the devices or during the measurement process. Here the performance of the algorithm to different levels of contrast is investigated with different levels of noise. The noise level is measured with signal-to-noise ratio (SNR), which is defined as

\[L_{\sigma}u^{\prime}(\sigma)\tau+\
abla\cdot(\tau\
abla u(\sigma))=0. \tag{7}\]8

where \(\mathcal{N}\) satisfies Gaussian white noise distribution. The noise is added to the data of \(\mathcal{E}(\sigma)\) for numerical examples with SNR = \(\infty\), 40 dB, and 20 dB. SNR = \(\infty\) for noise free. ### Numerical Study with a Simple Phantom

A phantom with a smaller circular inclusion included in a circular domain is used here for numerically studying the performance of the Levenberg-Marquardt method to various conductivity contrast. The radius of the circular DOI is 1.0 m. The circular inclusion with its center located at (\(-0.3\), \(0.3\)) has a radius of 0.2 m. The conductivities of the background and the inclusion are denoted as \(\sigma_{b}\) and \(\sigma_{i}\), respectively. The contrast of the conductivity is defined as \(\sigma_{i}/\sigma_{b}\). \(\alpha_{k}\) for the \(k\)-th iteration is set as \(\alpha_{k}=\alpha_{0}/a^{k}\) with \(\alpha_{0}=1.0\), \(a=2.0\), and \(\beta_{k}\) set as constant \(\beta_{0}=1e-3\). The mesh for the forward model is shown in Figure 0(a) which has 3854 vertices and 7522 cells. A different mesh, as shown in Figure 0(b), with 3671 vertices and 7156 cells is used to avoid inverse crime. The background conductivity is set to 2.0 S/m, and the one for the inclusion is set to a series of values listed as 4.0 S/m, 6.0 S/m, 8.0 S/m, 10.0 S/m, 12.0 S/m, 14.0 S/m, 16.0 S/m, and 18.0 S/m. Therefore, the contrast varies from 2.0 to 9.0. ### The Influences of \(\alpha_{k}\) and \(\beta_{k}\)

The different value of \(\alpha_{k}\) and \(\beta_{k}\) may provide a better contrast of the Levenberg-Marquardt method. Here, the parameters \(\alpha_{0}\) and \(\beta_{0}\) will be discussed. The conductivity of background is set as 2.0 S/m. The conductivity of interior domain is set as 10.0 S/m, 12.0 S/m, and 14.0 S/m. Two noise levels (SNR = 40 dB, 20 dB) are considered. The parameter \(\alpha_{0}\) is first set as a constant with 0.1, 0.5, 1.0 while \(\beta_{k}=1\times 10^{-3}\). The variation of the relative error with the iteration for different values of \(\alpha_{0}\) is given in \[L_{\sigma}u^{\prime}(\sigma)\tau+\
abla\cdot(\tau\
abla u(\sigma))=0. \tag{7}\]9 In the computation, each iteration takes \(\sim\)5 s, the whole 50 iterations can be finished in about 5 min. When \(\alpha_{0}=0.1\), the results given in Figure 2a show that the method can converge fast for both low and high contrast cases. However, the relative errors for the high contrast cases, e.g., \(\sigma=12.0\) S/m, 14.0 S/m, start to diverge after achieving to the minimum relative error.