## 4 Iterative Inverse Problem Solution  
We consider an iterative optimization algorithm denoted by \(Q\). The statement of our problem can be reduced to the iterative form:  
\[ \begin{cases}\mu_{0}=\mu_{a}^{0}\\ \mu_{n+1}=Q(\mu_{n};J_{R})\end{cases}  \tag{9}\]  
Naturally, the promise of the algorithm is to get us closer to the solution after each step in an iterative manner. The proof of convergence of any specific algorithm ensures that  
\[\lim_{n\rightarrow+\infty}Q(\mu_{n};J_{R})=\mu_{a}^{*} \tag{10}\]  
and can give even more information on the speed of convergence by deriving a theoretical formula of \(\|Q(\mu_{n};J_{R})-\mu_{a}^{*}\|\) as a bounded formula of n.  
In general, this is a hard formula to derive, and it is even more difficult when dealing with complex problems like DOT, with many multidimensional parameters. In practice, the convergence speed is influenced by many factors, related to the algorithm itself, and the configuration of the problem (physical reality and constraints). A numerical approach based on simulation and statistical analysis will prove to be very useful in tackling these kinds of hard situations, and can help us to gain more insight in the choice of optimization algorithm and all other practical purpose. As we consider in our study, a family of optimization algorithms based on gradient descent, we can point out the learning rate hyper parameter as the main factor of interest in this context. From a practical point of view, \(J_{R}\) depends on the structure of the problem, and, consequently, \(J_{R}\) depends on different factors like the nature of inclusions (their number, form, distribution...), the properties of the medium, and all other parameters that shape the above forward problem as stated in the previous section. In addition, it depends on the choice of the regularization and initial guess. Table 1 below gives an example of the parameters and hyperparameters that can be of interest in studying the practical optimization problem (including the iterative algorithm hyperparameters). A more focused statement of the iterative optimization algorithm, for the following study in the present paper, can be formulated as  
\[ \begin{cases}\mu_{0}=\mu_{a}^{0}\\ \mu_{n+1}=Q_{AM}(\mu_{n};n,\beta,\Theta)\end{cases}  \tag{11}\]  
where \(Q_{AM}\) describes the adaptive moment algorithm, \(\mu_{a}^{0}\) the initial guess, \(n\) the number of inclusions, \(\beta\) denotes the learning rate hyperparameter, and \(\Theta\) represents all the remaining parameters. Hereafter, we address our attention only to the number of inclusions (\(n\)), the learning rate \(\beta\), and the initial guess \(\mu_{a}^{0}\). In our implementation of the optimization problem, we used an objective function C defined as  
\[C(l;\beta,n,\mu_{a}^{0})=\frac{1}{2}(J_{R}(Q_{AM}(\mu_{l};n,\beta))+\epsilon_{ 0}+|J_{R}(Q_{AM}(\mu_{l};n,\beta)-\epsilon_{0}|))) \tag{12}\]  
We can easily show that  
\[C(l;\beta,n,\mu_{a}^{0})= \begin{cases}J_{R}(Q_{AM}(\mu_{l};n,\beta))&ifJ_{R}(Q_ {AM}(\mu_{l};n,\beta)>\epsilon_{0}\\ \epsilon_{0}&ifJ_{R}(Q_{AM}(\mu_{l};n,\beta)\leq\epsilon_{0}\end{cases}  \tag{13}\]  
We define the number of iterations to convergence by  
\[N_{Q_{AM}}=min(argmax_{l}(C(l;\beta,n,\mu_{a}^{0})),L_{max}) \tag{14}\]  
This formulation guarantees that our optimization algorithm will stop whenever \(J_{R}(Q_{AM}(\mu_{l};n,\beta)\) is lower than \(\epsilon_{0}\) or \(l\) is greater than \(L_{max}\), where \(\epsilon_{0}>0\) and \(L_{max}\geq 1\), are parameters used in iteration stopping criteria, which is explicitly set in this study to be either when the cost function is lower than \(\epsilon_{0}\) or the number of iteration exceeds \(L_{max}\). The aim of our numerical statistical study of convergence speed can then be brought down to the study of properties of the \(N_{Q_{AM}}\) probability distribution \(P(N_{Q_{AM}}|n,\beta,\mu_{a}^{0})\) using simulation tools. In the following study, we restrict our attention to the comparison of three algorithms based on the adaptive moment procedure.