## 2 Methodology of the Inversion
### Regularization Methods  
Mathematically, in the presence of noise, Equation (12) is no longer valid: strict equality becomes an approximation and inversion can be seen as a minimization problem by the least squares. The unknown \(\Omega\) is sought in such a way as to minimize the residual \(R\) by the introduction of a term of penalization:  
\[R^{2}=\|\mathbf{I}\,\Omega-\theta_{mes}\|_{2}^{2}+\alpha J(\Omega), \tag{14}\]  
where \(\alpha\) is a scalar called the \"coefficient of regularization\" and \(J(\Omega)\) is an independent function of the unknown \(\Omega\). The higher the coefficient of regularization is, the more stable the inversion will be. However, in return, the term \"penalization\" adds a bias to the initial problem. Thus, the greater the term of regularization is, the farther the final solution is from the good one. The difficulty therefore lies in the choice of the adjustment parameters in order to have the best compromise between stability and speed of the solution. Different methods enable choosing the \(\alpha\) parameter, for example, the UPRE (unbiased predictive risk estimator), GCV (generalized cross validation), L-Curve and discrepancy principle or Picard methods [17; 19; 20; 21; 22; 23]. Then, several types of regularization are possible depending on the choice of \(J\). In this paper, the following types will be studied: (i), regularization by the \(L_{2}\) standard (Tikhonov regularization [24]), and (ii), regularization by the \(L_{1}\) standard (Lasso method [25]). In the [PERSON] case, Equation (13) will take the following form:  
\[\Omega=\sum_{k=1}^{p}\frac{s_{k}^{2}}{s_{k}^{2}+\alpha}\frac{\mathbf{U}_{\cdot,k} ^{T}\theta_{mes}}{s_{k}}\mathbf{V}_{\cdot,k}. \tag{15}\]  
The comparison between the theoretical solution, given by Equation (13), and the solution of the Tikhonov regularization, given by Equation (15), shows that Tikhonov regularization has the effect of weighting each mode \(k\) of the SVD by a coefficient. In the \(L_{1}\) regularization, Equation (13) will take the following form:  
\[\left(\mathbf{V}\mathbf{S}^{T}\mathbf{S}\mathbf{V}^{T}+\alpha\left(\mathbf{J }_{(i)}\right)^{2}\right)\Omega_{(i)}=\mathbf{V}\mathbf{S}\mathbf{U}^{T} \theta_{mes}, \tag{16}\]  
where for each iteration \(i\), \(\mathbf{J}_{(i)}\) is a diagonal matrix that contains the \(L_{1}\)-norm of the solution of the previous iteration \(\Omega_{(i-1)}\). As can be seen in the definition of this matrix, each iteration uses the results from the source \(\Omega_{(i)}\) from the previous iteration. For the first iteration, it is therefore necessary to define the matrix \(\mathbf{J}_{(1)}\). The choice of this initialization does not matter, since from the first iteration, there will be correction according to the data of the inversion. However, it is unadvisable to take the null matrix since that would mean that the first iteration is not regularized. In this study, we take \(\mathbf{J}_{(1)}=Id\), the identity matrix. The problem given by Equation (16) is a simple linear equation, given by:  
\[\mathbf{I}_{\mathbf{L}_{1}}\,\Omega=\mathbf{V}\mathbf{S}\mathbf{U}^{T}\theta _{mes}. \tag{17}\]  
The matrix \(\mathbf{I}_{\mathbf{L}_{1}}\) is entirely determined since it involves only the data of SVD of the matrix of the basic problem, as well as the solution \(\Omega_{(i-1)}\) of the previous iteration (so known). Moreover, by its construction, this matrix is square and invertible. The resolution of the problem (17) can therefore be done via any inversion algorithm. Unlike [PERSON]'s regularization and truncation of the spectrum of SVD, the regularization by the \(L_{1}\)-norm allows penalizing each position of the source (node of discretization) with a different parameter. As it is defined, regularization by the \(L_{1}\)-norm strongly penalizes the \"zero\" zones (no source), and conversely, it lowly weights non-zero areas (source).