## 3 Data Completion and Models Correction or Enrichment
### Model Local Correction and Data Completion
Expression (15), with \(a_{e}<0\), represents a simple approximate method of diminishing the contribution of element \(\Omega_{e}\) to the structure's stiffness matrix \(\mathbf{K}\). For elements \(\Omega_{e^{\prime}}\) located in the undamaged region \(\Omega-\omega\), \(\Omega_{e^{\prime}}\subset\{\Omega-\omega\}\), there is no need for correction, and the associated coefficients will vanish, i.e., \(a_{e^{\prime}}=0\). Thus, it is expected that only the elements in the set \(\mathcal{E}\) concerned by the elements covering the damaged area \(\omega\) (\(\omega=\cup_{e\in\mathcal{E}}\Omega_{e}\)) will have non zero values of the \(a\)-coefficients, while the largest majority of the others (the ones concerning elements in the undamaged region \(\Omega-\omega\)) will vanish. Therefore, the minimization problem that enforces the expected sparsity is described as follows. \[\left\{ \begin{array}{l}\texttt{min}_{a_{1},\ldots,a_{\mathcal{E}}}\sum \limits_{e=1}^{\mathtt{E}}|a_{e}|\\ \text{subjected to : }\mathbf{K}\Delta\mathbf{U}+\Delta\mathbf{K} \mathbf{U}=\mathbf{0}\ \&\left(\tilde{\mathbf{U}}+\widetilde{\Delta\mathbf{U}}\right)=\mathbf{U}^{m }\end{array} \right.. \tag{16}\]  
Because of the linearized model employed, the solution cannot be good enough. On the other hand, trying to iterate by particularizing the nonlinear term at the previous iteration (as most usual linearization strategies consider) has, as a main drawback, the accumulative increase in the corrected elements with the iterations of the nonlinear optimization problem. To circumvent these difficulties, we consider the following iterative procedure (Algorithm 1):  
```
1 At first iteration, \(q=1\), solve the minimization problem (16) : \(\left\{ \begin{array}{l}\texttt{min}_{a_{1},\ldots,a_{\mathcal{E}}}\sum \limits_{e=1}^{\mathtt{E}}|a_{e}|\\ \text{subjected to : }\mathbf{K}\Delta\mathbf{U}+\Delta\mathbf{K} \mathbf{U}=\mathbf{0}\ \&\left(\tilde{\mathbf{U}}+\widetilde{\Delta\mathbf{U}}\right)=\mathbf{U}^{m }\end{array} \right.\)
2 Group, from the computed solution, all the elements with non-zero coefficients \(a_{e}^{q}\
eq 0\), in the set \(\tilde{\mathcal{E}}^{q}\): \(\tilde{\mathcal{E}}^{q}=\{e\in\{1,\ldots,\mathtt{E}\}\text{ such that }a_{e}^{q}>e\}\);
3 Extract a domain \(\omega^{q}\) that covers all the elements in \(\tilde{\mathcal{E}}^{q}\);
4 Re-distribute the sensors, in \(\omega^{q}\) (at this step, a reduction in the number of sensor is also possible) (This step is not essential in the resolution of the problem, it's just used to optimize the algorithm)
5 Solve the minimization problem (16) but by restricting the candidate elements to be corrected to the ones concerned by \(\omega^{q}\), referred by \(\mathcal{E}^{q}\), with \(\tilde{\mathcal{E}}^{q}\subset\mathcal{E}^{q}\), i.e.,: \(\left\{ \begin{array}{l}\texttt{min}_{\mathbf{a}^{q}\in\mathcal{E}^{q}} \sum\limits_{e\in\mathcal{E}^{q}}|a_{e}^{q}|\\ \text{subjected to : }\mathbf{K}\Delta\mathbf{U}+\Delta\mathbf{K}\mathbf{U}= \mathbf{0}\ \&\left(\tilde{\mathbf{U}}+\widetilde{\Delta\mathbf{U}}\right)=\mathbf{U}^{m }\end{array} \right.\)
6 Go back to step 2 with \(q\gets q+1\)
```  
**Algorithm 1**Sparse identification  
The converged solution allows computing both the model's local correction \(\Delta\mathbf{K}\) and the data's completion \(\mathbf{U}+\Delta\mathbf{U}\) for which its extraction at the sensor's location \(\widetilde{\mathbf{U}+\Delta\mathbf{U}}\) approaches measures \(\mathbf{U}^{m}\). In order to enhance the localization capabilities by enabling the most accurate correction, \(\mathbb{A}\) different loadings can be applied and are associated with \({}^{a}\mathbf{U}\) and \({}^{a}\mathbf{U}^{m}\), \(a=1,\ldots,\mathbb{A}\).