# Robust Structural Damage Detection Using Analysis of the CMSE Residual's Sensitivity to Damage
## 2 Theoretical Background
### ITR Method  
The CMSE-based structural damage identification can be treated as a linear inverse problem, whose discretization gives rise to the linear system as given by Equation (1) with a very ill-conditioned matrix \(\mathbf{C}^{*}\). This means that a small perturbation in measurements can lead to unrealistically large perturbations in the predicted damage level vector. For this case, the least-squares solution is not sufficient. To address this problem, the ITR method that was tested and that outperformed the generalized least-squares method to provide a proper sparse solution [24] is introduced with the following solution:  
\[\mathbf{\hat{a}}^{(k)}=\left\{\mathbf{C}^{*T}\mathbf{C}^{*}+\left[\varepsilon^{(k )}\right]^{2}\left[\mathbf{L}^{(k)}\right]^{T}\mathbf{L}^{(k)}\right\}^{-1} \mathbf{C}^{*T}\mathbf{b}^{*} \tag{17}\]  
where \(\varepsilon^{(k)}\) and \(\mathbf{L}^{(k)}\) are the iterative regularization parameter and regularization operator, respectively, in iteration \(k\). In the context of the identification of isolated damage, the actual damage level vector bears sparse features. Proper sparsity of the Tikhonov solution depends on both the regularization parameter and regularization operator that have to be optimally selected. In the ITR method, an adaptive strategy is presented to determine them. The regularization parameter is selected as the intermediate singular value of interval \([\sigma_{b},\ \sigma_{a}]\) as follows:  
\[\xi^{(k)}=\sigma_{p},\ p=\left[\frac{a+b}{2}\right],1\leq a\leq b\leq rank( \mathbf{C}^{*}) \tag{18}\]  
where \(\sigma_{a}\), \(\sigma_{b}\), and \(\sigma_{p}\) are the \(a\)-th, \(b\)-th and \(p\)-th non-zero singular values of the coefficient matrix \(\mathbf{C}^{*}\), respectively, and the square brackets can round the inside value to the nearest integer less than or equal to itself. Moreover, the regularization operator is selected as  
\[\mathbf{L}^{(k)}=\text{diag}\left\{\left[\overline{\alpha}_{1}^{(k-1)}+ \varepsilon\right]^{-1},\cdots,\ \left[\overline{\alpha}_{n}^{(k-1)}+\varepsilon\right]^{-1},\ \cdots,\ \left[ \overline{\alpha}_{N_{e}}^{(k-1)}+\varepsilon\right]^{-1}\right\} \tag{19}\]  
where \(\overline{\alpha}_{n}^{(k-1)}\) is the predicted and adjusted extent of damage of \(n\)-th structural element in iteration \(k-1\); and \(\varepsilon\) is a given small parameter in the case that the diagonal entries of \(\mathbf{L}^{(k)}\) approach infinity when \(\overline{\alpha}_{n}^{(k-1)}\) approaches \(0\). A detailed process of the ITR method can be found in the paper by [PERSON] et al. [24]  
In this paper, two modifications to the adaptive strategy of determining the iterative regularization parameter and regularization operator are introduced, although this is not the main aim of this study. First, it can be noticed that the regularization parameter is selected as the intermediate singular value of interval \([\sigma_{b},\ \sigma_{a}]\). When \(a=b\), the calibration of the regularization parameter is terminated although the iteration continues. In order to continuously calibrate the regularization parameter and accelerate the convergence of the iteration, a dichotomy method in logarithmic space is introduced  
\[\xi^{(k)}=\exp\left\{\frac{\left[\ln\left(\eta_{a}^{(k)}\right)+\ln\left(\eta _{b}^{(k)}\right)\right]}{2}\right\} \tag{20}\]  
where \(\left[\eta_{b}^{(k)},\ \eta_{a}^{(k)}\right]\) is the range for the regularization parameter selection in iteration \(k\), and in the initial iteration \(\eta_{b}^{(0)}=\sigma_{r}\) and \(\eta_{a}^{(0)}=\sigma_{1}\), where \(\sigma_{1}\) and \(\sigma_{r}\) are the first and last non-zero singular values of matrix \(\mathbf{C}^{*}\), respectively. Here the reason that the logarithmic space is used is that the singular values of matrix \(\mathbf{C}^{*}\) usually belong to a log-linear distribution.