## 9 Uncertainty Quantification for the Vibroacoustic Computational Model
#### 9.1.3 Stochastic Modeling of Random Matrix \([\mathbf{M}^{S}]\)  
Since there is no available information concerning the statistical dependency of \([\mathbf{M}^{S}]\) with the other random matrices of the problem, then random matrix \([\mathbf{M}^{S}]\) is independent of all the other random matrices. The deterministic matrix \([M^{S}]\) is positive definite, and can consequently be written as \([M^{S}]=[L_{M^{S}}]^{T}\left[L_{M^{S}}\right]\), in which \([L_{M^{S}}]\) is an upper triangular real matrix. Using the nonparametric probabilistic approach of uncertainties, the stochastic model of the positive-definite symmetric random matrix \([\mathbf{M}^{S}]\) is then defined by  
\[[\mathbf{M}^{S}]=[L_{M^{S}}]^{T}\left[\mathbf{G}_{M^{S}}\right]\left[L_{M^{S}} \right], \tag{76}\]  
where \([\mathbf{G}_{M^{S}}]\) is a \((N_{S}\times N_{S})\) random matrix belonging to ensemble \(\mathrm{SG}_{\epsilon}^{+}\) defined in Section 9.1.2 and whose probability distribution and generator of independent realizations depend only on dimension \(N_{S}\) and on the dispersion parameter \(\delta_{M^{S}}\). Stochastic Modeling of the Family \(\{[\mathbf{D}^{S}(\omega)],[\mathbf{K}^{S}(\omega)]\}_{\omega}\) of Random Matrices Indexed by \(\omega\)  
Since there is no available information concerning the statistical dependency of the family \(\{[\mathbf{D}^{S}(\omega)],[\mathbf{K}^{S}(\omega)]\}_{\omega}\) of random matrices with the other random matrices of the problem, then the family \(\{[\mathbf{D}^{S}(\omega)],[\mathbf{K}^{S}(\omega)]\}_{\omega}\) are independent of all the other random matrices. However, we will see below that the families of random matrices \(\{[\mathbf{D}^{S}(\omega)]\}_{\omega}\) and \(\{[\mathbf{K}^{S}(\omega)]\}_{\omega}\) will be statistically dependent. For stochastic modeling of \(\{[\mathbf{D}^{S}(\omega)]\}_{\omega}\) and \(\{[\mathbf{K}^{S}(\omega)]\}_{\omega}\) related to the linear viscoelastic structure, we propose to use the extension presented in [101], which is based on the Hilbert transform [102] in the frequency domain to express the causality properties (similarly to the transforms used in Section A.2 of Appendix A). The nonparametric probabilistic approach of uncertainties then consists of modeling the positive-definite symmetric \((N_{S}\times N_{S})\) real matrices \([D^{S}(\omega)]\) and \([K^{S}(\omega)]\) by random matrices \([\mathbf{D}^{S}(\omega)]\) and \([\mathbf{K}^{S}(\omega)]\) such that  
\[E\{[\mathbf{D}^{S}(\omega)]\}=[D^{S}(\omega)]\quad,\quad E\{[\mathbf{K}^{S}( \omega)]\}=[K^{S}(\omega)]\,, \tag{77}\]  
\[[\mathbf{D}^{S}(-\omega)]=[\mathbf{D}^{(S}\omega)]\quad,\quad\quad[\mathbf{K}^ {S}(-\omega)]=[\mathbf{K}^{S}(\omega)]\,. \tag{78}\]  
1. For \(\omega\geq 0\), the construction of the stochastic model of the family \(\{[\mathbf{D}^{S}(\omega)],[\mathbf{K}^{S}(\omega)]\}_{\omega\geq 0}\) of random matrices is carried out as follows: * Constructing the family \(\{[\mathbf{D}^{S}(\omega)]\}_{\omega\geq 0}\) of random matrices such that \[[\mathbf{D}^{S}(\omega)]=[L_{D^{S}}(\omega)]^{T}\left[\mathbf{G}_{D^{S}}\right] \left[L_{D^{S}}(\omega)\right],\] (79) where \([L_{D^{S}}(\omega)]\) is such that \[[D^{S}(\omega)]=[L_{D^{S}}(\omega)]^{T}\left[L_{D^{S}}(\omega)\right],\] (80) and where \([\mathbf{G}_{D^{S}}]\) is a \((N_{S}\times N_{S})\) random matrix belonging to ensemble \(\mathrm{SG}_{\epsilon}^{+}\), defined in Section 9.1.2. Its probability distribution and its generator of independent realizations depend only on dimension \(N_{S}\) and on the dispersion parameter \(\delta_{D^{S}}\) that allows the level of uncertainties to be controlled.