## 9 Uncertainty Quantification for the Vibroacoustic Computational Model
#### 9.1.1 Ensemble \(\mathrm{SG}_{0}^{+}\) of Random Matrices  
Let \(\mathbb{M}_{0}^{+}\) be the set of all the positive-definite symmetric \((m\times m)\) real matrices. Below, we summarize the construction [75; 89; 90] of an ensemble--denoted by \(\mathrm{SG}_{0}^{+}\)--of random matrices with values in \(\mathbb{M}_{0}^{+}\). An element of \(\mathrm{SG}_{0}^{+}\) is a positive-definite random matrix denoted by \([\mathbf{G}_{0}]\).  
**Definition of the available information for constructing ensemble \(\mathrm{SG}_{0}^{+}\)**. For a random matrix \([\mathbf{G}_{0}]\) belonging to \(\mathrm{SG}_{0}^{+}\), the available information consists of the mean value which is given and equal to the identity matrix, and an integrability condition that has to be imposed in order to ensure the decreasing of the probability density function around the origin,  
\[E\{[\mathbf{G}_{0}]\}=[I_{m}],\,E\{\log(\det{[\mathbf{G}_{0}]})\}=\chi\,, \tag{70}\]  
in which \(|\chi|\) is finite and where \([I_{m}]\) is the \((m\times m)\) identity matrix.  
**Probability density function of a matrix \([\mathbf{G}_{0}]\in\mathrm{SG}_{0}^{+}\)**. The probability density function, \([G]\ \mapsto\ p_{[\mathbf{G}_{0}]}([G\,])\), defined on \(\mathbb{M}_{0}^{+}\), of a random matrix \([\mathbf{G}_{0}]\) satisfies the usual normalization condition,\[\int_{\mathbb{M}_{0}^{+}}P_{[\mathbf{G}_{0}]}([G\,])\ \tilde{d}G=1\,, \tag{71}\]  
where the volume element \(\tilde{d}G\) is written as \(\tilde{d}G=2^{m(m-1)/4}\,\Pi_{1\leq j\leq k\leq m}\,dG_{jk}\). Let \(\delta\) be the positive real number defined by  
\[\delta=\left\{\frac{1}{m}E\{\|\,[\mathbf{G}_{0}]-[I_{m}]\,\|_{F}^{2}\}\right\} ^{1/2}\,, \tag{72}\]  
which will allow the dispersion of the probability model of random matrix \([\mathbf{G}_{0}]\) to be controlled, and where \(\|\,\mathcal{M}\,\|_{F}\) is the Frobenius matrix norm of the matrix \([\mathcal{M}]\) such that \(\|\,\mathcal{M}\,\|_{F}^{2}=\mathsf{tr}\{[\mathcal{M}]^{T}[\mathcal{M}]\}\). For \(\delta\) such that \(0<\delta<(m+1)^{1/2}(m+5)^{-1/2}\), the use of the maximum entropy principle under the constraints defined by Equations (70) and (71) yields for all \([G\,]\) in \(\mathbb{M}_{0}^{+}\),  
\[p_{[\mathbf{G}_{0}]}([G\,])=c_{0}\big{(}\text{det}\,[G\,]\big{)}^{c_{1}}\exp \{-c_{2}\,\mathsf{tr}[G\,]\}\,, \tag{73}\]  
in which \(c_{0}\) is the positive constant of normalization and where \(c_{1}=(m+1)(1-\delta^{2})/(2\delta^{2})\) and \(c_{2}=(m+1)/(2\delta^{2})\).  
**Generator of independent realizations of a random matrix \([\mathbf{G}_{0}]\)** in \(\text{SG}_{0}^{+}\). The generator of independent realizations (which is required to solve the random equations with the Monte Carlo method) is constructed using the following algebraic representation of any random matrix \([\mathbf{G}_{0}]\) that belongs to \(\text{SG}_{0}^{+}\),  
\[[\mathbf{G}_{0}]=[\mathbf{L}]^{T}\,[\mathbf{L}]\,, \tag{74}\]  
in which \([\mathbf{L}]\) is an upper triangular \((m\times m)\) random matrix such that:  
* the family of the random entries \(\{[\mathbf{L}]_{jj^{\prime}},j\leq j^{\prime}\}\) are independent random variables;
* for \(j<j^{\prime}\), the real-valued random variable \([\mathbf{L}]_{jj^{\prime}}\) is written as \([\mathbf{L}]_{jj^{\prime}}=\sigma_{m}U_{jj^{\prime}}\) in which \(\sigma_{m}=\delta(m+1)^{-1/2}\) and where \(U_{jj^{\prime}}\) is a real-valued Gaussian random variable with zero mean and variance equal to \(1\);
* for \(j=j^{\prime}\), the positive-valued random variable \([\mathbf{L}]_{jj}\) is written as \([\mathbf{L}]_{jj}=\sigma_{m}\sqrt{2V_{j}}\) in which \(V_{j}\) is a positive-valued Gamma random variable with probability density function \(\Gamma(a_{j},1)\) in which \(a_{j}=\ \frac{m+1}{2\delta^{2}}+\ \frac{1-j}{2}\).