# Development of a Vibroacoustic Stochastic Finite Element Prediction Tool for a CLT Floor
## 4 Stochastic Process
### Decomposition of the Random Elastic Tensor
#### 4.2.1 Calculation of Lagrange Multipliers  
To derive Lagrange multipliers introduced in the previous section, there are several different methods to generate the independent the random variable \(\mathbf{B}_{\lambda}\) with respect to the corresponding probability density function (Equation (15)), such as the Metropolis-Hastings method [30; 31], Gibbs method [32]. However, it should be noticed that the Metropolis-Hastings method demands an appropriate proposal distribution, which is sometimes difficult to choose, and the Gibbs method requires us to know the conditional distributions. As a consequence, it could be intricate to give a robust calculation without an adequate initial guess, especially for a high-dimension case. Therefore, another alternative algorithm is introduced in Reference [16] to generate the independent random variable \(\mathbf{B}_{\lambda}\). Random Number Generator  
Let \(\mathbf{u}\to\Phi(\mathbf{u},\lambda)\) be the potential function defined as:  
\[\Phi(\mathbf{u},\lambda)=<\lambda,g(\mathbf{u})>_{R^{N+1}}. \tag{17}\]  
Let \(\{(\mathbf{U}(r),\mathbf{V}(r))\},r\in\mathbb{R}^{+}\) be the Markov stochastic process defined on the probability space \((\Theta,T,P)\) indexed by \(\mathbb{R}^{+}\) with values in \(\mathbb{R}^{+}\times\mathbb{R}^{+}\), for \(r>0\), satisfying the following Ito stochastic differential equation (ISDE). \[\left\{ \begin{array}{c}d\mathbf{U}(r)=\mathbf{V}(r)dr,\\ d\mathbf{V}(r)=-\
abla_{u}\Phi(\mathbf{U}(r),\lambda)dr-\frac{1}{2}f_{0}\
abla(r)dr+ \sqrt{f_{0}}d\mathbf{W}(r),\end{array} \right. \tag{18}\]where \(W(r)\) is the normalized Wiener process defined on \((\Theta,T,)\) indexed by \(\mathbb{R}^{+}\) and with values in \(\mathbb{R}^{N}\). The probability distribution of the initial condition \(\mathbf{U}(0)\) and \(\mathbf{V}(0)\) are supposed to be known. The parameter \(f_{0}\) is a real positive number, which could dissipate the transition part of the response generated by the initial condition and ensures a reasonable fast convergence of the stationary solution corresponding to the invariant measure. When \(r\) tends to infinity, the solution \(\mathbf{U}(r)\) of ISDE converges to the probability distribution of the random variable \(\mathbf{B}_{\lambda}\). \[\lim_{r\rightarrow\infty}(\mathbf{U}(r))=\mathbf{B}_{\lambda}. \tag{19}\]  
\(n_{s}\) independent realization of \(\mathbf{B}_{\lambda}\) is denoted as \(\mathbf{B}_{\lambda}(\theta_{1}),\ \ldots,\ \mathbf{B}_{\lambda}(\theta_{n_{s}})\). Let \(r_{o}\) be the iteration step that the solution of ISDE converge. When \(r_{k}\geq r_{0}\), the ISDE could be written as the equation below. \[\left\{ \begin{array}{c}d\mathbf{U}(r_{k})=\mathbf{V}(r_{k})dr_{k},\\ dV(r_{k})=-\
abla_{u}\Phi(\mathbf{U}(r_{k}),\lambda)dr_{k}-\frac{1}{2}f_{0}V(r_{k })dr_{k}+\sqrt{f_{0}}d\mathbf{W}(r_{k}).\end{array} \right. \tag{20}\]  
Therefore, the independent realizations \(\mathbf{B}_{\lambda}(\theta_{l})\) can be presented by the stationary solution of ISDE. \[\mathbf{B}_{\lambda}(\theta_{l})=\mathbf{U}(r_{k},\theta_{l}). \tag{21}\]  
It is worthwhile to mention that the Ito stochastic differential equation defined by Equation (18) can be discretized by the Explicit Euler scheme to obtain an approximate solution. \[k=1,\ldots,\ M-1,\left\{ \begin{array}{c}\mathbf{U}^{k+1}=\mathbf{U}^{k}+\Delta r\bm {V}^{k},\\ \mathbf{V}^{k+1}=\left(1-\frac{f_{0}}{2}\Delta r\right)\mathbf{V}^{k}+\Delta r\mathbf{L}^{ k}+\sqrt{f_{0}}\Delta\mathbf{W}^{k+1},\end{array} \right. \tag{22}\]  
with the initial conditions:  
\[\mathbf{U}^{1}=\mathbf{u}_{0},\ \mathbf{V}^{1}=\mathbf{v}_{0}, \tag{23}\]  
where \(\Delta r\) is the iteration step and \(\Delta\mathbf{W}^{k+1}=\mathbf{W}^{k+1}-\mathbf{W}^{k}\) is a second-order Gaussian centered \(\mathbb{R}^{N}\)-valued random variable with a covariance matrix \(E\Big{\{}\Delta\mathbf{W}^{k+1}\Big{(}\Delta\mathbf{W}^{k+1}\Big{)}^{T}\Big{\}}= \Delta r\{I_{N}\}\), where \(\mathbf{W}^{1}=0_{N}\).