## 2 Non-Intrusive Method
### Point-Collocation Non-Intrusive Polynomial Approach  
Unlike the Non-Intrusive Spectral Projection (NISP) method discussed above, this method does not aim to determine the projection of the model output \(s(5)\) on a predefined stochastic subspace, but instead relies on the interpolation based on Equation (7). Then, \(P+1\) vectors \(\left(\overset{\rightarrow}{\xi}=\{\xi_{1},\xi_{2},\ldots,\xi_{n}\}_{k},k=0,1,2,\ldots,P\right)\) are chosen in the random space for a given PC expansion with \(P+1\) modes and the deterministic simulation code is evaluated at this point, which has a specific probability distribution. The discrete sum is taken over the number of output modes [15],  
\[P+1=\frac{(n+p)!}{n!p!}, \tag{13}\]  
where \(p\) and \(n\) are the order of polynomial chaos and the number of the random input dimension, respectively. The PC model obtained via Equation (13) is formed by polynomials up to a maximum degree \(P\) (also called order of the PC model): this truncation strategy is referred to as a \"total order expansion\". A linear system of equations can be obtained [16; 17; 18; 25; 26; 27], as shown below.  
\[\left[\begin{array}{c}\Psi_{0}\left[\overset{\rightarrow}{\xi_{0}}\right] \quad\Psi_{1}\left[\overset{\rightarrow}{\xi_{0}}\right]\quad\ldots\quad\Psi_ {P}\left[\overset{\rightarrow}{\xi_{0}}\right]\\ \Psi_{0}\left[\overset{\rightarrow}{\xi_{1}}\right]\quad\Psi_{1}\left[ \overset{\rightarrow}{\xi_{1}}\right]\quad\ldots\quad\Psi_{P}\left[ \overset{\rightarrow}{\xi_{1}}\right]\\ \vdots\quad\vdots\quad\ddots\quad\vdots\\ \Psi_{0}\left[\overset{\rightarrow}{\xi_{p}}\right]\quad\Psi_{1}\left[ \overset{\rightarrow}{\xi_{p}}\right]\quad\ldots\quad\Psi_{P}\left[ \overset{\rightarrow}{\xi_{p}}\right]\end{array}\right]\left\{\begin{array}{c} \alpha_{0}\\ \alpha_{1}\\ \vdots\\ \alpha_{p}\end{array}\right\}=\left\{\begin{array}{c}\alpha^{*}\left\{ \overset{\rightarrow}{\xi_{0}}\right\}\\ \alpha^{*}\left\{\overset{\rightarrow}{\xi_{0}}\right\}\\ \vdots\\ \alpha^{*}\left\{\overset{\rightarrow}{\xi_{0}}\right\}\\ \vdots\\ \alpha^{*}\left\{\overset{\rightarrow}{\xi_{0}}\right\}\end{array}\right\} \tag{14}\]  
We solved the linear system of Equations (14), which requires \(P+1\) deterministic function evaluations. If more than \(P+1\) samples are chosen, then the over-determined system of equations should be solved using the Least Squares method. [PERSON] et al. [16] investigated the effects on the results by the number of collocation points in a systematic way through the introduction of a parameter, the oversampling rate \(n_{\text{p}}\) defined below.  
\[n_{p}=\frac{number\ of\ samples}{P+1} \tag{15}\]They used oversampling rates of \(n_{p}=1,2,3,\) and 4 for the solution of a stochastic model problem with multiple uncertain variables to study the effect of the number of collocation points on the given accuracy of the polynomial chaos expansions. They suggested that a number of collocation points corresponding to two times the minimum number required, which means that an oversampling rate \(n_{p}\) of 2 yields a better approximation to the statistic at each polynomial degree.  
The open source packages Chaospy [28] and Scilab [15; 29; 30] were used for UQ analysis in the present work. The Chaospy [28] software package provides various stochastic tools for UQ analysis for the point-collocation NIPC method. Scilab software [15] was developed for spectral projection based on the Gaussian quadrature.