## 2 Non-Intrusive Method
### Non-Intrusive Spectral Projection  
We consider a model where both the input and output are real valued scalar random variables on a probability space \((\Theta,\Sigma,P)\).  
\[P_{\xi}(\xi)=\prod\limits_{i=1}^{N}p_{i}(\xi_{i}) \tag{4}\]  
Non-Intrusive Spectral Projection (NISP) [15] computes the projection coefficient of the random model output \(s(\xi)\) on a finite dimensional stochastic subspace \(f^{p}\) of \(\mathrm{L}_{2}\big{(}\Xi,p_{\xi}\big{)}\). The space equipped with the inner product \(\langle\cdot\rangle\),  
\[\langle u,v\rangle=\int_{\Xi}u(\xi)v(\xi)p_{\xi}(\xi)d\xi, \tag{5}\]  
is a Hilbert space. We denote \((p+1)\) as the dimension of \(f^{p}\) and let \(\big{\{}\Psi_{0}\{\xi\},\ldots,\Psi_{P}\{\xi\}\big{\}}\) be an orthogonal basis of \(f^{p}\).  
\[\big{\langle}\Psi_{i},\Psi_{j}\big{\rangle}=\int\ldots\int\Psi_{i}(\xi)\Psi_{ j}(\xi)p_{\xi}(\xi)d\xi_{1}\ldots d\xi_{N}=\big{\langle}\Psi_{i},\Psi_{j} \big{\rangle}\delta_{ij}. \tag{6}\]  
Non-Intrusive Spectral Projection provides a decomposition of \((s)^{|\;|\;1}\) into a finite generalized Polynomial Chaos (gPC) basis \(\big{\{}\Psi_{0}\{\xi\},\ldots,\Psi_{P}\{\xi\}\big{\}}\). Thus, it is the truncated Polynomial Chaos Expansion (PCE) of the probability space. The spectral projection of the output is shown below.  
\[s^{||}(\xi)=\sum\limits_{k=0}^{P}s_{k}\Psi_{k}(\xi)\in f^{p} \tag{7}\]  
Since \(s^{||}(\xi)\) is the projection of the output onto \(f^{p}\), for each index \(k\in\{0,\ldots,n\}\), the coefficients are given by  
\[s_{k}=\frac{\big{\langle}s,\Psi_{k}\big{\rangle}}{\langle\Psi_{k},\Psi_{k} \rangle},\;k\;=\;0,\,1,\,\ldots, \tag{8}\]  
such that the following is satisfied.  
\[s-s^{||}\bot f^{p}\Leftrightarrow\big{\langle}s-s^{||},\Psi_{k}\big{\rangle}= 0\;\,\text{for}\;0\leq k\leq P \tag{9}\]  
When the generalized Polynomial Chaos (gPC) basis is selected, the values \(\langle\Psi_{k},\Psi_{k}\rangle\) are known analytically and, therefore, only the computation of the scalar product remains:  
\[\langle\Psi_{k},\Psi_{k}\rangle s_{k}=\langle s,\Psi_{k}\rangle=\int_{\Xi}s( \xi_{1},\ldots,\xi_{N})\Psi_{k}p_{1}(\xi_{1})\ldots p_{N}(\xi_{N})d\xi_{1} \ldots d\xi_{N}. \tag{10}\]  
Here, \(\Xi\subset R\) is the support of the probability density function \(p_{\xi}\) of the basic random variable \(\xi\). The multidimensional integral of a function \(f(\xi)\equiv s(\xi)\Psi_{k}(\xi)\) over \(\Xi\) with a non-negative weight \(p_{\xi}\) has a product form, which can be shortened as shown below.  
\[\gamma f=\int\limits_{\Xi}f(\xi)p_{\xi}(\xi)d\xi \tag{11}\]When the dimensionality N of the integration is not too large, as is the case for many problems in practical engineering fields, deterministic cubature can be effectively used. A cubature is an approximation of the multidimensional integral \(\gamma f\) as a discrete sum:  
\[\gamma f\approx\sum_{i=1}^{N_{Q}}f(\xi^{(i)})w^{(i)}, \tag{12}\]  
where \(\xi^{(i)}\in\Xi\) and \(w^{(i)}\in R\), \(i=1\),..., \(N_{Q}\), are the nodes and weights of the chosen quadrature formula. Various quadrature rules can be used by means of a non-negative weight function \(p_{\xi}\). \(N_{Q}\) is the number of model resolutions.  
In further sections, the set \(\left\{\xi^{(i)}\right\}_{i=1}^{N+1}\) is called a sampling of \(\xi\) due to the very definition of the integral in Equation (12) and the quadrature node can be understood as particular realizations of a basic random variable \(\xi\).  
By combining Equations (8), (10), and (12), it becomes clear how NISP relies on a set of deterministic model resolutions, corresponding to some specific realizations of \(\xi\). Along this line, a deterministic simulation code can be used as a black-box, which associates each realization of the parameters to the corresponding model outputs.