## 2 Methods
### Polynomial Chaos Expansion  
The PCE method is based on the principle that a computational model can be obtained by an infinite series expansion of orthogonal polynomials as in Equation (1). In this equation, \(\mathcal{M}\) represent the computational model with the outputs of the system, \(\mathbf{Y}=\{Y_{1},\ldots,Y_{N}\}\), being dependent on the variables \(\mathbf{X}=\{X_{1},\ldots,X_{M}\}\). \[\mathcal{M}(\mathbf{X})=\sum_{j=0}^{\infty}\alpha_{j}\mathbf{Y}_{j}(\mathbf{X}) \tag{1}\]  
From a computational point of view, the infinite series should appropriately be truncated. The PCE method suggests that, as in Equation (2), the computational model can be expressed with a small error, \(\varepsilon\), by using a \(P+1\) terms in the series expansion:  
\[\mathcal{M}(\mathbf{X})\approx\mathcal{M}^{\text{PCE}}(\mathbf{X})\cong\sum _{j=0}^{P}\alpha_{j}\mathbf{\Psi}_{j}(\mathbf{X})+\varepsilon \tag{2}\]  
The \(\alpha\) terms in Equation (2) indicate unknown coefficients, and the \(\Psi\) vector are the multivariate orthogonal terms. Multivariate orthogonal (\(\mathbf{\Psi}_{i}(\mathbf{X})\)) terms are obtained by multiplying univariate orthogonal polynomials (\(\psi_{m_{i}^{j}}(x)\)) as in Equation (3). \(m_{i}^{j}\) in this equation represents the index of the univariate Hermite polynomial. \[\Psi_{j}(\mathbf{X})=\prod_{i=1}^{M}\psi_{m_{i}^{j}}(x_{i}) \tag{3}\]  
Depending on the probability distribution of the inputs of the computational model, different orthogonal polynomials may be used. Legendre polynomials are used for uniformly distributed inputs and Hermite polynomials are used for normally distributed inputs. The definition of the first few two-dimensional Hermite polynomials is expressed in terms of the standard variable \(\xi\) as in Equation (4) [35]. \[ \begin{split}\Psi_{0}(\xi)&=\psi_{0}(\xi_{1})\psi_{ 0}(\xi_{2})=1\\ \Psi_{1}(\xi)&=\psi_{1}(\xi_{1})\psi_{0}(\xi_{2})= \xi_{1}\\ \Psi_{2}(\xi)&=\psi_{0}(\xi_{1})\psi_{1}(\xi_{2})= \xi_{2}\\ \Psi_{3}(\xi)&=\psi_{2}(\xi_{1})\psi_{0}(\xi_{2})= \xi_{1}^{2}-1\\ \Psi_{4}(\xi)&=\psi_{1}(\xi_{1})\psi_{1}(\xi_{2})= \xi_{1}\xi_{2}\\ \Psi_{5}(\xi)&=\psi_{0}(\xi_{1})\psi_{2}(\xi_{2})= \xi_{2}^{2}-1\end{split}  \tag{4}\]  
There are different methods such as the Smolyak sparse grid [36], random sampling [37], tensor product quadrature [38] and regression to calculate unknown \(\alpha\) coefficients. The methods other than the regression method are spectral projection methods, which consist of projecting the computational model responses against each basis function using inner products. Unlike projection methods, an arbitrary number of results from the analysis can be used to calculate unknown coefficients in the regression method. In the present work, the regression method is used to calculate the unknown coefficients. The PCE series expansion in Equation (2) is expressed in matrix notation as in Equation (5):  
\[\mathbf{Y}=\Psi\mathbf{\alpha}+\mathbf{\epsilon} \tag{5}\]The expressions in Equation (5) are expressed in matrix form as in Equation (6) where \(\mathbf{e}\) are the residuals. \[\mathbf{Y}=\left[ \begin{array}{c}Y_{1}\\ Y_{2}\\ \vdots\\ Y_{N_{c}}\end{array} \right],\mathbf{\Psi}=\left[ \begin{array}{cccc}1&\Psi_{1}( \mathbf{X_{1}})&\ldots&\Psi_{p}(\mathbf{X_{1}})\\ 1&\Psi_{1}(\mathbf{X_{2}})&\ldots&\Psi_{p}(\mathbf{X_{2}})\\ \vdots&\vdots&\ddots&\vdots\\ 1&\Psi_{1}(\mathbf{X_{N_{s}}})&\ldots&\Psi_{p}(\mathbf{X_{N_{s}}})\end{array}  \right],\mathbf{\alpha}=\left[ \begin{array}{c}\alpha_{1}\\ \alpha_{2}\\ \vdots\\ \alpha_{N_{s}}\end{array} \right],\mathbf{e}=\left[ \begin{array}{c}e_{1}\\ e_{2}\\ \vdots\\ e_{N_{s}}\end{array} \right] \tag{6}\]  
The unknown coefficients can be expressed as in Equation (7) with the ordinary least square method:  
\[\mathbf{\alpha}=\left(\mathbf{\Psi}^{T}\mathbf{\Psi}\right)^{-1}\mathbf{\Psi}^{T} \mathbf{y} \tag{7}\]  
With the calculation of unknown coefficients, the estimation for a new design variable can be obtained as in Equation (8).