## 2 Methods
### The Low-Rank Approximation Method  
The LRA method is constructed as an alternative to the PCE method to overcome the curse of dimensionality for high-dimensional problems ([PERSON] et al., 2017). As discussed in the previous section, in the PCE methodology, the response of the problem is expanded onto a basis of orthonormal multivariate polynomials. LRA instead relies on a tensor product form of multivariate basis, which accelerates the solution procedure of the unknown coefficients. In this study, LRA implementations are conducted using UQLab, an open-source MATLAB toolbox for uncertainty quantification ([PERSON] et al., 2017). A computational model \(\mathcal{M}^{LRA}\), which represents a surrogate, maps the input variables to the response of the system. \(\mathcal{M}\) represents the physical behaviour of the system outputs, which are denoted as \(\mathbf{Y}=\{Y_{1},\ldots,Y_{N}\}\) and correspond to the variables \(\mathbf{X}=\{X_{1},\ldots,X_{M}\}\). By considering a rank-one function of the input vector \(\mathbf{X}\) in Equation (10),  
\[w(\mathbf{X})=\prod_{i=1}^{M}v^{(i)}(X_{i}) \tag{10}\]  
where \(w\) is product of univariate component of the input variables and the LRA model in general form is given in Equation (11):\[\mathcal{M}(\mathbf{X})\approx\mathcal{M}^{LRA}(\mathbf{X})\cong\sum_{l=1}^{R}b_{l} \Big{(}\prod_{i=1}^{M}v_{l}^{(i)}(X_{i})\Big{)} \tag{11}\]  
In this expression of the model, \(v_{l}^{(i)}\) are univariate functions of \(X_{i}\) in the \(l^{th}\) rank-one component and \(\{b_{l},l=1,2,\ldots,R\}\) are scalars defined as weighing factors. To approximate the response of the system, univariate functions \(v_{l}^{(i)}\) are expanded onto the polynomial basis \(\{\psi_{k}^{(i)},k\in\mathbb{N}\}\) that is orthonormal with respect to \(f_{x_{i}}\). \[\langle\psi_{j}^{(i)},\psi_{k}^{(i)}\rangle=\int_{\mathcal{D}_{x_{i}}}\psi_{j} ^{(i)}\psi_{k}^{(i)}x(i)f_{x_{i}}(x_{i})dx_{i}=\delta_{jk} \tag{12}\]  
\(\mathcal{D}_{x_{i}}\) represents a particular input variable while \(\delta_{jk}\) is the Kronecker delta, which is 1 if \(j=k\) and otherwise equal to 0. Accordingly, the polynomial basis representation leads to the equation,  
\[v_{l}^{(i)}(X_{i})=\sum_{k=0}^{p_{i}}z_{k,l}^{(i)}\psi_{k}^{(i)}(X_{i}) \tag{13}\]  
where \(\psi_{k}^{(i)}\) is the \(k\)-th degree univariate polynomial in the \(i\)-th input variable, \(p_{i}\) is the maximum degree of \(\psi_{k}^{i}\), and \(z_{k,l}^{i}\) is the coefficient of \(\psi_{k}^{i}\) in the \(l\)-th rank-one term. Then, using this expression in the LRA model in Equation (14),  
\[\mathcal{M}^{\text{LRA}}(\mathbf{X})=\sum_{l=1}^{R}b_{l}\Bigg{(}\prod_{i=1}^{ M}\Bigg{(}\sum_{k=0}^{p_{i}}z_{k,l}^{(i)}\psi_{k}^{(i)}(X_{i})\Bigg{)}\Bigg{)} \tag{14}\]  
the approximation for the system outputs is obtained. Two stages are needed to construct LRA; one is to define the polynomial basis in each dimension and the other is to determine the polynomial coefficients. As indicated in the previous section, Legendre polynomials are used for uniformly distributed inputs, and Hermite polynomials are used for normally distributed inputs. While several algorithms have been implemented for computing unknown coefficients in the related literature, in this work the alternated least-squares (ALS) minimisation scheme is used [41]. The key feature of the ALS minimisation method is that the coefficients are obtained within each dimension while keeping the coefficients in remaining dimensions constant, thus accelerating the solution process. The construction of the LRA methodology is based on both correction and updating steps in which the rank-one tensor is created in correction and the weighting factors are subsequently determined in the updating stages [12]. In the correction stage, assuming that \(\mathcal{R}_{r}\) represents the residual after the \(r\)-th iteration,  
\[\mathcal{R}_{r}(\mathbf{X})=\mathcal{M}(\mathbf{X})-\mathcal{M}_{r}^{LRA}( \mathbf{X}) \tag{15}\]  
the solution procedure is initiated by \(\mathcal{M}_{0}^{LRA}(\mathbf{X})=0\).