## 2 Materials and Methods
### Uncertainty Quantification
#### 2.3.2 Sparse Polynomial Chaos Expansion  
Not all uncertain input parameters carry equal weight in influencing the variance of the output. This inherent sparsity in the relationship between inputs and outputs can result in a substantial reduction in the number of terms needed in the expansion. Moreover, many models exhibit a pattern where they can be effectively represented by main effects and low-order interactions, adhering to the sparsity of effects principle. Consequently, a significant portion of coefficients in the polynomial chaos expansion (PCE) may approximate to zero [36]. To address this challenge and mitigate the computational cost associated with constructing a conventional PCE, an approach is to detect non-negligible terms in the PCE before its actual construction. To this end, we employed the sparse polynomial chaos expansion (sparse PCE) algorithm, as described in [PERSON] et al. 's work [36]. This approach efficiently identifies and retains the most influential terms, substantially reducing the computational burden while maintaining the accuracy necessary for robust uncertainty quantification. Hence, this algorithm focuses on retaining only the most impactful coefficients and their associated polynomials in the truncated series. For a comprehensive visual representation of this method, readers can refer to the schematic diagram available in [PERSON] et al. 's work [36]. The initial step of the algorithm involves creating an initial design of experiment, denoted as \(\chi~{}=~{}[\xi^{1},\xi^{2},...,\xi^{n}]\), where the user-defined constant \(n\) determines its size. The experiment's design is evaluated using the true model, and the resulting output for the quantity of interest is stored in \(\mathbf{y}~{}=~{}[y_{1},y_{2},...,y_{n}]^{T}\). In the first forward step, the algorithm creates and evaluates \(P+1\) independent one-predictor regression models using individual basis functions. The assessment is performed by solving a least-squares problem. To select the preferred basis function \(\mathbf{\
abla}_{j*}\) out of the \(P+1\) one-predictor regression models, the following criterion is applied:  
\[j*~{}=~{}\text{argmax},\frac{|\hat{u}_{j}|}{\sqrt{\text{Var}\big{[}\hat{u}_{j} \big{]}}},~{}~{}j~{}=~{}0,...,P, \tag{3}\]  
where \(\hat{u}_{j}\) represents the estimate of the coefficient and \(\text{Var}\big{[}\hat{u}_{j}\big{]}\) corresponds to its variance. This criterion favors coefficients with high weights (i.e., high estimates) in most of the samples (i.e., low variances), as they carry the most significant information. This selection procedure ensures effectiveness and robustness. The best-performing one-predictor model is stored in the final regression model, and it is removed from the pool of basis functions. The residual \(\hat{\mathbf{e}}\) is updated based on the difference between the deterministic model results \(\mathbf{y}\) and the results from the final regression model at the design of experiment points \(\hat{\mathbf{y}}\):  
\[\hat{\mathbf{e}}~{}=~{}\mathbf{y}-\hat{\mathbf{y}}. \tag{4}\]  
Following the first forward step, the stepwise regression algorithm enters a forward-backward loop. The forward step is similar to the first forward step, except that the one-predictor regression models are fitted on the residual \(\hat{\mathbf{e}}_{i}\) at iteration \(i\). This fitting on the residual enables the models to capture the effects that are still missing in the final regression model. After each forward step, a backward step is initiated. In this backward step, the confidence intervals of the regression coefficients included in the regression model are evaluated to determine if the importance of the coefficients already in the model is affected by the entry of a new coefficient.