## 3 Results  
In this section, we consider models with relatively atypical input distributions, for which the application of interpolation-based UQ methods is not straightforward. In particular, we consider input RVs following truncated normal and Gumbel distributions, the latter also known as the generalized extreme value distribution of type 1 or the log-Weibull distribution. We denote the truncated normal distribution with \(\mathcal{TN}\left(\mu,\sigma^{2},l,u\right)\), where \(\mu\) and \(\sigma^{2}\) refer to the mean value and the variance of a normal distribution, while \(l\) and \(u\) are the lower and upper truncation limits. The Gumbel distribution is denoted with \(\mathcal{G}\left(\ell,\beta\right)\), where \(\ell\) is the location parameter and \(\beta\) the scaling parameter. Dedicated nested interpolation nodes are not readily available with respect to the corresponding input PDFs, therefore, interpolation-based UQ methods have not been considered so far in the literature for such random inputs. Here, we compute Leja nodes tailored to truncated normal and Gumbel distributions simply by applying Equation (10) with weight functions that are equal to the corresponding PDFs. Exemplarily, Figure 1 shows the first 10 computed Leja points with respect to two truncated normal and two Gumbel distributions, each with different distribution parameters.  
Based on the numerical results presented in this section, we demonstrate that weighted Leja interpolation can reliably be used for such atypical input PDFs, in terms of both approximation  
Figure 1: A total of 10 first univariate Leja nodes with respect to different truncated normal and Gumbel probability distributions. The truncated normal distributions are denoted with \(\mathcal{TN}\left(\mu,\sigma^{2},l,u\right)\), where \(\mu\) and \(\sigma^{2}\) refer to the mean value and the variance of a normal distribution, while \(l\) and \(u\) are the lower and upper truncation limits. The Gumbel distributions are denoted with \(\mathcal{G}\left(\ell,\beta\right)\), where \(\ell\) is the location parameter and \(\beta\) the scaling parameter. The annotation above each node denotes the order of appearance in the Leja sequence, where “0” refers to the initial node: (**a**) \(\mathcal{TN}\left(\mu=0,\sigma^{2}=1,l=0,u=3\right)\), (**b**) \(\mathcal{TN}\left(\mu=0,\sigma^{2}=1,l=-2,u=3\right)\), (**c**) \(\mathcal{G}\left(\ell=3,\beta=4\right)\), (**d**) \(\mathcal{G}\left(\ell=0.5,\beta=2\right)\).  
and UQ. For further verification, we compare the performance of the weighted Leja interpolation against gPC approximations with numerically constructed orthogonal polynomials [24]. All models feature multiple input parameters, hence, we use adaptive algorithms resulting in sparse approximations. The dimension-adaptive Algorithm 1 based on weighted Leja nodes is implemented in our in-house developed DALI (Dimension-Adaptive Leja Interpolation) software [14; 37]. The sparse gPC approximations are constructed with a well-established, degree-adaptive algorithm based on least angle regression (LAR) [9] and implemented in the UQLab [50] software. Quasi-random experimental designs based on Sobol sequences are used in the LAR-gPC approach, while numerically constructed orthogonal polynomials are used to tackle the input PDFs [24].