## 3 Results
### Meromorphic Function  
We consider the 16-dimensional meromorphic function:  
\[g\left(\mathbf{y}\right)=\frac{1}{1+\mathbf{w}\cdot\mathbf{y}}, \tag{18}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Distribution** \\ \hline \(F_{\text{s}}\) & \(\mathcal{TN}\left(\mu=400,\sigma^{2}=35^{2},l=295,u=505\right)\) \\ \(P_{\text{d}}\) & \(\mathcal{TN}\left(\mu=500000,\sigma^{2}=50000^{2},l=350000,u=650000\right)\) \\ \(P_{1}\) & \(\mathcal{G}\left(\ell=559495,\beta=70173\right)\) \\ \(P_{2}\) & \(\mathcal{G}\left(\ell=559495,\beta=70173\right)\) \\ \(B\) & \(\mathcal{TN}\left(\mu=300,\sigma^{2}=3^{2},l=291,u=309\right)\) \\ \(D\) & \(\mathcal{TN}\left(\mu=20,\sigma^{2}=2^{2},l=14,u=26\right)\) \\ \(H\) & \(\mathcal{TN}\left(\mu=300,\sigma^{2}=5^{2},l=285,u=315\right)\) \\ \(F_{0}\) & \(\mathcal{TN}\left(\mu=30,\sigma^{2}=10^{2},l=0,u=60\right)\) \\ \(E\) & \(\mathcal{G}\left(\ell=208110,\beta=3275\right)\) \\ \(L\) & \(\mathcal{TN}\left(\mu=7500,\sigma^{2}=7.5^{2},l=7470,u=7530\right)\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameter distributions of the steel column function.  
Figure 3: Cost-error relation for the approximations of the steel column function. The approximations are constructed with a dimension-adaptive weighted Leja interpolation algorithm [37] and with a degree-adaptive LAR-gPC algorithm [9; 50]. The size of the validation sample is \(Q=10^{5}\). The reference expected value is computed with quasi-MC integration based on a Sobol sample size equal to \(10^{6}\): **(a)** RMS validation error, **(b)** Expected value relative error.  
where \(\mathbf{w}\) is a vector of positive weights. The weight vector in Equation (18) is given by \(\mathbf{w}=\frac{\hat{\mathbf{w}}}{\hat{\mathbf{z}}\|\hat{\mathbf{w}}\|_{1}}\), where \(\hat{\mathbf{w}}=\left(1,0.5,0.1,0.05,0.01,\ldots,5\cdot 10^{-8}\right)\)[10]. The input vector \(\mathbf{y}\) takes values in the image space of the random vector \(\mathbf{Y}=\left(Y_{1},Y_{2},\ldots,Y_{16}\right)\), where the single RVs are given as:  
\[Y_{n}\sim\begin{cases}\mathcal{T}\mathcal{N}\left(\mu=0,\sigma^{2}=1,l=0,u=3 \right),&n=1,3,\ldots,15,\\ \mathcal{T}\mathcal{N}\left(\mu=0,\sigma^{2}=1,l=-3,u=0\right),&n=2,4,\ldots,16.\end{cases}\]  
While a normal distribution truncated exactly at its (untruncated) mean value is not very likely to be encountered in practical applications, the selected truncation limits result in very atypical PDFs which fit very well within the concept of arbitrary input PDFs.  
Figure 4 shows the error-cost relations corresponding to the error metrics of Equations (14) and (15) for both approximations of the meromorphic function in Equation (18), i.e., using the dimension-adaptive weighted Leja interpolation algorithm and the degree-adaptive LAR-gPC algorithm. The results resemble those of Section 3.3, i.e., the Leja interpolation method had a clear advantage over the LAR-gPC approximation with respect to both error metrics. Regarding the \(\epsilon_{\text{RMS}}\) error, the difference between the two approximations was typically greater than one order of magnitude, while it increased for an increasing computational budget. Regarding the \(\epsilon_{\text{rel},\mathbb{E}}\) error, the two methods showed a comparable performance for costs up to 100 simulations, however, the Leja interpolation was again clearly superior as the simulation budget increased. Hence, in this numerical experiment, the weighted Leja interpolation was able to accurately approximate a model with very atypical input PDFs, and provide accurate estimations of statistics. Moreover, it was found to have a clear edge over the well-established LAR-gPC method.