## 3 Results
### Steel Column Limit State Function  
We consider the 10-dimensional parametric limit state function:  
\[g\left(\mathbf{y}\right)=F_{\mathrm{s}}-P_{\mathrm{t}}\left(\frac{1}{2 BD}+\frac{ F_{0}E_{\mathrm{b}}}{BDH\left(E_{\mathrm{b}}-P_{\mathrm{t}}\right)}\right), \tag{17}\]  
which relates the reliability of a steel column to its cost [51; 52]. On the right hand side of Equation (17), \(P_{\mathrm{t}}=P_{\mathrm{d}}+P_{1}+P_{2}\) is the total load and \(E_{\mathrm{b}}=\frac{\pi^{2}EBDH^{2}}{2L^{2}}\) is known as the Euler buckling load. Thereby, the parameter vector \(\mathbf{y}\) consists of ten parameters, namely the yield stress \(F_{\mathrm{s}}\) (in MPa), the dead weight load \(P_{\mathrm{d}}\) (in N), the variable loads \(P_{1}\) and \(P_{2}\) (in N), the flange breadth \(B\) (in mm), the flange thickness \(D\) (in mm), the profile height \(H\) (in mm), the initial deflection \(F_{0}\) (in mm), Young's modulus \(E\) (in MPa), and the column length \(L\) (in mm).  
In the original setting given in [51; 52], \(P_{1}\) and \(P_{2}\) are modeled with Gumbel distributions, \(E\) follows a Weibull distribution, \(F_{\mathrm{s}}\), \(B\), \(D\), and \(H\) are log-normally distributed, and \(P_{\mathrm{d}}\) and \(F_{0}\) are normally distributed. In this numerical experiment, we replace all log-normal and normal distributions by truncated normal distributions and use the Gumbel distribution to model the remaining three parameters. The corresponding distributions are given in Table 1. With the exception of the parameter \(L\), for which the truncation limits are set to \(\mu\pm 4\sigma\), all other truncated normal distributions have support in \([\mu-3\sigma,\mu+3\sigma]\). For all parameters that follow a truncated normal distribution, \(\mu\) and \(\sigma^{2}\) coincide with the original mean and variance values given in [51; 52]. Accordingly, the location and scale parameters of the Gumbel distributions are chosen such that the original mean values and variances given in [51; 52] are preserved.  
Figure 2: Cost-error relation for the approximations of the borehole model. The approximations are constructed with a dimension-adaptive weighted Leja interpolation algorithm [37] and with a degree-adaptive LAR-gPC (least angle regression-generalized polynomial chaos) algorithm [9; 50]. The size of the validation sample is \(Q=10^{5}\). The reference expected value is computed with quasi-MC (Monte Carlo) integration based on a Sobol sample size equal to \(10^{8}\): (**a**) RMS (root-mean-square) validation error, (**b**) Expected value relative error.  
We approximate the steel column function Equation (17) using the dimension-adaptive weighted Leja interpolation and the degree-adaptive, LAR-based gPC method. For both approximations and for both error metrics in Equations (14) and (15), the error-cost relation is presented in Figure 3. Once more, the Leja interpolation method performs very well for the given input PDFs. However, contrary to the results of Section 3.3, no advantage is observed over the LAR-gPC method. With respect to the \(\epsilon_{\text{RMS}}\) error, the performance of both methods is comparable. This can be probably attributed to a reduced regularity of the QoI defined in Equation (17), in comparison to the QoI given in Equation (16). The Leja interpolation method has a slight edge for costs greater than 200 simulations, however, the difference is minor. With the exception of costs greater than 800 simulations, the LAR-gPC method is superior to the Leja interpolation in terms of the \(\epsilon_{\text{rel,E}}\) error. Overall, the weighted Leja interpolation is again able to yield accurate approximations and statistics and it may be regarded as competitive to the LAR-gPC method.