## 3 Results
### Borehole Model  
We consider the 8-dimensional parametric function:  
\[g\left(\mathbf{y}\right)=\frac{2\pi T_{\text{u}}\left(H_{\text{u}}-H_{\text{I }}\right)}{\ln\left(\frac{r}{r_{\text{w}}}\right)\left(1+\frac{T_{\text{u}}}{T _{\text{l}}}+\frac{2 LT_{\text{u}}}{\ln\left(\frac{r}{r_{\text{w}}}\right)r_{ \text{w}}^{2}K_{\text{w}}}\right)}, \tag{16}\]  
which models the water flow through a borehole [51]. The water flow is given in \(\text{m}^{3}/\text{y}\) and the parameter vector is \(\mathbf{y}=\left(r_{\text{w}},r,T_{\text{u}},H_{\text{u}},T_{\text{l}},H_{ \text{l}},L,K_{\text{w}}\right)\), where \(r_{\text{w}}\) and \(r\) respectively denote the radius of the borehole and the radius of influence (in m), \(T_{\text{u}}\) and \(T_{\text{l}}\) the transmissivity of the upper and lower aquifer (in m\({}^{2}/\text{y}\)), \(H_{\text{u}}\) and \(H_{\text{l}}\) the potentiometric head of the upper and lower aquifer (in m), \(L\) the length of the borehole (in m), and \(K_{\text{w}}\) the hydraulic conductivity of the borehole (in m\(/\text{y}\)).  
Regarding the parameter distributions, we follow the setting given in [51], where specific value ranges and probability distributions are given for each parameter and use this information to recast all parameters as to follow truncated normal distributions. In this numerical experiment, the input parameters are modeled as follows.  
* The parameter \(r_{\text{w}}\) originally follows the normal distribution \(\mathcal{N}\left(\mu=0.1,\sigma=0.0161812^{2}\right)\). The distribution is now truncated to the range \(\left[0.05,0.15\right]\);
* The parameter \(r\) originally follows the log-normal distribution \(\mathcal{LN}\left(\mu_{\text{LN}}=7.71,\sigma_{\text{LN}}=1.0056\right)\). Therefore, the parameter \(r\) has a mean value equal to \(\exp\left(\mu_{\text{LN}}+\frac{\sigma_{\text{LN}}^{2}}{2}\right)\) and a variance equal to \(\left(\exp\left(\sigma_{\text{LN}}^{2}\right)-1\right)\)\(\exp\left(2\mu_{\text{LN}}+\sigma_{\text{LN}}^{2}\right)\). The corresponding truncated normal distribution is defined by these mean and variance values, as well as by the truncation range \(\left[100,50000\right]\)]
* The remaining parameters are originally uniformly distributed, such that \(T_{\text{u}}\in\left[63070,115600\right]\), \(H_{\text{u}}\in\left[990,1110\right]\), \(T_{\text{l}}\in\left[63.1,116\right]\), \(H_{\text{l}}\in\left[700,820\right]\), \(L\in\left[1120,1680\right]\), and \(K_{\text{w}}\in\left[9855,12045\right]\). Assuming a uniform distribution with support in \(\left[a,b\right]\), the corresponding truncated normal distribution is given as \(\mathcal{TN}\left(\mu=\frac{a+b}{2},\sigma^{2}=\frac{\left(b-a\right)^{2}}{12},l=a,u=b\right)\), i.e., the mean value and variance of the normal distribution correspond to those of the original uniform distribution, while the truncation limits coincide with the uniform distribution's support boundaries.  
The borehole model of Equation (16) is approximated by both the dimension-adaptive weighted Leja interpolation and the degree-adaptive LAR-gPC expansion. Figure 2 shows the error-cost relation for both approximations, with respect to the errors in Equations (14) and (15). Regarding the \(\epsilon_{\text{RMS}}\) error, Leja interpolation outperformed the LAR-gPC method by approximately one order of magnitude over the whole cost-range. Leja interpolation was also advantageous with respect to the \(\epsilon_{\text{rel},\mathbb{E}}\) error, again always outperforming the LAR-gPC method and stagnating much sooner. We note that this error stagnation was observed due to the fact that both methods reached the accuracy of the quasi-MC estimate. Therefore, for the case of the borehole model, the weighted Leja interpolation approach was not only found to perform well for the considered truncated normal input PDFs, but was also superior to the LAR-gPC method for both considered error metrics.