## 4 Numerical Results
### Analytical Test Cases
#### 4.1.1 Problem 1: Park (1991) Function (4-D)  
The Park (1991) function is a 4-D benchmark problem. It has been extended by [PERSON] (2008) for multi-fidelity problems, and the high- and low-fidelity versions are given in Equations (27) and (28), respectively. For the presented problem, the design variable domain is defined as \(x_{i}\in[0,1)\).  
\[f_{hf}(\mathbf{x})=\frac{x_{1}}{2}\Bigg{[}\sqrt{1+\big{(}x_{2}+x_{3}^{2}\big{)} \frac{x_{4}}{x_{1}^{2}}}-1\Bigg{]}+(x_{1}+3x_{4})\exp[1+\sin(x_{3})] \tag{27}\]  
\[f_{lf}(\mathbf{x})=\bigg{[}1+\frac{\sin(x_{1})}{10}\bigg{]}f(\mathbf{x})-2x_ {1}+x_{2}^{2}+x_{3}^{2}+0.5 \tag{28}\]  
Uncertainty quantification is performed using a surrogate model, and model training is conducted to determine the best parameters for the surrogate model to be used.  
A surrogate model was created for single and multi-fidelity versions of the PCE and the LRA methods using different numbers of high-fidelity simulations results. The global and local accuracy metrics calculated at \(10^{5}\) test points and the results of the performed surrogate models are demonstrated in Figure 3.  
As is observed from Figure 3, an accurate surrogate model can be established by using the LRA method with only 10 high-fidelity simulations results. In contrast, when  
Figure 3: Surrogate model error metrics for [PERSON] (1991) function (\(N_{lf}=35\)).  
the PCE method is considered, a locally and globally accurate surrogate model can be established within approximately 35 high-fidelity simulations results. For the same simulation conditions (i.e., sampling strategy and number of simulations results), the PCE method therefore requires three times more data to achieve similar accuracy. In addition, by using the MF-PCE and MF-LRA methods, in which 35 low-fidelity simulations results are used, a multi-fidelity surrogate model can be created with the same number of high-fidelity simulations results. However, when few high-fidelity simulations results are used, the MF-LRA method gives better results for this problem. It is observed that, as the number of high-fidelity simulations increases, the MF-PCE method gives slightly more accurate results both globally and locally in comparison with the MF-LRA method. Single-fidelity simulations results demonstrate that a multi-fidelity surrogate model can be established with the MF-LRA method by using a smaller amount of low-fidelity simulations data. Moreover, for normalised \(RMSE\) results, one can conclude that, even the single-fidelity version of LRA performs similarly to the multi-fidelity type of models. In addition, according to the \(NMAE\), it can be inferred from the results that single-fidelity PCE gives erroneous results whereas MF-PCE gives competing results with MF-LRA.  
Uncertainty quantification was performed by assuming that the variables defined in the Park equation change uniformly in the range of [0,1]. The probability density function (PDF) for the surrogate models, where 20 high-fidelity results are used to observe the difference between the methods, are presented in Figure 4.  
The PDF values are calculated by the MC method using \(10^{7}\) sampling points and shown in Figure 4. It is observed in the figure that the results of the LRA and MF-LRA methods are close to the MC method performance. Although the surrogate model implementation results for the MF-PCE method produces similar accuracy values as the LRA and MF-LRA surrogate models, the behaviour of the confronted surrogates in terms of probability density function is quite different compared to the MC method. With regards to the presented results, there are significant differences between the PCE and the MC methods in terms of the uncertainty quantification metrics.