## 4 Numerical Results
### Analytical Test Cases
#### 4.1.3 Problem 3: Sobol'-Levitan (1999) Function (30-D)  
The Sobol'-Levitan (1999) function is an n-D single-fidelity benchmark problem as defined in Equation (31) [49]. The function is defined on the hypercube \(x_{i}\in[0,1]\) for all dimensions.  
\[\begin{split} f(\mathbf{x})=\exp\left[\sum_{i=1}^{d}b_{i}x_{i} \right]-I_{d}+c_{0},\text{ where }\\ I_{d}=\prod_{i=1}^{d}\frac{\exp(b_{i})-1}{b_{i}}\end{split} \tag{31}\]  
The value of the elements in the \(b\) vector \((b_{1},...,b_{d})\) indicates the significance of the corresponding \(x\) variables. The Sobol'-Levitan function is extended for multi-fidelity applications by defining two different \(b\) vectors for multi-fidelity applications. Within the scope of the study, the 30-D version of the Sobol'-Levitan (1999) function is used, and the b vectors used for the low- and high-fidelity versions are provided in Equation (32).  
\[\begin{split} b_{lf}&=[1.25\times ones(1,15),0.9 \times ones(1,15)],\text{ for low-fidelity version }\\ b_{hf}&=[1.25\times ones(1,15),1.0\times ones(1,15 )],\text{ for high-fidelity version }\end{split} \tag{32}\]  
A surrogate model is established with single- and multi-fidelity versions of the LRA and PCE methods using a different number of high-fidelity simulation results. The surrogate model accuracy metrics calculated using the \(10^{6}\) test data of the established surrogate models are given in Figure 7.  
As observed in Figure 7, despite the increase in the number of high-fidelity analyses, no improvement is observed in the accuracy values of the PCE method. The main reason for this is that the number of unknown coefficients that need to be calculated for the estimation of the PCE method is very high and it requires a large number of high-fidelity simulations results for the estimation of the unknown coefficients. Similarly, the MF-PCE method also has low accuracy values, as errors in the single-fidelity version result in decreased accuracy values of the MF-PCE method. It is observed from the accuracy metrics of the LRA method that a globally accurate surrogate model can be created with a small number of high-fidelity simulations results. It has been observed that, with the help of the widespread information obtained from the low-fidelity model, the surrogate model established with the MF-LRA method is more accurate than the LRA method with few high-fidelity simulations results. The inconvenience observed in this particular benchmark problem is that the established surrogate models have locally high erroneous values at least at one point in the design space.  
Figure 6: Comparison of probability density function (\(N_{hf}=50\), \(N_{lf}=200\)).  
Uncertainty analysis is carried out by assuming uniform distribution for the variables of the equation. The obtained PDF values for uncertainty analysis are shown in Figure 8. As observed in Figure 8, the PDF functions obtained by the PCE and MF-PCE methods are quite different than the PDF values obtained by the MC method. The PDF values obtained by the LRA and MF-LRA methods are similar to the MC results, and it is observed that these methods successfully model the tail regions of the PDF.