## 3 Computational Experiments  
Real-world engineering applications are expected to be solved within limited computational budgets. For uncertainty quantification studies, the number of simulations increases drastically when propagating the input uncertainties to an output response through time consuming simulation processes. On the contrary, very limited amount of high fidelity data may be available to be used in surrogate modelling and uncertainty quantification studies. In this study, the goal is to investigate the cost and accuracy performance of the LRA method in multi-fidelity modelling with respect to the PCE method; in order to isolate the effects of surrogate performance, all accompanying computational strategies are kept the same, including design of experiments, local/global accuracy metrics, UQ metrics, and fidelity cost assignment criteria. * Design of Experiments (DoE): Halton Sampling The key component of the surrogate modelling process is the design of experiments for the simulations. As already known from the associated studies, surrogate model predictions strictly depend on the applied design of experiments (DoE) set. In this study, several widely used DoE sampling types are demonstrated for a reliable performance assessment of the considered surrogates. To incorporate all the characteristics of the data over the sampling space, samples must be distributed homogeneously as much as possible. If this is achieved, the performance of the surrogate can be evaluated without the biased effect of the selected sampling strategy. With this in mind, Monte Carlo (MC) [43], Latin Hypercube (LHS) [44], Optimized Latin Hypercube (OLHS) [45] and Halton sequence (HS) sampling [46] strategies are compared in Figure 2 to visualise behavioural differences for several conventional sampling types. As indicated in the two-dimensional sampling strategy visualisations, Latin Hypercube sampling has the ability to fill the design space successfully. In contrast, Monte Carlo sampling may have some accumulated data areas that may result in learning deficits for the implemented surrogate model. The Halton sequence distributes the samples homogeneously over the design space. As the number of samples is increased, HS keeps the previous sampling set the same, while adding the new samples homogeneously distributed to the vacant regions of the design space. In the present work, Halton sequence is therefore selected as a DoE strategy in an effort to prevent potential bias effects from the sampling strategy. * Local/Global accuracy metrics: In order to compare the accuracy of surrogate models, error metrics are calculated using the predicted values of the surrogate models and their corresponding analytical results at the same test points. The proposed metrics are selected to offer a comprehensive assessment of the surrogates both in terms of local and global characteristics of the modelling ability of the real functions. These metrics include coefficient of determination (\(R^{2}\)), root-mean square error (\(RMSE\)), and maximum absolute error (\(MAE\)); their formulas and normalised versions are later given in Equations (22)-(24). The coefficient of determination is a statistical measure of how well the regression estimate converges to the actual data points. \(R^{2}\) takes a value between 0 and 1, with a high \(R^{2}\) value indicating that the model fits well with the data used. Using the test data-set, the formulation is expressed as follows, where \(\bar{y}\) is the mean of test data-set, \(y_{i}\) is the exact value of the function and \(\hat{y}_{i}\) represents the surrogate model prediction. \[R^{2}=1-\frac{\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i}^{N}(y_{i}-\bar{y })^{2}} \tag{22}\]  
The \(RMSE\) metric expresses the standard deviation of the prediction error and measures how well the predicted values match the actual values in absolute terms. The \(RMSE\) value ranges from 0 to \(\infty\), with smaller values indicating that the method makes a more accurate prediction.