## 5 Discussion and Conclusions
Since the surrogate modelling methods considered in the present study use orthogonal polynomials that show smooth behaviour in the design space, it can be observed from the \(NMAE\) metric that the success of capturing local features is low. These methods are generally preferred in uncertainty and sensitivity analysis due to their ability to deal with the general behaviour of the design space. However, because of their inefficient ability to capture local features, they are generally not preferred for optimisation. Considering the comparison of the multi-fidelity versions of the methods with the single-fidelity methods, when using the same number of high-fidelity analyses, the multi-fidelity surrogate versions produce more accurate results by making use of low-fidelity model information. When the high-fidelity simulations required to construct a surrogate model with similar accuracy using the LRA and MF-LRA methods are compared, it is observed that the number of high-fidelity simulations required in the LRA method is two to three times the number of simulations needed in the MF-LRA method. Considering the single- and multi-fidelity applications of the PCE method, the number of high-fidelity simulations required in the PCE method for low-dimensional problems is two to three times the number of simulations required by the MF-PCE method. In high-dimensional problems, it is observed that an accurate model cannot be obtained with the multi-fidelity PCE method. In this respect, a more accurate surrogate model can be established with fewer simulations in low- and high-dimensional problems with the LRA method compared to the PCE method. Moreover, with the multi-fidelity version of the LRA method, accurate surrogate models are constructed with only few simulations results. The number of unknowns increases exponentially with the problem dimension in both single- and multi-fidelity PCE methods. Thus, an accurate surrogate model can be established for low-dimensional problems. However, in high-dimensional problems, an accurate surrogate model cannot be established with a low number of simulations. Moreover, the accuracy level of the low-fidelity surrogate model directly affects the ability to model the response of the multi-fidelity surrogate model. The importance of the accuracy level of the low-fidelity model is especially noticeable when the number of high-fidelity analyses are increased in the multi-fidelity surrogate, and it causes the multi-fidelity model to have lower accuracy values than the single-fidelity model, as observed in the Sobol'-Levitan benchmark function and 32-D sonic boom uncertainty quantification problem. When examining the results of the uncertainty analysis tackled with different methods, it is observed that the PDF results of the LRA method are in good agreement with the MC results, although a small number of the analyses results are used in all applications of the single- and multi-fidelity LRA method. In many applications, when using the same number of results of the analyses, the PDF values obtained by the PCE method produce incompatible results with the MC method. A remarkable point in the results is that the single- and multi-fidelity LRA methods can model the tail region of the PDF very accurately, which has crucial importance in reliability studies. Small probabilities of exceedence used in the reliability analysis can be obtained more accurately by the LRA method which makes the LRA method more advantageous than the PCE method in reliability studies. In addition, the success of the LRA method in modelling the general behaviour of the computational model can be observed from the mean, standard deviation, skewness, and kurtosis values shared in the results of the engineering problem, which can also be observed in PDF functions. While the mean and standard deviation are obtained correctly with the single- and multi-fidelity PCE method, parameter estimates that explain the behaviour of the PDF function such as skewness and kurtosis contain high errors.