## 5 Discussion and Conclusions  
In the uncertainty quantification studies, the required number of realisations become a computational burden due to the excessive number of unknown coefficients involved. Depending on the selected procedure for modelling the response, the curse-of-dimensionality  
Figure 16: Comparison of probability density function (**a**) Results of all surrogate models (**b**) Results of LRA and MF-LRA (\(N_{hf}=500\), \(N_{lf}=1000\)). Figure 15: Surrogate model error metrics for sonic boom problem 2 (\(N_{lf}=1000\)). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Mean (dB)** & \multicolumn{2}{c}{**Standard Dev.**} & \multicolumn{1}{c}{**Skewness**} & \multicolumn{1}{c}{**Kurtosis**} \\ \hline MC & 83.5511 & 0.5062 & \(-\)0.1590 & 3.1699 \\ PCE & 83.5649 & 0.7311 & \(-\)0.0565 & 3.8712 \\ LRA & 83.5486 & 0.5000 & \(-\)0.0357 & 2.8596 \\ MF-PCE & 28.9314 & 40.087 & \(-\)0.6869 & 5.0330 \\ MF-LRA & 83.5507 & 0.5010 & \(-\)0.0710 & 2.9130 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison of descriptive statistics of sonic boom loudness. may cause time-consuming computations and out-of-memory problems. Considering those drawbacks, the Low-Rank Approximation (LRA) method becomes a promising surrogate model, in which the unknown coefficients grow only linearly with respect to the input dimensions. In the present study, the LRA method is investigated and compared with widely used PCE methodology within an application-oriented framework. Their ability to capture the uncertainty propagation of sonic boom prediction with four uncertain variables is assessed for low-dimensional engineering problems. The LRA method is observed to generate more accurate results compared to the PCE method, even for higher dimensions and when very limited amount of data are available. Thus, the multi-fidelity extension of Low-Rank Approximation method is proposed by the authors and applied to uncertainty quantification of high-dimensional supersonic aircraft design. The methodology already derived is demonstrated on several multi-fidelity uncertainty quantification benchmarks and applied to high-dimensional real-world engineering problems. Since the quantification of the uncertainty strictly depends on the success of the surrogate, the LRA and PCE methods are first compared in terms of surrogate modelling capabilities and through local and global accuracy metrics, with the purpose of displaying the surrogate performance of the LRA and PCE methods. Moreover, to demonstrate the efficiency of proposed multi-fidelity extensions for improving the convergence history, single-fidelity versions are taken into consideration for comparison. As it is indicated by the accuracy metrics of the single- and multi-fidelity surrogate models for the benchmark problems, an accurate surrogate model can be achieved with a small number of high-fidelity simulations by using the LRA method. When using the PCE method, however, many more high-fidelity simulations are required to set up a single- or multi-fidelity surrogate model with a similar accuracy. The number of high-fidelity simulations required in the single-fidelity PCE method is more than two to three times the number of high-fidelity simulations required in the LRA method for the 4-D Park and 4-D sonic boom uncertainty quantification implementations. However, as the problem dimension increases, it is observed that this ratio is five times in the 8-D borehole function, and much higher than five times in the 30-D Sobol'-Levitan function and 32-D sonic boom uncertainty quantification problem. The performance difference is attributed to the fact that the LRA method contains fewer unknown coefficients than that of the PCE method due to the univariate polynomials that the LRA method employs. It is observed from the analytical test cases that the single- and multi-fidelity PCE and LRA methods are successful in capturing global features of the response.