# Multi-Fidelity Gradient-Based Strategy for Robust Optimization in Computational Fluid Dynamics
## 5 Results
### Preliminary Assessment of the UQ Methods
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**UQ Method** & **CFD Solves** & **Adjoint Solves** & \(E[J]\) & \(\mathbf{var[J]}\) & \(\mathbf{err\%}\) \(E[J]\) & \(\mathbf{err\%}\) \(\mathbf{var[J]}\) & **Time (s)** \\ \hline MC & \(1.0\times 10^{5}\) & 0 & 3.508 & 0.6724 & - & 0.0\% & \(3.0\times 10^{6}\) \\ BK & 60 & 0 & 3.507 & 0.6724 & \(-0.03\%\) & 0.0\% & \(1.8\times 10^{3}\) \\ GEK (discrete adjoint) & 15 & 15 & 3.506 & 0.6724 & \(-0.05\%\) & 0.0\% & \(9.0\times 10^{2}\) \\ GEK (continuous adjoint) & 15 & 15 & 3.467 & 0.6972 & \(-1.16\%\) & 3.69\% & \(6.8\times 10^{2}\) \\ MoM (discrete adjoint) & 1 & 1 & 3.451 & 0.6464 & \(-1.63\%\) & \(-3.87\%\) & \(6.0\times 10\) \\ MoM (continuous adjoint) & 1 & 1 & 3.451 & 0.6757 & \(-1.63\%\) & 0.49\% & \(4.5\times 10\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Accuracy and computational efficiency of the UQ solvers. different results for the standard deviation. The slightly lower errors obtained for the continuous adjoint method is the effect of the compensation of errors for the case at hand. Since the MoM requires only one direct and one adjoint CFD calculation, its computational cost is essentially \(1/15\) of the GEK sampling. Overall, the first-order MoM provides a reasonably accurate estimate of the lower order statistics of the QoI and a very good tradeoff between cost and accuracy for the present problem. Its accuracy is however expected to decrease for more severe uncertainty ranges. For this reason, the MoM is categorized as a lower fidelity model than BK or GEK. The UQ results for MC, BK, and GEK can be used to carry out a global sensitivity analysis and to identify the random parameters contributing the most to the variance of the QoI by means of an analysis of variance (ANOVA) decomposition. Specifically, exact or surrogate-based MC samples are employed to calculate the Sobol indexes [71] in the parameter space defined by all uncertain inputs. For this purpose, we used the SALib Python library [72]. Only the discrete GEK results (reported in Figure 4) are considered here, the MC, BK, and continuous GEK methods leading to similar conclusions. The figure reports the first-order Sobol indexes with respect to each uncertain parameter, as well as the sum of the higher order indexes, corresponding to the interaction between parameters when these are changed simultaneously. Sobol total indexes are also included in order to further evaluate the relative importance of the various input parameters. The inlet total pressure \(P_{T,in}\) appears to be by far the most influential parameter, consistent with the much larger range of its pdf. Nevertheless, the interactions terms are very significant, due to the highly nonlinear nature of the compressible CFD problem, with geometric parameters \(c\) and \(d\) contributing by more than 10% to the total variance and parameters \(a\) and \(b\) by more than 2%. For this reason, in the RDO calculations of the following section, we treat all six input parameters as random variables.