# Multi-Fidelity Gradient-Based Strategy for Robust Optimization in Computational Fluid Dynamics
## 5 Results
### RDO Results  
In this section, the UQ methods are coupled with a SMOGA RDO optimizer. The objective is to investigate how different approximations of the statistical models of \(J\), i.e., of the RDO cost functions, may affect the RDO process. The BK and GEK UQ solvers were used in the same setting as Section 5.1, i.e., they are trained for each new design by using 60 and 15 samples in the uncertain space, respectively. This choice provides a good tradeoff between accuracy and computational cost. More sophisticated approaches involving an adaptive refinement of the UQ  
Figure 4: Global sensitivity analysis with Sobol indexes. and will be the object of future work. Secondly, an MF SMOGA RDO is carried out, and the results are compared with single fidelity RDO. For the single fidelity surrogates, an initial LHS DOE of \(10\times n_{des}=40\) UQ runs was carried out prior to the first generation. Then, the first population was randomly initialized in the design space, and the GA was allowed to evolve over 80 generations. MOEI infills were conducted every five generations. For the MF surrogate, the initial DOE contained 40 LF continuous MoM and 10 HF discrete GEK UQ runs. Low-fidelity MOEI infills were added every five generations except for the last infills, for which the tolerance criterion suggested an HF infill. The evolution of the SMOGA search in the objective space \(E[J]\) vs. \(var[J]\) is depicted in Figure 5 for various RDO optimizers: for each strategy, designs belonging to the first generation, the generations after the first and the second MOEI infill, and the last one are highlighted in red, while all other designs are depicted in grey. For the present inverse design problem, the Pareto front collapses onto a single point corresponding to simultaneous minimization of both objectives \(E[J]\) and \(var[J]\). The progression toward the optimal point is similar for all UQ techniques and optimization surrogates. For more clarity, Figure 6 shows the locations of selected GA populations in the objective space. For each technique, four subplots are provided showing in red all points calculated respectively at the SMOGA first generation, at the generation after the first MOEI infill, at the generation after the second MOEI infill, and at the last generation. Moreover, in grey, as a background for each subplot, the reader can find all points calculated with each technique during the SMOGA optimization. It appears that kriging, GEK, the MoM, and MF evolve similarly, as they quickly reach the area near the optimum after only two MOEI infills. Figure 5: Objective space. Figure 6: Evolution of the convergence for BK, GEK, MoM, and MF. Red color highlights designs belonging to the first generation, generations after the first and the second MOEI infill, and the last one. Other designs are in grey color. The optimal solutions calculated with each technique are reported in Table 4. Despite the different accuracy of the UQ techniques in use, all RDO loops converge to very similar solutions identified through the expected values of the four design parameters \([a,b,c,d]\), which differ only on the sixth decimal digit or less for BK, GEK, and MF and on the fifth decimal digit for the MoM. The table also displays the values of the objective functions computed on the surrogate models. BK-BK, GEK-BK, and MF surrogates lead to close estimates of the objective functions, with discrepancies of less than 1% on the mean and less than 2% for the variance. The CPU cost of the different RDO algorithms on a single processor personal computer are reported in the same table. Due to the reduced number of samples used in the UQ runs, the GEK algorithm allows gaining a factor of two to 2.5 over the BK. A further reduction of a factor of 15 or larger is obtained using the MoM, but caution must be taken in systematically preferring such a method for RDO problems, because of its lower accuracy.