# Multi-Fidelity Gradient-Based Optimization for High-Dimensional Aeroelastic Configurations
## 3 Results
### Summary and Comparison
This is due to the shape parameterization, as evidenced in the stress contours. Thus, more designable sections across the wing (as well as variable dihedral) may potentially prevent this. In order to quantify the benefits of the multi-fidelity optimizer compared to its single-fidelity counterpart, Table 9 lists the optimization costs and savings for each shape-only case. To achieve this, the best optimum from either optimizer is first identified to serve as the \"true\" optimum. This yields a total percent reduction in penalty-constrained objective when compared to the starting design. Specific percentages of this total reduction, ranging from 50 to 99%, are then chosen. For each optimizer, the number of high-fidelity function (and gradient) evaluations required to reach each reduction is then identified (for example, for rigid P1a, each optimizer required three function calls to achieve 50% of the total objective reduction). The cost savings associated with using the multi-fidelity optimizer compared to single-fidelity can then be computed as a percentage of the single-fidelity optimizer cost. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Case** & **Model** & **Threshold (\%)** & **Reduction (\%)** & **NF (SF)** & **NF (MF)** & **Savings (\# Evals)** & **Savings (\% of SF)** \\ \hline \multirow{6}{*}{P1a} & & 50.0000 & 45.5603 & 3 & 3 & 0 & 0.0000 \\  & & 75.0000 & 68.3404 & 7 & 5 & 2 & 28.5714 \\  & & 90.0000 & 82.0085 & 8 & 5 & 3 & 37.5000 \\  & & 95.0000 & 86.5645 & 14 & 9 & 5 & 35.7143 \\  & & 98.0000 & 89.2981 & 15 & 12 & 3 & 20.0000 \\  & & 99.0000 & 90.2093 & 16 & 16 & 0 & 0.0000 \\ \cline{2-8}  & & 50.0000 & 47.5303 & 3 & 3 & 0 & 0.0000 \\  & & 75.0000 & 71.2954 & 4 & 4 & 0 & 0.0000 \\  & & 90.0000 & 85.5545 & 5 & 5 & 0 & 0.0000 \\  & & 95.0000 & 90.3075 & 7 & 6 & 1 & 14.2857 \\  & & 98.0000 & 93.1593 & 11 & 8 & 3 & 27.2727 \\  & & 99.0000 & 94.1099 & 15 & 14 & 1 & 6.6667 \\ \hline \multirow{6}{*}{P1b} & & 50.0000 & 38.4992 & 5 & 4 & 1 & 20.0000 \\  & & 75.0000 & 57.7489 & 11 & 10 & 1 & 9.0909 \\  & & 90.0000 & 69.2986 & 13 & 12 & 1 & 7.6923 \\ \cline{1-1}  & & 95.0000 & 73.1486 & 15 & 16 & \(-1\) & \(-6.6667\) \\ \cline{1-1}  & & 98.0000 & 75.4585 & 19 & n/a & n/a & n/a \\ \cline{1-1}  & & 99.0000 & 76.2285 & 24 & n/a & n/a & n/a \\ \hline \multirow{6}{*}{P1b} & & 50.0000 & 46.6153 & 3 & 3 & 0 & 0.0000 \\  & & 75.0000 & 69.9230 & 3 & 3 & 0 & 0.0000 \\ \cline{1-1}  & & 90.0000 & 83.9076 & 6 & 4 & 2 & 33.3333 \\ \cline{1-1}  & & 95.0000 & 88.5691 & 8 & 5 & 3 & 37.5000 \\ \cline{1-1}  & & 98.0000 & 91.3660 & 11 & 6 & 5 & 45.4545 \\ \cline{1-1}  & & 99.0000 & 92.2983 & 15 & 7 & 8 & 53.3333 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Cost summary for airfoil optimization cases. For the rigid P1a case, the multi-fidelity optimizer provided cost savings from 75 to 98% of total reduction, ranging from 20 to roughly 40%. However, by the end of the optimization, roughly where a 99% reduction was met, there were no longer any cost benefits. This is generally consistent with the metrics shown in Figures 11d-13d, which suggest benefits early in the optimization process, but less as the optimization progresses. Depending on the trust region size, it may also suggest that the minimum trust region size could be increased so that the optimizer more quickly switches to high-fidelity only. The other rigid case, P1b, also showed cost savings for smaller thresholds, but was ultimately unable to achieve the 98 and 99% thresholds. Interestingly, the aeroelastic cases differed in that the savings appeared for somewhat larger thresholds. Potentially, this could mean that the different low-fidelity model is better correlated near the optimum, where shocks in the flow would be weaker. Figure 17 illustrates representative characteristics of the multi-fidelity optimization for airfoil design. Considering the comparison to the single-fidelity optimization in Figure 11 and Table 9, the multi-fidelity approach provided a cost benefit.