# Multi-Fidelity Gradient-Based Strategy for Robust Optimization in Computational Fluid Dynamics
## 5 Results
### Preliminary Assessment of the UQ Methods  
We select one of the nozzle geometries in the design space by assigning the geometric coefficients' fixed normal pdfs with the standard deviation equal to 1% of the mean. The pdf parameters for the geometric variables are provided in Table 2. The operation and thermodynamic parameters are assigned the same pdf as in Table 1. The pdfs are sampled in order to build BK and GEK surrogates, while the MoM is applied by computing the Qol and its gradient at the expected value of the input parameters. A summary of the UQ results is given in Table 3, where we report the approximated mean and variance of the QoI \(J\) according to the various UQ methods. A reference calculation based on MC integration over \(10^{5}\) samples is carried out to provide a reference solution. For the present inexpensive test problem, the CPU time required for MC sampling is approximately \(3\times 10^{6}\) seconds on a personal computer having an Intel(R) Xeon(R) CPU E5-1620 v3 at 3.50 GHz. Results are also reported for MC sampling on the BK surrogate. The latter uses 60 function evaluations at points selected according to a Latin hypercube sampling (LHS) of the six-dimensional uncertain space. The sample size corresponds to the empirical rule \(N=10n_{unc}\). This is already sufficient to achieve extremely low errors with respect to the MC mean and variance, while reducing the overall computational cost of the UQ by three orders of magnitude. The GEK UQ based on the discrete adjoint solver provides an accuracy similar to BK by using only 15 samples. Despite the additional adjoint solves for gradient computations, the computational cost is reduced by more than 1/3 with respect to BK. The accuracy of GEK is confirmed by inspection of Figure 3, showing the full empirical pdfs of \(J\) computed with PC, BK, and GEK. A very good agreement between MC, BK, and GEK is observed. The accuracy is less satisfactory for the continuous adjoint approach, due to the less accurate computation of gradients. This reflects numerical errors introduced by the discretization of the adjoint equations and the treatment of boundary conditions, which are not completely consistent with the discretization errors introduced by the direct CFD solver. On the other hand, the continuous adjoint solver is developed independently on the direct solver, and in this sense, it is non-intrusive. The computational cost of GEK samples using the direct or continuous adjoint is approximately the same. The MoM method is obviously less accurate than the other UQ solvers. Nevertheless, the errors on both the computed mean and variance remain very reasonable, despite the presence of a shock in the divergent nozzle(where the Taylor-series expansion is not defined) and the relatively large uncertainty range on the reservoir pressure (approximately 20%). The shock is in practice regularized by numerical diffusion both in the direct and the adjoint solvers, while the uncertainty ranges of all other parameters are reasonably small. While the discrete and continuous-adjoint MoM calculations provide strictly the same results for \(E[J]\) (which does not use gradient information), they predict \begin{table}
\begin{tabular}{c c} \hline \hline
**Design Parameter** & **pdf** \\ \hline \(a\) & \(\sim\)\(N(1.75,0.0175)\) \\ \(b\) & \(\sim\)\(N(0.699,0.00699)\) \\ \(c\) & \(\sim\)\(N(0.8,0.008)\) \\ \(d\) & \(\sim\)\(N(4.00,0.004)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Design parameters for the quasi-1D nozzle used for the assessment of the UQ techniques. Figure 3: Assessment of UQ methods: approximate pdf of QoI \(J\) for MC, BK, and GEK.