# Multi-Fidelity Gradient-Based Strategy for Robust Optimization in Computational Fluid Dynamics
## 4 Robust Design Optimization Strategy
### Multi-Fidelity Methods for RDO  
Although the BK-based SMOGA allows a considerable reduction of function calls during the RDO loop, the overall computational cost remains significant for high-dimensional design and uncertain spaces. While the computational cost is higher for BK or GEK solvers than for MoM, the former may provide an accurate estimate of the QoI statistics, which is not the case for the first-order MoM. With the aim of achieving a compromise between the accuracy of kriging-surrogate UQ solvers and the computational efficiency of the MoM, in this section, we introduce an advanced SMOGA, based on a multi-fidelity surrogate model of the design space replacing the preceding external BK surrogate (MF-SMOGA). In the MF surrogate, the MoM UQ solver is used as the low-fidelity (LF) model, and the GEK UQ solver is the high-fidelity (HF) one. In the present implementation, an initial DoE is run at the first iteration of the SMOGA, where the HF and LF sampling points are generated independently. The auto-regressive correlation:  
\[f_{HF}(\boldsymbol{\xi})=\rho f_{LF}(\boldsymbol{\xi})+\delta(\boldsymbol{\xi}) \tag{22}\]  
is used for correcting the discrepancy between the LF and HF models. More specifically, we follow the implementation proposed in [57], and we use the LF model as a basis function for the regression term expression of a universal kriging model; therefore, the term \(m(\boldsymbol{\xi})\) in Equation (5) becomes:  
\[m(\boldsymbol{\xi})=\sum_{i}f_{i}(\boldsymbol{\xi})\beta_{i}+f_{LF}\beta_{\rho} \tag{23}\]  
where \(\beta_{\rho}\) is an estimate of the coefficient \(\rho\) of Equation (22) by means of a GP regression. Assuming that the LF and HF models are independent, the mean \(\mu_{HF}\) and variance \(\sigma_{HF}^{2}\) of the high-fidelity model are given by:  
\[ \begin{cases}\mu_{HF}=\rho\mu_{LF}+\mu_{\delta}\\ \sigma_{HF}^{2}=\rho^{2}\sigma_{LF}^{2}+\sigma_{\delta}^{2}\end{cases}  \tag{24}\]  
with \(\mu_{LF}\), \(\sigma_{LF}^{2}\), \(\mu_{\delta}\), and \(\sigma_{\delta}^{2}\) the means and variances of the LF model and of the discrepancy function \(\delta\), respectively. In order to improve the MF surrogate accuracy during SMOGA convergence, adaptive infill based on the MOEI criterion is used. In this case, however, either the LF or the HF model can be used for the infill. In the following calculations, we adopt the strategy of [42]: for the infill, priority is given to the less expensive LF model, and the HF one is used only when improvement achieved with the lower level of fidelity is below a given tolerance, \(tol=10^{-4}\) in the present calculations. Either way, re-sampling at the same location is avoided. Pseudo-code for the MF model is provided below in Algorithm 2; for more information about it, the reader is addressed to [57]. ```
1. Initialization:  LHS LF DOE with \(n_{LF,init}\) samples \(\mathcal{S}_{DOE,LF}=\left\{s^{(1)}_{LF},s^{(2)}_{LF},...,s^{(n_{LF,init})}_{LF}\right\}\)  LHS HF DOE with \(n_{HF,init}<<n_{LF,init}\) samples \(\mathcal{S}_{DOE,HF}=\left\{s^{(1)}_{HF},s^{(2)}_{HF},...,s^{(n_{HF,init})}_{ HF}\right\}\)  While \(i<n_{LF,init}\)  Select design \(s^{(i)}_{LF}\) in \(\mathcal{S}_{DOE,LF}\)  Run MoM solver to compute the statistics \(\mu(s^{(i)}_{LF})\), \(\sigma^{2}(s^{(i)}_{LF})\)  Train LF surrogates \(\mathcal{S}_{LF,\mu}(\mathcal{S}_{DOE,LF})\), \(\mathcal{S}_{LF,\sigma^{2}}(\mathcal{S}_{DOE,LF})\)  While \(i<n_{HF,init}\)  Select design \(s^{(i)}_{HF}\) in \(\mathcal{S}_{DOE,HF}\)  Run GEK solver to compute the statistics \(\mu(s^{(i)}_{HF})\), \(\sigma^{2}(s^{(i)}_{HF})\)  Use Equation (22) to construct MF surrogates: \(\mathcal{S}_{MF,\mu}(\mathcal{S}_{DOE,HF})\), \(\mathcal{S}_{MF,\sigma^{2}}(\mathcal{S}_{DOE,HF})\) 2.