# Multi-Fidelity Gradient-Based Strategy for Robust Optimization in Computational Fluid Dynamics
## 3 Uncertainty Quantification Methods
### Bayesian Kriging and Gradient-Enhanced Kriging
#### 3.1.2 Gradient Enhanced Kriging  
Bayesian kriging can be accelerated by adding gradient information. The formulation described hereafter is the form of so-called gradient-enhanced co-kriging, denoted GEK for simplicity, where the gradient data are added as covariables. This approach has been shown to be simpler and more robust for indirect gradient-enhanced kriging, where the gradient information is included in the surrogate by adding finite difference-based values at small distances from the sample locations, and then, a standard kriging is performed on the resulting augmented sample [22; 23; 24].  
For the sake of clarity, we first describe GEK in the one-dimensional case: \(J=J(\xi)\). Assuming that the derivatives of \(dJ/d\xi\) are known at the \(M\) sampling points, the observation vector is redefined by concatenating the gradient information to the function values:  
\[\mathbf{J}^{\star}=\left({J_{1}}^{\star},{J_{2}}^{\star},\ldots,{J_{M}}^{\star}, \frac{dJ}{d\xi^{\star}}\right|_{1}^{\star},\frac{dJ}{d\xi^{\star}}\right|_{2}^ {\star},\ldots,\frac{dJ}{d\xi^{\star}}\bigg{|}_{M}^{\star}\bigg{)}^{T} \tag{14}\]The covariance matrix is subsequently modified to account for the observational errors on the derivatives, also assumed to be uniform and uncorrelated. This results in a diagonal matrix of dimension \(2M\times 2M\):  
\[\mathbb{R}=diag(\underbrace{\epsilon^{2},\ldots,\epsilon^{2}}_{M\text{times}} \underbrace{\varsigma^{2},\ldots,\varsigma^{2}}_{M\text{times}}) \tag{15}\]  
where \(\varsigma^{2}\) is the variance of the (identically distributed) observation errors on the gradients. Finally, the prior covariance is reformulated as in [63]:  
\[\mathbb{P}=\begin{bmatrix}P_{00}&P_{01}\\ P_{10}&P_{11}\end{bmatrix} \tag{16}\]  
with \(P_{00}=cov(h_{ij},\theta)\),\(P_{01}=\frac{\partial cov(h_{ij},\theta)}{\partial h_{ij}}\),\(P_{10}=-\frac{\partial cov(h_{ij},\theta)}{\partial h_{ij}}\), and \(P_{11}=\frac{\partial^{2}cov(h_{ij},\theta)}{\partial h_{ij}^{2}}\).  
In the above, \(P_{00}\) is the covariance of the function values, \(P_{11}\) is the covariance of the derivatives, and \(P_{01}\) and \(P_{10}\) are the cross-covariances.  
The GEK formulation is extended to \(M\)-dimensional problems in a straightforward manner: the observation vector is expanded by adding \(N\)\(M\)-dimensional gradients, which results in the \(N\times(M+1)\) compiled observations [64]. The prior covariance \(\mathbb{P}\) becomes:  
\[\mathbb{P}=\begin{bmatrix}P_{00}&P_{01}&\cdots&P_{0M}\\ P_{10}&P_{11}&\cdots&P_{1M}\\ \vdots&\vdots&\ddots&\vdots\\ P_{M0}&P_{M1}&\cdots&P_{MM}\end{bmatrix} \tag{17}\]  
where:  
\[P_{ij}^{kl}=\frac{\partial^{2}cov(\tilde{\varsigma}_{k,j},\tilde{\varsigma}_ {l,i})}{\partial\tilde{\varsigma}_{k}\partial\tilde{\varsigma}_{l}} \tag{18}\]  
Due to the use of additional gradient information, GEK achieves a given level of accuracy with a number of samples that is consistently lower than BK. However, the solution quality may be highly dependent on the accuracy of the computed gradients [24], which must be properly accounted for when building the surrogate.