## 3 Reconstruction Algorithms
### Algorithm
\begin{table}
\begin{tabular}{l c c} \hline \hline  & **Range** & **Law** \\ \hline Bump 1: \(\Xi_{1}\) & \([-0.0025c;+0.0025c]\) & \(\mu_{1}^{\ddagger}\): Beta law with \(\alpha=6\), \(\beta=6\) \\ Bump 2: \(\Xi_{2}\) & \([-0.0025c;+0.0025c]\) & \(\mu_{2}^{\ddagger}\): Beta law with \(\alpha=2\), \(\beta=2\) \\ Bump 3: \(\Xi_{3}\) & \([-0.0025c;+0.0025c]\) & \(\mu_{3}^{\ddagger}\): Beta law with \(\alpha=2\), \(\beta=2\) \\ Bump 4: \(\Xi_{4}\) & \([-0.0025c;+0.0025c]\) & \(\mu_{4}^{\ddagger}\): Beta law with \(\alpha=2\), \(\beta=2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Range of each input parameter. Figure 8: Picture depicting the lift \(C_{l}\) and the drag \(C_{d}\) of an airfoil. Figure 7: Black lines: Maximum and minimum deformation of the RAE2822 profile. Red: Maximum deformation of the third bump alone. Blue: Minimum deformation of the third bump alone. This image is taken from [PERSON] et al. [12]. \[K(\Xi,\Xi^{\prime})=\exp\left(-\frac{1}{2}\sum_{i=1}^{4}\frac{(\[PERSON]_{i}-\[PERSON]_{i}^{ \prime})^{2}}{\gamma_{i}^{2}}\right)\]

has been chosen, where \(\[PERSON]=(\[PERSON]_{1},\[PERSON]_{2},\[PERSON]_{3},\[PERSON]_{4})\) and \(\[PERSON]^{\prime}=(\[PERSON]_{1}^{\prime},\[PERSON]_{2}^{\prime},\[PERSON]_{3}^{\prime},\[PERSON]_{4}^{ \prime})\) are inputs of the function \(g^{\dagger}\), and where \(\gamma=(\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4})\) are the parameters of the kernel. These parameters are chosen to minimize the variance between the ground truth data defined by the \(N\) observations and their Kriging metamodel \(g^{\dagger}\). The response surfaces in the \((\Xi_{1},\Xi_{3})\) plan for two values of \((\Xi_{2},\Xi_{4})\) are shown in Figure 9.

One seeks to determine \(\overline{P}_{\mathcal{A}}(x)\coloneqq\sup_{\mu\in\mathcal{A}}\mathbb{P}_{ \Xi\sim\mu}[g^{\dagger}(\Xi)\leq x]\), where the admissible set \(\mathcal{A}\) is defined as follows:

\[\mathcal{A}=\left\{ \begin{array}{c}\left.\begin{array}{c}\Xi\in\mathcal{X }=\mathcal{X}_{1}\times\mathcal{X}_{2}\times\mathcal{X}_{3}\times\mathcal{X} _{4}\\ g:\mathcal{X}\mapsto\mathcal{Y}\text{ is known equal to }g^{\dagger}\\ \mu=\mu_{1}\otimes\mu_{2}\otimes\mu_{3}\otimes\mu_{4}\\ \mathbb{E}_{\Xi\sim\mu}[g(\Xi)]=\text{LD}\end{array}\right\}.\end{array} \right. \tag{17}\]

A priori, finding \(\overline{P}_{\mathcal{A}}(x)\) is not computationally tractable because it requires a search over a infinite-dimensional space of probability measures defined by \(\mathcal{A}\). Nevertheless, as described briefly in Remark 1, it has been shown in [PERSON] et al. (2017) that this optimisation problem can be reduced to a finite-dimensional one, where now the probability measures are products of finite convex combinations of Dirac masses. **Remark 2**. : _The ground truth law \(\mu^{\dagger}\) of each input variable given in Table 1 is only used to compute the expected value \(\mathbb{E}_{\Xi\sim\mu}[g(\Xi)]=\text{LD}\). This expected value is computed with \(10^{4}\) samples._

**Remark 3**. : _The admissible set \(\mathcal{A}\) from (17) can be understood as follows:_

* _One knows the range of each input parameter_ \((\Xi_{i})_{i=1,\ldots,\mathcal{A}}\)_;_
* \(g\) _is exactly known as_ \(g=g^{\dagger}\)_;_
* \((\Xi_{i})_{i=1,\ldots,4}\) _are independent;_
* _One only knows the expected value of_ \(g\)_:_ \(\mathbb{E}_{\Xi\sim\mu}[g(\Xi)]\)_._

The optimisation problem of determining \(\overline{P}_{\mathcal{A}}(x)\) for each chosen \(x\) was solved using the Differential Evolution algorithm of [PERSON] (1975) within the _mytic_ optimisation framework ([PERSON], 1976). Ten iterations of Algorithm 1 have been performed using \(\mathcal{E}=1\times 10^{4}\). The evolution of \(\overline{P}_{\mathcal{A}}(x)\) as function of the iteration count, \(n\), is shown in Figure 10. At \(n=0\)--see Figure 10a--two consistent points are present at \(x=57.51\) and \(x=67.51\). At this step, \(\operatorname{WA}^{(0)}=35289\).