## 3 Reconstruction Algorithms
### Algorithm
In practice, an underlying task is, for any individual \(x\), reducing the calculation of \(\overline{P}_{\mathcal{A}}(x)\) to a tractable finite-dimensional optimisation problem. Central enabling results here are the reduction theorems of ([PERSON] et al. [7], Section 4), which loosely speaking, say that if, for each \(g\), \(\{\mu\mid(\mu,g)\in\mathcal{A}\}\) is specified by a system of \(m\) equality or inequality constraints on expected values of arbitrary test functions under \(\mu\), then for the determination of \(\overline{P}_{\mathcal{A}}(x)\) it is sufficient to consider only distributions \(\mu\) that are convex combinations of at most \(m+1\) point masses; the optimisation variables are then the \(m\) independent weights and \(m+1\) locations in \(\mathcal{X}\) of these point masses. If \(\mu\) factors as a product of distributions (i.e., \(\boldsymbol{\Xi}\) is a vector with independent components), then this reduction theorem applies componentwise._

As a function of the performance threshold \(x\), \(\overline{P}_{\mathcal{A}}(x)\) is an increasing function, and so it is potentially advantageous to determine \(\overline{P}_{\mathcal{A}}(x)\) jointly for a wide range of \(x\) values using the algorithm developed above. Indeed, determining \(\overline{P}_{\mathcal{A}}(x)\) for many values of \(x\), rather than just one value, is desirable for multiple reasons:

1. Since numerical optimisation to determine \(\overline{P}_{\mathcal{A}}(x)\) may be affected by errors, computing several values of \(\overline{P}_{\mathcal{A}}(x)\) could lead to validate their consistency as the function \(x\mapsto\overline{P}_{\mathcal{A}}(x)\) must be increasing;
2. The function \(\overline{P}_{\mathcal{A}}(x)\) can be discontinuous. Thus, by computing several values of \(\overline{P}_{\mathcal{A}}(x)\), one can highlight potential discontinuities and can identify key threshold values of \(x\mapsto\overline{P}_{\mathcal{A}}(x)\). ### Test Case

For the application of Algorithm 1 to OUQ, we study the robust shape optimization of the two-dimensional RAE2822 airfoil [10] (Appendix A6) using ONERA's CFD software _elsA_[11]. The following example is taken from [PERSON] et al. [12]. The shape of the original RAE2822 is altered using four bumps located at four different locations: 5%, 20%, 40%, and 60% of the way along the chord \(c\) (see Figure 7). These bumps are characterised by B-splines functions. The lift-to-drag ratio \(\frac{C_{l}}{C_{d}}\) of the RAE2822 wing profile (see Figure 8) at Reynolds Number \(Re=6.5\times 10^{6}\), Mach number \(M_{\infty}=0.729\) and angle of attack \(\alpha=2.31^{\circ}\) is chosen as the performance function \(g^{\ddagger}\) with inputs \(\Xi=(\Xi_{1},\Xi_{2},\Xi_{3},\Xi_{4})\), where \((\Xi_{i})_{i=1..4}\) is the amplitude of each bump. They will be considered as random variables over their respective range given in Table 1.

The corresponding flow values are the ones described in test case #6 together with the wall interferences corrections formulas given in [13] (Chapter 6) and in [14] (Section 5.1). Moreover, we will assume that \((\Xi_{i})_{i=1..4}\) are mutually independent. An ordinary Kriging procedure has been chosen to build a metamodel (or response surface) of \(g^{\ddagger}\), which is identified with the actual response function \(g^{\ddagger}\) in the subsequent analysis. A tensorised grid of 9 equidistributed abscissas for each parameter is used. The model is then based on \(N=9^{4}=6561\) observations. In that respect, a Gaussian kernel