## 3 Reconstruction Algorithms
### Algorithm
This can be seen on Figure 5 where the worst quality is almost constant over 100 steps while the sum of areas strongly decreases; see Figure 5 and Figure 5 respectively. At each step, the algorithm is adding a new point by splitting the biggest rectangle. One can note on Figure 5 that the minimum of the quality is not constant. It means that when the algorithm added a new data point, the point with the worst quality was not consistent any more and had to be recomputed. In summary, in this case, we obtain more points but with lower quality values. #### 4.3.2 Case \(\mathcal{E}\gg 1\)

We now consider the case \(\mathcal{E}=10^{4}\gg 1\). This choice corresponds to the case where one wishes to redo the worst quality point over split. This can be seen on Figure 6 where the sum of areas stays more or less the same over 100 steps while the minimum of the quality surges; see Figure 6 and Figure 6 respectively. There is no new point. The algorithm is only redoing the worst quality point to improve it. To sum up, we obtain fewer points with higher quality values. Figure 4: Evolution of \(F^{(n)}\) and the \(\infty\)- and \(1\)-norms of the error \(F^{\dagger}-F^{(n)}\) as functions of the iteration count, \(n\), for a discontinuous ground truth \(F^{\dagger}\). Figure 5: Evolution of \(F^{(n)}\) and the minimum of the quality and the total area as functions of the iteration count, \(n\), for a discontinuous ground truth \(F^{\dagger}\) with \(\mathcal{E}=10^{-4}\). Figure 6: Evolution of [PERSON]F^{(n)}\) and the minimum of the quality and the total area as functions of the iteration count, \(n\), for a discontinuous ground truth \(F^{\dagger}\) with \(\mathcal{E}=10^{4}\). ## 5 Application to Optimal Uncertainty Quantification

### Optimal Uncertainty Quantification

In the _optimal uncertainty quantification_ paradigm proposed by [PERSON] et al. [7] and further developed by, e.g., [PERSON] et al. [8] and [PERSON] et al. [9], upper and lower [PERSON] on the performance of an incompletely specified system are calculated via optimisation problems. More concretely, one is interested in the probability that a system, whose output is a function \(g^{\dagger}\colon\mathcal{X}\to\mathbb{R}\) of inputs \(\[PERSON]{\Xi}\) distributed according to a probability measure \(\mu^{\dagger}\) on an input space \(\mathcal{X}\), satisfies \(g^{\dagger}(\boldsymbol{\Xi})\leq x\), where \(x\) is a specified performance threshold value. We emphasise that although we focus on a scalar performance measure, the input \(\boldsymbol{\Xi}\) may be a multivariate random variable.

In practice, \(\mu^{\dagger}\) and \(g^{\dagger}\) are not known exactly; rather, it is known only that \((\mu^{\dagger},g^{\dagger})\in\mathcal{A}\) for some admissible subset \(\mathcal{A}\) of the product space of all probability measures on \(\mathcal{X}\) with the set of all real-valued functions on \(\mathcal{X}\). Thus, one is interested in

\[\underline{P}_{\mathcal{A}}(x)\coloneqq\inf_{(\mu,g)\in\mathcal{A}}\mathbb{P} _{\boldsymbol{\Xi}\sim\mu}[g(\boldsymbol{\Xi})\leq x]\quad\text{and}\quad \overline{P}_{\mathcal{A}}(x)\coloneqq\sup_{(\mu,g)\in\mathcal{A}}\mathbb{P} _{\boldsymbol{\Xi}\sim\mu}[g(\boldsymbol{\Xi})\leq x].\]

The inequality

\[0\leq\underline{P}_{\mathcal{A}}(x)\leq\mathbb{P}_{\boldsymbol{\Xi}\sim\mu^{ \dagger}}[g^{\dagger}(\boldsymbol{\Xi})\leq x]\leq\overline{P}_{\mathcal{A}} (x)\leq 1\]

is, by definition, the tightest possible bound on the quantity of interest \(\mathbb{P}_{\boldsymbol{\Xi}\sim\mu^{\dagger}}[g^{\dagger}(\boldsymbol{\Xi}) \leq x]\) that is compatible with the information used to specify \(\mathcal{A}\). Thus, the optimal UQ perspective enriches the principles of worst- and best-case design to account for distributional and functional uncertainty. We concentrate our attention hereafter, without loss of generality, on the least upper bound \(\overline{P}_{\mathcal{A}}(x)\). **Remark 1**. : _The main focus of this paper is the dependency of \(\overline{P}_{\mathcal{A}}(x)\) on \(x\).