## 3 Reconstruction Algorithms
### Algorithm
As \(\operatorname{WA}^{(0)}\geq\mathcal{E}\), at next step \(n=1\), the algorithm adds a new point at the middle of the biggest rectangle--see Figures 10b and 11b. After \(n=10\) steps, eight points are now present in total with a minimum quality increasing from 5000 to 11,667 and with a total area decreasing from 7.05 to 0.84; see Figure 11a and Figure 11b respectively. The number of iterations in this complex numerical experiment has been limited to 10 because obtaining new or improved data points consistent throughout the optimization algorithm may take up to two days (wall-clock time on a personal computer equipped with an Intel Core i5-6300 HQ processor with 4 cores and 6 MB cache memory) for one single point. This running time is increased further for data points of higher quality. Nevertheless, this experiment shows that the proposed algorithm can be used for real-world examples in an industrial context. Figure 10: Evolution of \(\overline{P}_{\mathcal{A}}(x)\) as function of the iteration count, \(n\). ## 6 Concluding Remarks

In this paper we have developed an algorithm to reconstruct a monotonically increasing function such as the cumulative distribution function of a real-valued random variable, or the least upper bound of the performance criterion of a system as a function of its performance threshold. In particular, this latter setting has relevance to the optimal uncertainty quantification (OUQ) framework of [7] we have in mind for applications to real-world incompletely specified systems. The algorithm uses imperfect pointwise evaluations of the target function, subject to partially controllable one-sided errors, to direct further evaluations either at new sites in the function's domain or to improve the quality of evaluations at already-evaluated sites. It allows for some flexibility at targeting either strategy through a user-defined \"exchange rate\" parameter, yielding an approximation of the target function with a few high-quality points or alternatively more lower-quality points. We have studied its convergence properties and have applied it to several examples: known target functions that are either continuous and discontinuous, and a performance function for aerodynamic design of a well-documented standard profile in the OUQ setting. Algorithm 1 is reminiscent of the classical PAVA approach to isotonic regression that applies to statistical inference with order restrictions. Examples of its use can be found in shape constrained or parametric density problems as illustrated in e.g., [6]. Possible improvements and extensions of our algorithm include weighting the areas \(a_{i}^{(n)}\) as they are summed up to form the total weighted area \(\text{WA}^{(n)}\) driving the iterative process, in order to optimally enforce both the addition of \"steps\" \(s_{i}^{(n)}\) in the reconstruction function \(F^{(n)}\) of Definition 1, and the improvement of their \"heights\" \(y_{i}^{(i)}\). This could be achieved considering for example the following alternative definition \(i_{+}^{(n)}=\arg\max_{i}\{(I^{(n)}-i-1)a_{i}^{(n)}\}\) in Algorithm 1, which results in both adding a step to the \(i_{+}^{(n)}\)-th current one and possibly improving all subsequent evaluations \(y_{i}^{(n+1)}\), \(i>i_{+}^{(n)}\). We may further envisage to adapt the ideas elaborated in this research to the reconstruction of convex functions by extending the notion of consistency. These perspectives shall be considered in future works. Conceptualization, L.B. and T.J.S. ; methodology, L.B. and T.J.S. ; software, L.B. ; validation, J.-L.A., E.S., and T.J.S. ; formal analysis, L.B., J.-L.A., E.S., and T.J.S. ; investigation, L.B. ; resources, L.B., J.-L.A., E.S., and T.J.S. ; data curation, L.B. ; writing-original draft preparation, [PERSON]; writing--review and editing, [PERSON], [PERSON]-L.A., [PERSON], and T.J.S. ; visualization, L.B. ; supervision, [PERSON] and T.J.S. ; project administration, T.J.S. ; funding acquisition, L.B., E.S., and T.J.S. All authors have read and agreed to the published version of the manuscript.