## 2 Notation and Problem Description  
In the following, the \"ground truth\" response function that we wish to reconstruct is denoted \(F^{\dagger}\colon[a,b]\to\mathbb{R}\) and has inputs \(x\in[a,b]\subset\mathbb{R}\). It is assumed that \(F^{\dagger}\) is monotonically increasing and non-constant on \([a,b]\). In contrast, \(G\colon[a,b]\times\mathbb{R}_{+}\to\mathbb{R}\) denotes the numerical process used to obtain an imperfect pointwise observation \(y\) of \(F^{\dagger}(x)\) at some point \(x\in[a,b]\) for some numerical parameter \(q\in\mathbb{R}_{+}\). Here, on a heuristic level, \(q>0\) stands for the \"quality\" of the noisy evaluation \(G(x,q)\). The main aim of this paper is to show the effectiveness of the proposed algorithm for the adaptive reconstruction of \(F^{\dagger}\), which could be continuous or not, from imperfect pointwise observations \(G(x_{i},q_{i})\) of \(F^{\dagger}\), where we are free to choose \(x_{i+1}\) and \(q_{i+1}\) adaptively-based on \(x_{j}\), \(q_{j}\), and \(G(x_{j},q_{j})\) for \(j\leq i\)  
First, we associate with \(I\) imperfect pointwise observations \(\{x_{i},y_{i}\coloneqq G(x_{i},q_{i})\}_{i=1}^{I}\subset[a,b]\times\mathbb{R}\), positive numbers \(\{q_{i}\}_{i=1}^{I}\subset\mathbb{R}_{+}\) which we will call _qualities_. The quality \(q_{i}\) quantifies the confidence we have in the pointwise observation \(y_{i}\) of \(F^{\dagger}(x_{i})\) using the numerical process \(G(x_{i},q_{i})\). The higher this value, the greater the confidence. We divide this quality as the product of two different numbers \(c_{i}\) and \(r_{i}\), \(q_{i}=c_{i}\times r_{i}\), with the following definitions:  
* _Consistency_\(c_{i}\in\{0,1\}\): This describes the fact that two successive points must be monotonically consistent with respect to each other. That is, when one takes two input values \(x_{2}>x_{1}\), one should have \(y_{2}\geq y_{1}\) as \(y\) must be monotonically increasing. There is no consistency associated with the very first data point as it does not have any predecessor. * _Reliability_\(r_{i}\in\mathbb{R}_{+}\): This describes how confident we are about the numerical value. Typically, it will be related to some error estimator if one is available, or the choice of optimisation parameters. It is expected that the higher the reliability, the closer the pointwise observation is to the true value, on average. Typically, if the observation \(y_{i+1}=G(x_{i+1},q_{i+1})\) is consistent with regard to the observation \(y_{i}=G(x_{i},q_{i})\) where \(x_{i+1}>x_{i}\), the quality \(q_{i+1}\) associated with \(y_{i+1}\) will be equal to \(q_{i+1}=r_{i+1}\in\mathbb{R}_{+}^{*}\) since \(c_{i+1}=1\) in this case. If the value is not consistent, we have \(q_{i+1}=r_{i+1}\times c_{i+1}=0\). Finally, if \(x=a\) there is no notion of consistency as there is no point preceding it. Thereby, the quality associated with this point is only equal to its reliability. Moreover, we associate to these pointwise observations a notion of area, illustrated in Figure 1 and defined as follows. Consider two consecutive points \(x_{i}\) and \(x_{i+1}\) with their respective observations \(y_{i}\) and \(y_{i+1}\), the area \(a_{i}\) for these two points is  
\[a_{i}=(x_{i+1}-x_{i})\times(y_{i+1}-y_{i})\,. \tag{3}\]  
Thus, we can define a vector \(\boldsymbol{a}=\{a_{i}\}_{i=1}^{I-1}\) which contains all the computed areas for the whole dataset. In addition, we can assure that if we take two points \(x_{1}\) and \(x_{2}>x_{1}\) with \(y_{1}=F^{\dagger}(x_{1})\) and \(y_{2}=F^{\dagger}(x_{2})\)--namely the error at these point is equal to zero, the graph of ground truth function \(F^{\dagger}\) must lie in the rectangular area spanned by the two points \((x_{1},F^{\dagger}(x_{1}))\) and \((x_{2},F^{\dagger}(x_{2}))\).