## 3 Reconstruction Algorithms
### Algorithm
That is,

\[\leq\text{A}^{(n)}\!\left(\frac{2(I^{(n)}-1)-1}{2(I^{(n)}-1)} \right),\]2

with \(\sigma^{2}=1\) and \(\mu(x)\) is chosen as \(\mathbb{P}[0\leq\epsilon(x)\leq 0.1\cdot F^{\dagger}(x)]=0.9\). Thus, the mean \(\mu\) is different for each \(x\in[a,b]\). As we have access to the ground truth function and for validation purpose, the quality value associated with a numerical point is the inverse of the relative error. Moreover, we assume that the initial points are consistent. For illustrative purposes, we set the parameter \(\mathcal{E}=15\) for the examples considered below. ### \(F^{\dagger}\) Is a Continuous Function

First, consider the function \(F^{\dagger}\in C^{0}([1,2],[1,2])\) defined as follows:

\[F^{\dagger}(x)= \begin{cases}F^{\dagger}_{1}(x)&\text{if }x\in[1,\frac{3}{2}]\,,\\ F^{\dagger}_{2}(x)&\text{if }x\in[\frac{3}{2},2]\,,\end{cases} \]

with

\[F^{\dagger}_{1}(x) =a_{1}\exp(x^{3})+b_{1}\,, \tag{16}\] \[F^{\dagger}_{2}(x) =a_{2}\exp((3-x)^{3})+b_{2}\,,\]

where:

\[a_{1}=-\frac{1}{2(\exp(1)-\exp(27/8))}\,,\quad b_{1}=\frac{3-2\exp(19/8)}{2(1- \exp(19/8))}\,,\quad a_{2}=-a_{1}\,,\quad b_{2}=2a_{1}\exp(27/8)+b_{1}\,.\]

The target function \(F^{\dagger}\) and the reconstructions \(F^{(n)}\) obtained through the algorithm for several values of the step \(n\) are shown in Figure 3. For each \(n\), the reconstruction \(F^{(n)}\) is increasing and the initial points are consistent. The \(\infty\)-norm and \(1\)-norm of the error appear to converge to zero with approximate rates \(-0.512\) and \(-0.534\) respectively. ### \(F^{\dagger}\) Is a Discontinuous Function

Now, consider the discontinuous function \(F^{\dagger}\) defined as follows:

\[F^{\dagger}(x)= \begin{cases}F_{1}^{\dagger}&\text{if }x\in[1,\frac{3}{2}]\,,\\ F_{2}^{\dagger}&\text{if }x\in(\frac{3}{2},2]\,,\end{cases} \]

where \(F_{1}^{\dagger}\) and \(F_{2}^{\dagger}\) are given by (16), and:

\[a_{1} =-\frac{1}{2(\exp(1)-\exp(27/8))}\,, b_{1} =\frac{3-2\exp(19/8)}{2(1-\exp(19/8))}\,,\] \[a_{2} =\frac{2}{5(\exp(8)-\exp(27/8))}\,, b_{2} =\frac{10-8\exp(37/8)}{5(1-\exp(37/8))}\,.\]

Here, \(F^{\dagger}\) is piecewise continuous on \([1,\frac{3}{2}]\) and \(]^{\frac{3}{2}},2]\). In this case, one can apply Proposition 4. The target function \(F^{\dagger}\) and the reconstructions \(F^{(n)}\) obtained through the algorithm for several values of the step \(n\) are shown in Figure 4. Observe that the approximation quality, as measured by the \(\infty\)-norm of the error \(F^{\dagger}-F^{(n)}\), quite rapidly saturates and does not converge to zero. This is to be expected for this discontinuous target \(F^{\dagger}\), since closeness of two functions in the supremum norm mandates that they have approximately the same discontinuities in exactly the same places. The 1-norm error, in contrast, appears to converge at the rate \(-0.561\). Regarding computational cost, the number of calls to the numerical model is lower when \(F^{\dagger}\) is continuous than when it is discontinuous. For both examples above and for the same number of data points, the number of evaluations of the numerical model (analytical formula in the present case) in the discontinuous case is about six times higher than the number of evaluations in the continuous case. This is because the algorithm typically adds more points near discontinuities and the effort of making them consistent increases the number of calls to the model. Figure 3: 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 smooth ground truth \(F^{\dagger}\). ### Influence of the User-Defined Parameter \(\mathcal{E}\)

We consider the case in which \(F^{\dagger}\) is discontinuous, as in Section 4.2. We will show the influence of the choice of the parameter \(\mathcal{E}\) on the reconstruction function \(F^{(n)}\). #### 4.3.1 Case \(\mathcal{E}\ll 1\)

Let us consider the case \(\mathcal{E}=10^{-4}\ll 1\). This choice corresponds to the case where one wishes to split over redo the worst quality point.