## 2 Materials and Methods
### Surrogate Fourier Modeling with surf
** The major drawback of the previous approach is that sampling \(\rho^{D}\) points from \(f\) can be extremely expensive, since the number of points grows exponentially with the number of dimensions. Hence, a way to reduce the number of sampling operations of \(f\) is employed by surf. The main idea is to use \(\sigma\ll\rho^{D}\) points of \(f\) to generate a first surrogate of \(\hat{f}\), which is then used for the sampling and the remaining of the surf procedure. To construct \(\hat{f}\), \(\sigma\) points \(\vec{z}_{0},\ldots\vec{z}_{\sigma-1}\) are randomly selected in \([\ell,u]^{D}\). The selection procedure can be, for example, a uniform selection in the search space. To evaluate \(\hat{f}\) on a point \(\vec{z}\in[\ell,u]^{D}\), the following procedure is employed: * If \(\vec{z}\) is in the convex hull defined by the points \(\vec{z}_{0},\ldots\vec{z}_{\sigma-1}\), then \(\hat{f}(\vec{z})\) is obtained by a linear interpolation; * otherwise, a linear interpolation is not possible and \(\hat{f}(\vec{z})\) is defined as \(\hat{f}(\vec{z}^{\prime})\), where \(\vec{z}^{\prime}\) is the point among \(\vec{z}_{0},\ldots\vec{z}_{\sigma-1}\) nearest to \(\vec{z}\). Once \(\hat{f}\) is constructed, we can sample \(\rho^{D}\) points from it and proceed with the previously define construction of a smooth surrogate (of \(\hat{f}\) in this case). 3. **Parameters of surf. ** The surf algorithm requires the following settings: * \(\sigma\), which is the number of samples from \(f\) used to build \(\hat{f}\); * \(\rho\), which is the \"density\" of samples from \(\hat{f}\) to obtain the \(\rho^{D}\) points used to calculate the DFT; and * \(\gamma\), which controls the number of low frequencies preserved. Concerning the first settings, in this work, we assume \(\sigma=100\), generated using quasi random sampling using Sobol sequences [29], one of the five methods natively supported by surf along with pseudo-random generators, chaotic sequences, quantum random numbers, and entropic point packing [13]. Quasi-random sampling is surf's default because it was shown to be the most effective approach when a few samples can be collected [13]. The \(\rho\) value should be selected carefully in order to not exceed the resources of the machine. In this work, we investigate the PE of a model with 3 missing parameters using \(\rho=100\), leading to the calculation of \(1\times 10^{6}\) interpolated values. Finally, the \(\gamma\) parameter is particularly sensitive: as a matter of fact, it has a relevant impact on the optimization because a \(\gamma\) value too small can make the reconstruction of the original global minimum impossible. Since the shape and characteristics of the fitness landscape are unknown, the optimal \(\gamma\) cannot be determined using analytical approaches. One possible solution for this problem is trial and error: repeat the optimization with an increasing number of \(\gamma\) values, observing which setting leads to the best solution. Alternatively, we can co-evolve the optimal \(\gamma\) by extending the candidate solutions' representation, as described in the next section.