# Application and Evaluation of Surrogate Models for Radiation Source Search
## 4 Training Points  
The basis functions, \(\Psi_{j},\ j=1,\ldots,N\), should be employed in conjunction with a complimentary choice of training points, \(q^{m},\ m=1,\ldots,M\), to accurately approximate the ray-tracing model. Training points derived from Gaussian quadrature are often employed, due to their accuracy with relatively few points. However, these points are not nested and hence cannot be re-employed when increasing discretization levels. This is made apparent by Figure 5a,b, in which we plot the Gauss-Legendre and Clenshaw-Curtis points for levels \(\ell=0,...,4\), where the number of training points are given by \(R(\ell)=2^{\ell}+1\). We note that each level of Gauss-Legendre points do not overlap with the level above it, whereas the Clenshaw-Curtis points do overlap. This becomes important when attempting to improve a grid without re-evaluating the high-fidelity model at every point on the updated grid. We consider four separate sets of training points and we compare how the surrogate models previously discussed perform when employing each set of points. Whereas there is a whole class of model-based design methods that have been studied for use with various surrogate modeling techniques [37], we consider model-free techniques. Additionally, methods for experimental design could be employed to improve surrogate performance, but we leave this as future work. The first set of points is Gauss-Legendre (GL), which we use to obtain the surrogate model results discussed in Section 5. The Gauss-Legendre points can be obtained by solving for the roots of the Legendre polynomials. We compile \(rRMSE\) results for the surrogate models employing these training points in Table 4. We also compile in Table 4 the surrogate model \(rRMSE\) when randomly selected training points are used. These randomly selected points are drawn from a uniform distribution over \([0,1]\). Clenshaw-Curtis (CC) quadrature points are specified by the extrema of Chebyshev polynomials, which are typically defined on the interval \([-1,1]\). These points, when mapped to \([0,1]\), are given by  
\[q_{\ell}^{r}=\frac{1}{2}\Big{[}1-\cos\frac{(r-1)\pi}{R(\ell)-1}\Big{]},\ r=1,...,R( \ell), \tag{55}\]  
where \(R(\ell)=2^{\ell}+1\). As shown in Figure 5, these points are not equally spaced and cluster around the endpoints of the interval. This effectively avoids producing spurious oscillations, termed the Runge's phenomenon, when interpolating. Furthermore, we note that these points are nested, so that the training points in level \(R(\ell)\) are also in level \(R(\ell+1)\), which is important when more points must be added for increased accuracy of the surrogate model. In practice, it has been shown that, for most functions, Clenshaw-Curtis quadrature performs as well as Gaussian quadrature [38] in most applications. Latin hypercube points--obtained via latin hypercube sampling (LHS)--were designed to be quasi-random while adequately exploring a multidimensional space [39; 40]. For a \(p\) dimensional parameter space, each dimension is divided into \(\mathcal{M}\) equally probable sections. Then, \(\mathcal{M}\) sample points are randomly placed so that each sample is the only point in its axis-aligned hyperplane. This sampling scheme is favorable in high dimensions--e.g., \(p\geq 10\)--since increased samples are not required for increased dimensions. In addition, these points can be nested, as with Clenshaw-Curtis. We note that the discrete projection-based Legendre surrogate model employing Gauss-Legendre sampling outperform the same surrogate model employing the other sampling methods. Conversely, the regression-based Legendre, MARS, RBF, Gaussian processes, and neural networks surrogate models favor random sampling of the parameter space. Latin hypercube sampling has been shown to perform about as well as simple random Monte Carlo sampling strategies [39; 40].