## 2 Surrogate Modeling by Nested Kriging
### Domain Definition  
Due to imperfect approximation of \(U_{F}(F)\) by \(\mathbf{s}_{l}(F)\) it is necessary to extend the latter so that all (or most) of the designs \(U_{F}(\mathbf{f})\) are included. In [51], the extension is implemented using the vectors normal to \(\mathbf{s}_{l}(F)\). We denote by \(\{\mathbf{v}_{n}{}^{(k)}(\mathbf{f})\}\), \(k=1,\,\ldots,\,n-N\), an orthonormal basis of vectors normal to \(\mathbf{s}_{l}(F)\) at \(\mathbf{f}\)  
Figure 1: Fundamentals of the nested kriging concept (here, shown for two performance figures and three-dimensional parameter space): (**a**) objective space \(F\), (**b**) parameter space \(X\), the reference designs, the optimum design manifold \(U_{F}(F)\), and the first-level model image \(\mathbf{s}_{l}(F)\). The manifolds \(U_{F}(F)\) and \(\mathbf{s}_{l}(F)\) do not coincide for all \(\mathbf{f}\in F\) but the agreement is perfect for all reference designs \(\mathbf{x}^{(l)}\), which are optimal in the sense of (1).  
Additional notation is also used: \(x_{\max}=\max\{x^{(k)},\,k=1,\,\ldots,\,p\}\), \(x_{\min}=\min\{x^{(k)},\,k=1,\,\ldots,\,p\}\), and \(\mathbf{x}_{d}=\mathbf{x}_{\max}-\mathbf{x}_{\min}\), which quantifies the parameter variations within \(\mathbf{s}_{I}(F)\). The extension coefficients are then defined as  
\[\mathbf{\alpha}(\mathbf{f})=[\alpha_{1}(\mathbf{f})\,\ldots\,\alpha_{n-N}(\mathbf{f})]^{T}=0.5T \bigg{[}\mathbf{x}_{d}\mathbf{v}_{n}^{(1)}(\mathbf{f})|\,\,\ldots\,\,|\mathbf{x}_{d}\mathbf{v}_{n}^ {(n-N)}(\mathbf{f})|\bigg{]}^{T} \tag{3}\]  
where \(T\) is a user-defined thickness parameter. The surrogate model domain \(X_{S}\) is allocated between the manifolds \(M_{+}\) and \(M_{-}\)  
\[M_{\pm}=\left\{\mathbf{x}\in X:\mathbf{x}=\mathbf{s}_{I}(\mathbf{f})\pm\sum\
olimits_{k=1}^{n -N}\alpha_{k}(\mathbf{f})\mathbf{v}_{n}^{(k)}(\mathbf{f})\right\} \tag{4}\]  
We have  
\[X_{S}=\left\{\begin{array}{c}\mathbf{x}=\mathbf{s}_{I}(\mathbf{f})+\sum\limits_{k=1}^{n -N}\lambda_{k}\alpha_{k}(\mathbf{f})\mathbf{v}_{n}^{(k)}(\mathbf{f}):\mathbf{f}\in F,\\ -1\leq\lambda_{k}\leq 1,\,k=1,\ldots,n-N\end{array}\right\} \tag{5}\]  
A graphical illustration of the manifolds \(M_{+}\), \(M_{-}\), and the domain \(X_{S}\) has been provided in Figure 2. The second-level (i.e., the actual) surrogate is implemented as a kriging interpolation model over the domain \(X_{S}\). The training data is \(\{\mathbf{x}_{B}^{(k)},\mathbf{R}(\mathbf{x}_{B}^{(k)})\}_{k=1,\,\ldots,\,NB}\), where \(\mathbf{x}_{B}^{(k)}\in X_{S}\) are uniformly allocated samples, whereas \(\mathbf{R}\) stands for the response of the EM-simulation model of the antenna. Detailed information about the design of experiments (sampling) procedure can be found in [51] (see also Section 3.3).  
As demonstrated in [51], confining the surrogate model domain to \(X_{S}\), the volume of which is significantly smaller than the volume of \(X\), brings in considerable computational benefits. The major advantage is a possibility of constructing reliable surrogates using a small number of training samples and without formally restricting the ranges of antenna parameters and operating conditions the model is valid for. Even more importantly, these benefits are especially pronounced in higher-dimensional spaces where modeling within the conventional domain \(X\) is often computationally prohibitive.