## 2 Surrogate Modeling by Nested Kriging
### First-Level Surrogate  
The nested kriging framework takes advantage of the correlations between antenna geometry parameters corresponding to the designs that are optimum with respect to the typical figures of interest [49]. For example, antenna re-design for various operating frequencies normally entails synchronized adjustment of the parameters. The parameter sets being away from these correlation patterns are of poor quality and excluding them from the modelling process may be beneficial from the point of view of the computational efficiency. This is in opposition to conventional domains determined by the lower and upper bounds for design parameters, which are indiscriminative with respect to the aforementioned correlations. In nested kriging, identification of the regions containing high-quality designs is realized using a set of reference designs and the first-level surrogate as elaborated on below.  
We denote by \(\boldsymbol{x}=[x_{1}\ldots\ x_{n}]^{T}\) the vector of antenna parameters. The (conventional) parameter space \(X\) is defined by the lower and upper bounds \(\boldsymbol{l}=[l_{1}\ldots\,l_{n}]^{T}\) and \(\boldsymbol{u}=[u_{1}\ldots\,u_{n}]^{T}\), so that \(l_{k}\leq x_{k}\leq u_{k}\) for \(k=1,\ldots,n\). Furthermore, we denote by \(f_{k},k=1,\ldots,N\), the figures of interest pertinent to the design task at hand, operating conditions, or material parameters (e.g., operating frequency, bandwidth, or substrate permittivity assuming that the antenna is to be re-designed for various substrates). The ranges \(f_{k,\min}\leq f_{k}^{(0)}\leq f_{k,\max}\), \(k=1,\ldots,N\), define the objective space \(F\), i.e., the region over which the surrogate model is to be valid.  
The design optimality is understood, for a given objective vector \(\mathbf{f}\in F\), in the sense of minimizing the functional \(U(\mathbf{x}\mathbf{f})\), where \(U\) is the scalar merit function which quantifies the design quality. In other words, the optimum design \(\mathbf{x}^{*}\) is given as  
\[\mathbf{x}^{*}=U_{F}(\mathbf{f})=\underset{\mathbf{x}}{\mathrm{argmin}}U(\mathbf{x},\mathbf{f}) \tag{1}\]  
We also define  
\[U_{F}(F)=\{U_{F}(\mathbf{f}):\mathbf{f}\in F\} \tag{2}\]  
which is the manifold consisting the designs being optimum in the sense of (1) for all \(\mathbf{f}\in F\).  
For the sake of computational efficiency, it is advantageous to construct the surrogate model in the vicinity of \(U_{F}(F)\). Within the nested kriging framework, the first approximation of this set is obtained using the reference designs \(\mathbf{x}^{(l)}=[x_{1}{}^{(j)}\,\ldots\,x_{n}{}^{(j)}]^{T}\,j=1,\,\ldots,\,p\), optimized with respect to the objective vectors \(\mathbf{f}^{(l)}=[f_{1}{}^{(j)}\,\ldots\,f_{N}{}^{(j)}];\mathbf{f}^{(l)}\) should be allocated in a possibly uniform manner in \(F\). As mentioned before, \(\mathbf{x}^{(l)}\) may be available from the previous design work on the same structure or obtained specifically for the purpose of surrogate model construction.  
The first-level surrogate \(\mathbf{s}_{l}(\mathbf{f})\): \(F\to X\) is identified as a kriging interpolation model [51] using \([\mathbf{f}^{(l)},\mathbf{x}^{(l)}],j=1,\,\ldots,\,p\), as the training set. Figure 1 shows a graphical illustration of these concepts. Note that \(\mathbf{s}_{l}(F)\) (the image of \(F\)) approximates \(U_{F}(F)\) but it does not coincide with it because the number of the reference designs is normally limited. It should be noted that the geometry of the reference set in the objective space is generally different than that in the parameter space; however, the reference points in the parameter space are arranged along the manifold \(U_{F}(F)\), i.e., the object of the same dimensionality as \(F\).