# Performance-Driven Yield Optimization of High-Frequency Structures by Kriging Surrogates
## 3 Surrogate-Based Yield Optimization with Domain Confinement
### Yield Optimization of Multi-Band Antennas  
The directions that affect the antenna characteristics to the highest degree are selected by pre-optimizing two supplementary designs: (i) the design that maximizes the antenna symmetrical fractional bandwidths, and (ii) the design that minimizes the antenna reflection at\(f_{0,k}\), \(k=1,\,\ldots\,,\,N\), (i.e., the resonant frequencies). These designs are rendered by solving the following [63].  
\[\mathbf{x}^{(1)}=\operatorname*{argmin}_{\mathbf{x}}\{-\min\{B_{1}(\mathbf{x}),\ldots,B_{ N}(\mathbf{x})\}\} \tag{4}\]  
\[\mathbf{x}^{(2)}=\operatorname*{argmin}_{\mathbf{x}}\{\max\{\left|S_{11}(\mathbf{x},f_{01 })\right|,\ldots,\left|S_{11}(\mathbf{x},f_{0N})\right|\}\} \tag{5}\]  
In (4), \(B_{1}(\mathbf{x})=2\min\{f_{0k}-f_{1k}(\mathbf{x}),\,f_{2k}(\mathbf{x})-f_{0k}\}\), \(k=1,\,\ldots\,,\,N\), (symmetric portion of the bandwidth), whereas \(f_{1k}\) and \(f_{2k}\) denote the frequencies for which \(\,|S_{11}\,|\,\) assumes \(-10\) dB level (the lower and higher frequencies around the \(k\)th resonance). The trust-region gradient algorithm [64] is employed to solve problems (4) and (5), and the antenna response sensitivities are updated by applying the Broyden formula [65]. Consequently, the optimization cost equals approximately \(1.5n\) EM analyses (\(n\) being the number of designable parameters).  
Figure 3 describes surrogate domain \(X_{S}\) and its establishment with the use of the reference designs, whereas Figure 4 illustrates this process graphically. The surrogate domain \(X_{S}\) is small; still, it encompasses the most consequential directions of antenna response variations that directly affect the yield, thereby allowing for significant cost reductions. The yield is optimized directly by solving (3) (i.e., the surrogate is used instead of EM simulations), similarly as in Algorithm 1.  
Let \(\mathbf{s}(t)=[s_{1}(t)\,\ldots\,s_{n}(t)]^{\top}\) be a \(t\)-parameterized curve such that the following is the case:  
\[s_{j}(t)=a_{j0}+a_{j1}t+a_{j2}t^{2}\]  
for \(0\leq t\leq 1\), so that \(\mathbf{s}(0)=\mathbf{x}^{(1)}\), \(\mathbf{s}(0.5)=\mathbf{x}^{(0)}\), and \(\mathbf{s}(1)=\mathbf{x}^{(2)}\).  
The model coefficients can be found as follows.  
\[\begin{bmatrix}a_{10}&\text{L}&a_{n0}\\ a_{11}&\text{L}&a_{n1}\\ a_{12}&\text{L}&a_{n2}\end{bmatrix}=\begin{bmatrix}1&0&0\\ 1&0.5&0.25\\ 1&1&1\end{bmatrix}^{+1}\begin{bmatrix}\mathbf{x}^{(1)}\mathbf{x}^{(1)}\mathbf{x}^{\top}\\ \mathbf{x}^{(2)}\mathbf{x}^{\top}\end{bmatrix}\]  
Let \(S(t)\) be the interval with the center at \(\mathbf{s}(t)\) and the size \(\delta_{t}=[\delta_{1}\,\ldots\,\delta_{n}]^{\top}\), where \(\delta_{i}\) is a small multiplicity of the maximum design deviation \(\delta_{\text{max}}\), e.g, \(2\delta_{\text{max}}\).  
Surrogate model domain \(X_{S}\) is defined as the set-theory union of the intervals \(S(t)\) for \(0\leq t\leq 1\).  
\[X_{S}=\bigcup_{\delta\leq t\leq 1}S(t)\]  
\(X_{S}\) contains the reference designs \(\mathbf{x}^{(0)}\), \(\mathbf{x}^{(2)}\), and \(\mathbf{x}^{(2)}\) and a vicinity of the entire curve \(\mathbf{s}(t)\) of size \(\delta\).  
Figure 3: Surrogate domain definition using the reference designs. Multi-band antennas case.