# Performance-Driven Yield Optimization of High-Frequency Structures by Kriging Surrogates
## 3 Surrogate-Based Yield Optimization with Domain Confinement
### Yield Optimization of Microwave Couplers  
In this case, the design requirements include the power split error and the bandwidth. We define \(f_{11L}(\mathbf{x}),f_{11H}(\mathbf{x}),f_{41L}(\mathbf{x})\), and \(f_{41H}(\mathbf{x})\) (the frequencies that are bandwidth boundaries) and also \(l_{1}(\mathbf{x})=|\,S_{21}(\mathbf{x}f_{0})\,|\,\) and \(l_{2}(\mathbf{x})=|\,S_{31}(\mathbf{x}f_{0})\,|\,\) (the levels of the respective coupler responses at its operating frequency). These, in turn, are gathered in the vector \(\mathbf{F}(\mathbf{x})=[f_{11L}(\mathbf{x}),f_{11H}(\mathbf{x}),f_{41L}(\mathbf{x}),f_{41H}(\mathbf{x} )\,l_{1}(\mathbf{x})\,l_{1}(\mathbf{x})]^{T}\), for which Jacobian \(\mathbf{J}_{F}\) can be derived from the Jacobian \(\mathbf{J}_{S}\) of the circuit response (evaluated through finite differentiation). The linear expansion model of \(\mathbf{F}\) at the nominal design \(\mathbf{x}^{(0)}\) is as follows [66]. \[\mathbf{L}_{F}(\mathbf{x})=\left[L_{1}(\mathbf{x})L_{2}(\mathbf{x})\ldots L_{6}(\mathbf{x})\right] ^{T}=\mathbf{F}(\mathbf{x}^{(0)})+\mathbf{J}_{F}(\mathbf{x}^{(0)})\cdot(\mathbf{x}-\mathbf{x}^{(0)}) \tag{6}\]  
As before, we also have two supplementary designs \(\mathbf{x}^{(1)}\) and \(\mathbf{x}^{(2)}\) yielded by solving the following. \[\mathbf{x}^{(1)}=\underset{\mathbf{x}}{\mathrm{argmin}}\left\{L_{5}(\mathbf{x})-L_{6}(\bm {x})+\beta_{2}\left\|\left[ \begin{array}{c}L_{1}(\mathbf{x})\\ \vdots\\ L_{4}(\mathbf{x})\end{array} \right]-\left[ \begin{array}{c}L_{1}(\mathbf{x}^{(0)})\\ \vdots\\ L_{4}(\mathbf{x}^{(0)})\end{array} \right]\right\|^{2}\right\} \tag{7}\]  
\[\mathbf{x}^{(2)}=\underset{\mathbf{x}}{\mathrm{argmin}}\left\{ \begin{array}{c}-2 \mathrm{min}\{f_{0}-\mathrm{max}\{L_{1}(\mathbf{x}),L_{3}(\mathbf{x})\},\mathrm{min}\{ L_{2}(\mathbf{x}),L_{4}(\mathbf{x})\}-f_{0}\}\\ +\beta_{1}\cdot\left[L_{5}(\mathbf{x})-L_{6}(\mathbf{x})\right]^{2}\end{array} \right\} \tag{8}\]  
Both (7) and (8) are subject to \(|\,|\,\mathbf{x}-\mathbf{x}^{(0)}\,|\,\leq D\) (\(D\) being selected by the user; here, we set \(D=0.5\) mm). Solving (7) and (8) requires merely \(n\) EM analyses of the coupler (it is the cost of estimating \(\mathbf{J}_{F}\) in (6)). These designs serve to find the directions corresponding to the maximal variations of the coupler power split and bandwidth, respectively. Figure 5 provides a brief description of the surrogate domain definition using these reference designs, whereas a graphical illustration of these procedure is provided in Figure 6.  
Figure 4: Yield optimization of multi-band antennas using performance-driven surrogates: (**a**) reflection responses of an exemplary narrow-band antenna at the nominal design \(\mathbf{x}^{(0)}\), maximum bandwidth design \(\mathbf{x}^{(1)}\), and best matching (at\(f_{0}\)) design \(\mathbf{x}^{(2)}\). These designs determine the directions of the most significant response changes (from the point of view of the target operating bandwidth); (**b**) The reference designs \(\mathbf{x}^{(0)}\) through \(\mathbf{x}^{(2)}\) form a path (a parameterized curve \(s(t)\)). The union of intervals \(S(\mathbf{t})\) (cf. (15)) form the surrogate model domain \(X_{5}\). Let \(S(\mathbf{t})=[S_{i}(t)\...\ S_{n}(t)]^{T}\) be a \(\mathbf{t}\)-parameterized surface such that  
\[S(\mathbf{t})=S([t_{1}\ t_{2}]^{T})=\mathbf{x}^{(0)}+t_{1}\mathbf{v}_{1}+t_{2}\mathbf{v}_{2}\]  
for \(-1\leq t_{1}\), \(t_{2}\leq 1\). We have \(S([0\ 0]^{T})=\mathbf{x}^{(0)}\), \(S([1\ 0]^{T})=\mathbf{x}^{(1)}\), and \(S([0\ 1]^{T})=\mathbf{x}^{(2)}\). Let \(S(\mathbf{t})\) be the interval with the centre at \(S(\mathbf{t})\) and the size \(\delta=[\delta_{1}\...\ \delta_{n}]^{T}\), where \(\delta_{i}\) is a multiplicity of the maximum design deviation \(\delta_{\max}\), e.g., \(2\delta_{\max}\).