# Performance-Driven Yield Optimization of High-Frequency Structures by Kriging Surrogates
## 1 Introduction  
High-frequency systems are normally designed in the nominal sense, with possible deviations of geometry and material parameters (e.g., due to fabrication inaccuracies) being generally neglected. Yet, uncertainties may have detrimental effects on the performance. As a consequence, it is important to develop procedures for their evaluation. Among possible types of uncertainties, those pertinent to imperfect manufacturing [1, 2] play the most important role in practice. They are stochastic, which makes statistical analysis imperative for their evaluation [3, 4, 5]. Diminishing the impact of parameter deviations requires a maximization of appropriately defined figures of merit such as response variance or the yield [6, 7, 8]. In the case of high-frequency systems, yield seems to be more suitable because performance specifications are often expressed in a minimax form, particularly for lower/upper acceptance thresholds for \(S\)-parameters, etc. [9, 10, 11, 12]. As mentioned earlier, the evaluation of the effects of parameter tolerances involves statistical analysis [13, 14]. For reliability, it is normally carried out using full-wave electromagnetic (EM) analysis, which is associated with significant computational costs. Nevertheless, it is mandatory whenever simpler models (e.g., equivalent circuits) fail insufficiently to account for cross-coupling and similar effects [15, 16, 17, 18]. Improving the computational efficiency of statistical analysis can be achieved using simplistic yet inaccurate methods(e.g., worst-case analysis [19; 20; 21]) or surrogate-assisted techniques [22; 23], where repetitive EM simulations are replaced by an evaluation of a fast replacement model, usually prepared beforehand. Popular modeling methods include polynomial approximation [24], neural networks (NNs) [25; 26], or polynomial chaos expansion (PCE) [27; 28; 29; 30; 31; 32; 33]. Despite their advantages, surrogate-based methods are affected by the curse of dimensionality, resulting in excessive costs of model rendition for more complex systems. The alleviation of these difficulties is partially possible by the employment of hybrid techniques (e.g., PC kriging [6; 34; 35]), dimensionality reduction [36], multi-resolution methods (space mapping [37; 38; 39; 40] and co-kriging [41; 42]), or model order reduction [43]. Reducing the effects of manufacturing tolerances is even more essential than their evaluation. The relevant procedures are often referred to as robust design (also yield-driven design, tolerance-aware optimization, etc.) [1; 44; 45; 46; 47]. In practical terms, this can be accomplished by improving the statistical merit functions of choice, such as the yield. Unfortunately, yield maximization is a CPU-heavy task to the extent of being prohibitive when directly executed at the level of EM simulation models. Surrogate-assisted procedures offer viable workarounds [6; 24; 25; 26; 27; 28]. Some of the most popular modeling methods utilized in this context are polynomial approximations [24], space mapping [48], NNs [49], and PCE [50]. As for statistical analysis, the bottleneck is a potentially high cost of the surrogate model setup, related to the dimensionality of the parameter space and parameter ranges. A partial mitigation has been offered by sequential approximation optimization (SAO) [51], where the metamodel is rendered along the optimization path, within the limited-volume domains centered at the current design produced by the robust design procedure. Another option is to employ response-feature technology [52]. This method capitalizes on the reduced nonlinearity of the relationship between the appropriately selected characteristic points of the system's responses and geometry parameters of the structure under design. The latter enables the construction of accurate metamodels using small training datasets [53]. In this work, we discuss a method for reduced-expense yield maximization of antenna and microwave components.