Yield Optimization Based on Adaptive Newton-Monte Carlo and Polynomial Surrogates

Abstract

In this paper we present an algorithm for yield estimation and optimization consisting of Hessian-based optimization methods, an adaptive Monte Carlo (MC) strategy, polynomial surrogates, and several error indicators. Yield estimation is used to quantify the impact of uncertainty in a manufacturing process. Since computational efficiency is one main issue in uncertainty quantification, we propose a hybrid method, where a large part of a MC sample is evaluated with a surrogate model, and only a small subset of the sample is reevaluated with a high-fidelity finite element model. In order to determine this critical fraction of the sample, an adjoint error indicator is used for both the surrogate error and the finite element error. For yield optimization we propose an adaptive Newton-MC method. We reduce computational effort and control the MC error by adaptively increasing the sample size. The proposed method minimizes the impact of uncertainty by optimizing the yield. It allows one to control the finite element error, surrogate error, and MC error. At the same time it is much more efficient than standard MC approaches combined with standard Newton algorithms.

A blackbox yield estimation workflow with Gaussian process regression applied to the design of electromagnetic devices

“…[17,Chap. 3] and [8], i.e., Q ω (p) := S ω (p). In this case, q is an affine linear function, but this is no requirement for the following yield estimation methods.…”

“…This allows to evaluate the performance feature specifications (2) and thus the safe domain (3) without solving a PDE for each sample point. The stochastic collocation hybrid approach proposed by [8] showed that the computational effort can be reduced significantly while ensuring the same accuracy and robustness as with a classic MC method. Nevertheless, there are a few drawbacks.…”

“…As mentioned before, the choice of the critical sample points is crucial, for efficiency and accuracy of this approach. In [7] and [8] adjoint error indicators are used. Here, we take advantage of the GPR that provides an error indicator in the point p in the form of the standard deviation σ GPR (p).…”

“…The benchmark problem, an academic example, on which we perform the numerical tests is a simple dielectrical waveguide, cf. [8]. We consider two uncertain geometrical parameters, the length of the dielectrical inlay p 1 , the length of the offset p 2 (see Fig.…”

“…In [7], a hybrid approach is proposed, using radial basis functions (RBF) for the surrogate model and an adjoint error indicator to choose the critical sample points. In [8] a similar hybrid approach is proposed, using stochastic collocation with polynomial basis functions and also an adjoint error indicator. In this paper we combine these ideas and propose a hybrid approach using Gaussian process regression (GPR) for both, building the surrogate model and obtaining an error indicator in form of the prediction standard deviation given by the GPR.…”