Stochastic Collocation With Non-Gaussian Correlated Process Variations: Theory, Algorithms, and Applications

Abstract

Stochastic spectral methods have achieved great success in the uncertainty quantification of many engineering problems, including electronic and photonic integrated circuits influenced by fabrication process variations. Existing techniques employ a generalized polynomial-chaos expansion, and they almost always assume that all random parameters are mutually independent or Gaussian correlated. However, this assumption is rarely true in real applications. How to handle non-Gaussian correlated random parameters is a long-standing and fundamental challenge. A main bottleneck is the lack of theory and computational methods to perform a projection step in a correlated uncertain parameter space. This paper presents an optimization-based approach to automatically determinate the quadrature nodes and weights required in a projection step, and develops an efficient stochastic collocation algorithm for systems with non-Gaussian correlated parameters. We also provide some theoretical proofs for the complexity and error bound of our proposed method. Numerical experiments on synthetic, electronic and photonic integrated circuit examples show the nearly exponential convergence rate and excellent efficiency of our proposed approach. Many other challenging uncertainty-related problems can be further solved based on this work.

Chance-Constrained and Yield-Aware Optimization of Photonic ICs With Non-Gaussian Correlated Process Variations

“…If the parameters ξ are non-Gaussian correlated, the computation is more difficult. In such cases, Ψ α (ξ) can be constructed by the Gram-Schmidt approach in [28], [29] or the Cholesky factorization in [45], [46]. The main difficulty lies in computing high order moments of ξ, which can be well resolved by the functional tensor train approach proposed in [46].…”

“…In this section, we build the surrogate model for f (x, ξ) and {y i (x, ξ)} n i=1 by using generalized polynomial chaos [47] and our recent developed uncertainty quantification solver [28], [29]. Once the surrogate models are constructed, we can perform deterministic optimization.…”

“…Recently, uncertainty quantification methods based on generalized polynomial chaos have gained great success in modeling the uncertainty caused by various process variations in electronic and photonic ICs [16]- [27]. A novel stochastic collocation approach was further proposed in [28], [29] to handle non-Gaussian correlated process variations, which shows significantly better accuracy and efficiency than [30] due to the smooth basis functions and an optimization-based quadrature rule. These techniques can construct stochastic surrogate models with a small number of simulation samples, but their power in yield optimization has not been well explored or exploited despite recent robust optimization methods [31] based on generalized polynomial chaos.…”

“…Paper Contributions. Leveraging the chance-constrained optimization [32] and our recent uncertainty quantification solvers [28], [29], this paper presents a data-efficient technique to optimize photonic ICs with non-Gaussian correlated process variations. Instead of just optimizing the yield, we optimize a target performance metric while enforcing the probability of violating some design rules to be smaller than a user-defined threshold.…”

“…In many cases, both the objective function and constraints are only available through an expensive black-box simulator. To reduce the simulation time, we build a surrogate model based on the recent uncertainty quantification solver [28], [29]. We propose a new three-stage process: firstly we generate quadrature points for the design variables; secondly, we compute the quadrature points for the uncertainty parameter; thirdly we re-optimize the quadrature points in their joint variable space.…”