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

Abstract

Stochastic spectral methods have achieved a great success in the uncertainty quantification of many engineering problems, including variation-aware electronic and photonic design automation. State-of-the-art techniques employ generalized polynomial-chaos expansions and assume that all random parameters are independent or Gaussian correlated. This assumption is rarely true in real applications. How to handle non-Gaussian correlated random parameters is a long-standing and fundamental challenge: It is not clear how to choose basis functions and to perform a projection step in a correlated uncertain parameter space. This paper first presents a new set of basis functions to well capture the impact of non-Gaussian correlated parameters and then proposes an automatic and optimization-based quadrature method to perform projection-based stochastic collocation with a few simulation samples in the correlated parameter space. We further provide some theoretical proofs for the complexity and error bound of our proposed method. The numerical experiments on several synthetic, electronic, and photonic integrated circuit examples show the nearly exponential convergence rate of our approach and its significant ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$700 \times $ </tex-math></inline-formula> – <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$6000 \times $ </tex-math></inline-formula> ) speedup than Monte Carlo. Many other open problems with non-Gaussian correlated uncertainties can be further solved based on this paper.