[Paper Review] High-Dimensional Uncertainty Quantification of Electronic and Photonic IC with Non-Gaussian Correlated Process Variations

Chunfeng Cui, Zheng Zhang|arXiv (Cornell University)|Jan 31, 2019

Probabilistic and Robust Engineering Design63 references4 citations

TL;DR

This paper proposes a high-dimensional uncertainty quantification framework for electronic and photonic ICs with non-Gaussian correlated process variations, using a tensor-based construction of smooth, orthonormal basis functions and a sparse solver with adaptive sampling. It achieves thousands of times speedup over Monte Carlo while accurately capturing complex output distributions with multiple peaks.

ABSTRACT

Uncertainty quantification based on generalized polynomial chaos has been used in many applications. It has also achieved great success in variation-aware design automation. However, almost all existing techniques assume that the parameters are mutually independent or Gaussian correlated, which is rarely true in real applications. For instance, in chip manufacturing, many process variations are actually correlated. Recently, some techniques have been developed to handle non-Gaussian correlated random parameters, but they are time-consuming for high-dimensional problems. We present a new framework to solve uncertainty quantification problems with many non-Gaussian correlated uncertainties. Firstly, we propose a set of smooth basis functions to well capture the impact of non-Gaussian correlated process variations. We develop a tensor approach to compute these basis functions in a high-dimension setting. Secondly, we investigate the theoretical aspect and practical implementation of a sparse solver to compute the coefficients of all basis functions. We provide some theoretical analysis for the exact recovery condition and error bound of this sparse solver in the context of uncertainty quantification. We present three adaptive sampling approaches to improve the performance of the sparse solver. Finally, we validate our methods by synthetic and practical electronic/photonic ICs with 19 to 57 non-Gaussian correlated variation parameters. Our approach outperforms Monte Carlo by thousands of times in terms of efficiency. It can also accurately predict the output density functions with multiple peaks caused by non-Gaussian correlations, which are hard to capture by existing methods.

Motivation & Objective

• Address the lack of efficient uncertainty quantification methods for high-dimensional, non-Gaussian correlated process variations in IC design.
• Overcome the limitations of existing stochastic spectral methods that assume independence or Gaussian correlation.
• Develop a scalable framework that maintains accuracy and efficiency in high-dimensional settings.
• Enable accurate prediction of complex output density functions with multiple peaks caused by non-Gaussian correlations.
• Reduce simulation cost through adaptive sampling and sparse solver techniques in high-dimensional parameter spaces.

Proposed method

• Propose a new class of smooth, orthonormal basis functions derived via Cholesky factorization to model non-Gaussian correlated uncertainties.
• Use functional tensor train decomposition to efficiently compute basis functions in high-dimensional parameter spaces.
• Implement an ℓ₀-minimization sparse solver with theoretical guarantees on recovery error and convergence.
• Introduce three adaptive sampling strategies to select informative samples and reduce simulation cost.
• Leverage orthonormality of basis functions to enable closed-form computation of mean and variance of output distributions.
• Apply compressed sensing theory to ensure stable and accurate coefficient recovery with minimal samples.

Experimental results

Research questions

• RQ1 How can non-Gaussian correlated process variations in high-dimensional IC design be accurately modeled using generalized polynomial chaos?
• RQ2 What is an efficient and scalable method to compute basis functions for high-dimensional, non-Gaussian correlated random parameters?
• RQ3 Can sparse solvers with adaptive sampling achieve high accuracy with significantly reduced simulation cost in high-dimensional UQ problems?
• RQ4 How does the proposed framework perform in capturing complex output distributions with multiple peaks, which are challenging for traditional methods?
• RQ5 What theoretical guarantees can be provided for the accuracy and convergence of the sparse solver in this context?

Key findings

• The proposed framework achieves over 1,000x speedup compared to Monte Carlo simulations in electronic and photonic ICs with 19 to 57 variation parameters.
• The method accurately captures output density functions with multiple peaks, a capability that is difficult for existing methods due to non-Gaussian correlations.
• Theoretical analysis confirms the exact recovery condition and provides error bounds for the sparse solver under the (s, κs)-RIP condition.
• Adaptive sampling strategies significantly improve the performance of the sparse solver by reducing the number of required simulations.
• The tensor-based basis function computation enables scalable handling of high-dimensional problems beyond the reach of prior methods.
• The framework maintains high accuracy even when parameters follow non-Gaussian distributions such as Gaussian mixture and Gamma distributions.