[论文解读] 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 Design参考文献 63被引用 4

一句话总结

本文提出了一种用于电子与光子集成电路的高维不确定性量化框架，针对非高斯相关工艺偏差，采用基于张量的光滑正交基函数构造方法与自适应采样稀疏求解器。该方法相较蒙特卡洛方法实现数千倍的速度提升，同时准确捕捉具有多个峰值的复杂输出分布。

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.

研究动机与目标

• 解决集成电路设计中高维、非高斯相关工艺偏差下高效不确定性量化方法的缺失问题。
• 克服现有随机谱方法在假设独立性或高斯相关性时的局限性。
• 开发一种可扩展的框架，在高维场景下保持精度与效率。
• 实现对由非高斯相关性引起的复杂输出密度函数（具有多个峰值）的精确预测。
• 通过自适应采样与稀疏求解器技术，在高维参数空间中降低仿真成本。

提出的方法

• 提出一类通过Cholesky分解导出的新型光滑正交基函数，用于建模非高斯相关不确定性。
• 采用函数奇偶张量列车分解，在高维参数空间中高效计算基函数。
• 实现ℓ₀-最小化稀疏求解器，并提供恢复误差与收敛性的理论保证。
• 提出三种自适应采样策略，以选择信息丰富的样本，降低仿真成本。
• 利用基函数的正交性，实现输出分布均值与方差的闭式计算。
• 应用压缩感知理论，确保在最小样本数下实现稳定且精确的系数恢复。

实验结果

研究问题

• RQ1 如何利用广义多项式混沌方法，准确建模高维集成电路设计中的非高斯相关工艺偏差？
• RQ2 在高维、非高斯相关随机参数下，如何实现高效且可扩展的基函数计算方法？
• RQ3 在高维不确定性量化问题中，结合自适应采样的稀疏求解器能否在显著降低仿真成本的同时实现高精度？
• RQ4 所提出的框架在捕捉具有多个峰值的复杂输出分布方面表现如何，此类分布对传统方法构成挑战？
• RQ5 在此背景下，稀疏求解器的精度与收敛性可提供哪些理论保证？

主要发现

• 所提框架在具有19至57个变异参数的电子与光子集成电路中，相较蒙特卡洛仿真实现超过1,000倍的速度提升。
• 该方法能准确捕捉具有多个峰值的输出密度函数，而现有方法因非高斯相关性而难以实现此能力。
• 理论分析确认了精确恢复条件，并在(s, κs)-RIP条件下提供了稀疏求解器的误差界。
• 自适应采样策略显著提升了稀疏求解器的性能，大幅减少了所需仿真次数。
• 基于张量的基函数计算方法实现了对先前方法难以处理的高维问题的可扩展处理。
• 当参数服从非高斯分布（如高斯混合分布与伽马分布）时，该框架仍保持高精度。