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Beyond the Yield Barrier: Variational Importance Sampling Yield Analysis

Yanfang Liu, Lei He, Wei W. Xing

TL;DR

This work addresses the computational bottleneck of yield estimation in IC design by introducing Variational Importance Sampling (VIS), a rigorous variational framework that analyzes and improves importance-sampling-based yield estimators. VIS proves that the optimal OMSV lies beyond the failure boundary and yields closed-form updates for the mean and covariance (True OMSV and Full SSS), while extending to skewed and mixed proposals (Skew Normal and Mixture of Skew Normals). Building on VIS, BEYOND combines True OMSV, Full SSS, and MSN to achieve up to 29.03x speedups and improved accuracy across multiple circuits, with additional benefits when calibrating state-of-the-art yield optimization (Variational-ASAIS). The approach demonstrates strong performance on SRAM and analog circuits, including ablation and robustness analyses, suggesting practical impact for fast, reliable yield analysis in modern IC design, especially in high-variance, high-dimensional settings.

Abstract

Optimal mean shift vector (OMSV)-based importance sampling methods have long been prevalent in yield estimation and optimization as an industry standard. However, most OMSV-based methods are designed heuristically without a rigorous understanding of their limitations. To this end, we propose VIS, the first variational analysis framework for yield problems, enabling a systematic refinement for OMSV. For instance, VIS reveals that the classic OMSV is suboptimal, and the optimal/true OMSV should always stay beyond the failure boundary, which enables a free improvement for all OMSV-based methods immediately. Using VIS, we show a progressive refinement for the classic OMSV including incorporation of full covariance in closed form, adjusting for asymmetric failure distributions, and capturing multiple failure regions, each of which contributes to a progressive improvement of more than 2x. Inheriting the simplicity of OMSV, the proposed method retains simplicity and robustness yet achieves up to 29.03x speedup over the state-of-the-art (SOTA) methods. We also demonstrate how the SOTA yield optimization, ASAIS, can immediately benefit from our True OMSV, delivering a 1.20x and 1.27x improvement in performance and efficiency, respectively, without additional computational overhead.

Beyond the Yield Barrier: Variational Importance Sampling Yield Analysis

TL;DR

This work addresses the computational bottleneck of yield estimation in IC design by introducing Variational Importance Sampling (VIS), a rigorous variational framework that analyzes and improves importance-sampling-based yield estimators. VIS proves that the optimal OMSV lies beyond the failure boundary and yields closed-form updates for the mean and covariance (True OMSV and Full SSS), while extending to skewed and mixed proposals (Skew Normal and Mixture of Skew Normals). Building on VIS, BEYOND combines True OMSV, Full SSS, and MSN to achieve up to 29.03x speedups and improved accuracy across multiple circuits, with additional benefits when calibrating state-of-the-art yield optimization (Variational-ASAIS). The approach demonstrates strong performance on SRAM and analog circuits, including ablation and robustness analyses, suggesting practical impact for fast, reliable yield analysis in modern IC design, especially in high-variance, high-dimensional settings.

Abstract

Optimal mean shift vector (OMSV)-based importance sampling methods have long been prevalent in yield estimation and optimization as an industry standard. However, most OMSV-based methods are designed heuristically without a rigorous understanding of their limitations. To this end, we propose VIS, the first variational analysis framework for yield problems, enabling a systematic refinement for OMSV. For instance, VIS reveals that the classic OMSV is suboptimal, and the optimal/true OMSV should always stay beyond the failure boundary, which enables a free improvement for all OMSV-based methods immediately. Using VIS, we show a progressive refinement for the classic OMSV including incorporation of full covariance in closed form, adjusting for asymmetric failure distributions, and capturing multiple failure regions, each of which contributes to a progressive improvement of more than 2x. Inheriting the simplicity of OMSV, the proposed method retains simplicity and robustness yet achieves up to 29.03x speedup over the state-of-the-art (SOTA) methods. We also demonstrate how the SOTA yield optimization, ASAIS, can immediately benefit from our True OMSV, delivering a 1.20x and 1.27x improvement in performance and efficiency, respectively, without additional computational overhead.
Paper Structure (23 sections, 16 equations, 10 figures, 7 tables, 1 algorithm)

This paper contains 23 sections, 16 equations, 10 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Problem Definition
  4. Monte Carlo Yield Estimation
  5. Importance Sampling Yield Estimation
  6. MN-OMSV
  7. Proposed Approach
  8. Variational Analysis of IS Yield Estimation
  9. True OMSV
  10. Full SSS
  11. Skew Normal Distribution
  12. Mixture of Skew Normal Distribution
  13. Complexity and Implementation
  14. Calibration of SOTA Yield Optimization
  15. Experimental Results
  16. ...and 8 more sections

Figures (10)

  • Figure 1: Illustration of progressive refinement of the classic MN-OMSV using VIS.
  • Figure 2: The structure of SRAM column circuit
  • Figure 3: Failure rate estimation with FoM on 6T-SRAM
  • Figure 4: Failure rate estimation with FoM on OTA
  • Figure 5: Operational Transconductance Amplifier Circuit
  • ...and 5 more figures