Importance Sampling Reuse for Cell Failure Rate Estimation
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Solution Overview
Problem
Traditional methods for estimating cell failure rates in memory arrays, such as Monte-Carlo analysis and sensitivity analysis, are inefficient and inaccurate at low failure rates, especially in large arrays with stringent failure rate control requirements, due to the sparse distribution of samples in the tail regions and exponential increase in simulations with the number of independent variables.
Innovation Solution
The method employs importance sampling reuse by performing uniform sampling to find failing samples, determining their center of gravity, and recomputing importance sampling weight ratios for new origins, allowing for efficient estimation of cell failure rates and device yield through mixture importance sampling, which concentrates sampling in failure regions and reduces computational overhead.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If Monte-Carlo analysis is used to estimate cell failure rate, then accurate results are obtained, but the number of iterations becomes very large due to random sampling of the entire probability space
Solution Approach 1:
The patent applies local quality by concentrating sampling efforts specifically in the failure region (tail of the distribution) rather than uniformly sampling the entire probability space. Importance sampling functions are constructed to generate samples preferentially in the failure region, allowing accurate failure rate estimation with fewer iterations compared to standard Monte-Carlo analysis.
Solution Approach 2:
The patent changes the sampling distribution parameters by introducing importance sampling functions that modify the probability density to concentrate samples in the failure region. This parameter transformation allows the sampling process to target critical regions more efficiently, reducing the number of iterations needed for accurate failure rate estimation.
2Reliability
If the cell failure rate decreases to achieve higher yield, then device quality improves, but the number of samples required for accurate analysis increases exponentially
Solution Approach 1:
The patent applies local quality by concentrating sampling efforts specifically in the failure region (tail of the distribution) rather than uniformly sampling the entire probability space. Importance sampling functions are constructed to generate samples preferentially in the failure region, allowing accurate failure rate estimation with fewer iterations compared to standard Monte-Carlo analysis.
Solution Approach 2:
The patent performs preliminary action by first identifying the failure region through an initial uniform sampling phase, then using this information to construct importance sampling functions tailored to that specific failure region. This preliminary identification allows subsequent sampling to be highly efficient, avoiding the need for exponentially increasing sample sizes as failure rates decrease.
3Measurement precision
If grid analysis approach is used to accurately estimate failure rate, then precise results are obtained, but the number of simulations increases exponentially with the number of independent variables
Solution Approach 1:
The patent changes the sampling distribution parameters by introducing importance sampling functions that modify the probability density to concentrate samples in the failure region. This parameter transformation allows the sampling process to target critical regions more efficiently, reducing the number of iterations needed for accurate failure rate estimation.
Solution Approach 2:
The patent transitions from a grid-based approach in parameter space to a sampling-based approach in the probability space, effectively changing the dimensionality of the problem. By using importance sampling functions that operate in the probability space rather than systematically exploring parameter grids, the method avoids the exponential complexity associated with high-dimensional grid analysis.
Data Source
AI summary
A mechanism is provided for reusing importance sampling for efficient cell failure rate estimation of process variations and other design considerations. First, the mechanism performs a search across circuit parameters to determine failures with respect to a set of performance variables. For a single failure region, the initial search may be a uniform sampling of the parameter space. Mixture importance sampling (MIS) efficiently may estimate the single failure region. The mechanism then finds a center of gravity for each metric and finds importance samples. Then, for each new origin corresponding to a process variation or other design consideration, the mechanism finds a suitable projection and recomputes new importance sampling (IS) ratios.