Efficient parametric yield estimation of analog/mixed-signal circuits via Bayesian model fusion

Abstract

Parametric yield estimation is one of the most critical-yetchallenging tasks for designing and verifying nanoscale analog and mixed-signal circuits. In this paper, we propose a novel Bayesian model fusion (BMF) technique for efficient parametric yield estimation. Our key idea is to borrow the simulation data from an early stage (e.g., schematic-level simulation) to efficiently estimate the performance distributions at a late stage (e.g., post-layout simulation). BMF statistically models the correlation between early-stage and late-stage performance distributions by Bayesian inference. In addition, a convex optimization is formulated to solve the unknown late-stage performance distributions both accurately and robustly. Several circuit examples designed in a commercial 32 nm CMOS process demonstrate that the proposed BMF technique achieves up to 3.75× runtime speedup over the traditional kernel estimation method.

Indirect performance sensing for on-chip analog self-healing via Bayesian model fusion

“…On the other hand, f OLD (x) and f NEW (x) cannot be exactly identical due to process shift. To statistically encode the "common" information between f OLD (x) and f NEW (x), we define a Gaussian distribution as our prior distribution for each new model coefficient α NEW,k : (5) where α OLD,k and λ 2 ⋅α OLD,k 2 are the mean and variance of the Gaussian distribution respectively, and λ is a parameter that can be determined by cross-validation [8]- [9]. The prior distribution in (5) has a two-fold meaning.…”

“…Based on Bayes' theorem [8]- [9], the uncertainties of the new coefficients {α NEW,k ; k = 1, 2, …, K} after knowing the data {(x (n) , f NEW (n) ); n = 1, 2, …, N} can be mathematically described by the following posterior distribution: 7…”

“…Similar to the parameter λ in (5), the value of σ 0 can be determined by cross-validation [8]- [9]. Since the modeling error associated with the n-th data point (x (n) , f NEW (n) ) is simply one sampling point of the random variable ε NEW that follows the Gaussian distribution in (8), the probability of observing the n-th data point is: Assuming that all data points {(x (n) , f NEW (n) ); n = 1, 2, …, N} are independently generated, we can write the likelihood function pdf(f NEW | α NEW ) as:…”

“…If the posterior distribution pdf(α NEW | f NEW ) reaches its maximum value at {α* NEW,k ; k = 1, 2, …, K}, these values {α* NEW,k ; k = 1, 2, …, K} are the optimal estimation of the new coefficients, since these coefficient values are most likely to occur. Such a method is referred to as the MAP estimation in the literature [8]- [9].…”

Efficient moment estimation with extremely small sample size via bayesian inference for analog/mixed-signal validation

“…There is a recent work [8] that considers a similar problem, but for performance modeling. Another recently published technique [9] solves a similar problem for post-layout performance distribution estimation, but with mildly small number of samples (50 or more).…”