Adaptive Stochastic Collocation Method (ASCM) for Parameterized Statistical Timing Analysis with Quadratic Delay Model

Abstract

In this paper, we propose an Adaptive Stochastic Collocation Method for block-based Statistical Static Timing Analysis (SSTA). A novel adaptive method is proposed to perform SSTA with delays of gates and interconnects modeled by quadratic polynomials based on Homogeneous Chaos expansion. In order to approximate the key atomic operator MAX in the full random space during timing analysis, the proposed method adaptively chooses the optimal algorithm from a set of stochastic collocation methods by considering different input conditions. Compared with the existing stochastic collocation methods, including the one using dimension reduction technique and the one using Sparse Grid technique, the proposed method has 10x improvements in the accuracy while using the same order of computation time. The proposed algorithm also show great improvement in accuracy compared with a moment matching method. Compared with the 10,000 Monte Carlo simulations on ISCAS85 benchmark circuits, the results of the proposed method show less than 1% error in the mean and variance, and nearly 100x speeds up.9th International Symposium on Quality Electronic Design 0-7695-3117-2/08 $25.00

Statistical reliability analysis under process variation and aging effects

“…Inspired by the work of [24], we propose a fast pruning algorithm which prunes redundant timing nodes and edges and thus improves the subsequent repeated analysis.…”

“…It is observed in [24] that the distributions of the arrival time at the different inputs of a node may be distantly separated. The statistical maximum of them, which is the distribution of arrival time at the output of the node, will also be separated from some of the inputs.…”

Binning optimization based on SSTA for transparently-latched circuits

“…The sample-based SSTA performs DSTA (deterministic STA) on a set of sample points, and fits the distribution of T work by using the deterministic results at all sample points. The samplebased stochastic collocation method (SCM) for FF-clocked circuits is proposed in [13] and [14] where the variations are assumed to be Gaussian distributed. In Section V, we introduce the gSCM [15] to SSTA for latched circuits under process variations of arbitrary distribution.…”

Timing yield driven clock skew scheduling considering non-Gaussian distributions of critical path delays

“…Equations (11) and (13) have been widely used as problem formulations by previous works [3,9,10] in yield driven clock skew scheduling. Here we provide the statistical interpretations for them based on our proposed formulation in (9).…”

“…In order to consider the statistical difference of each critical path delays during clock skew scheduling, the yield optimization problem is transformed to a minimum cost-to-time ratio problem in [9,10] by introducing variance 1 of each path delay distribution into the corresponding deſnition of the feasible region for skew, and adopts the minimum balancing algorithm for skew assignment (referred to as "PROP"). However, both EVEN and PROP cannot accurately handle non-Gaussian critical path delays shown by latest statistical static timing analysis [11][12][13] on circuits with nanometer technology.…”