A Modified Nested Sparse Grid Based Adaptive Stochastic Collocation Method for Statistical Static Timing Analysis

Xu LUO  Fan YANG  Xuan ZENG  Jun TAO  Hengliang ZHU  Wei CAI  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E92-A   No.12   pp.3024-3034
Publication Date: 2009/12/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E92.A.3024
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on VLSI Design and CAD Algorithms)
Category: Device and Circuit Modeling and Analysis
process variations,  statistical static timing analysis,  extended Gauss-Hermite quadrature,  nested sparse grid,  stochastic collocation method,  

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In this paper, we propose a Modified nested sparse grid based Adaptive Stochastic Collocation Method (MASCM) for block-based Statistical Static Timing Analysis (SSTA). The proposed MASCM employs an improved adaptive strategy derived from the existing Adaptive Stochastic Collocation Method (ASCM) to approximate the key operator MAX during timing analysis. In contrast to ASCM which uses non-nested sparse grid and tensor product quadratures to approximate the MAX operator for weakly and strongly nonlinear conditions respectively, MASCM proposes a modified nested sparse grid quadrature to approximate the MAX operator for both weakly and strongly nonlinear conditions. In the modified nested sparse grid quadrature, we firstly construct the second order quadrature points based on extended Gauss-Hermite quadrature and nested sparse grid technique, and then discard those quadrature points that do not contribute significantly to the computation accuracy to enhance the efficiency of the MAX approximation. Compared with the non-nested sparse grid quadrature, the proposed modified nested sparse grid quadrature not only employs much fewer collocation points, but also offers much higher accuracy. Compared with the tensor product quadrature, the modified nested sparse grid quadrature greatly reduced the computational cost, while still maintains sufficient accuracy for the MAX operator approximation. As a result, the proposed MASCM provides comparable accuracy while remarkably reduces the computational cost compared with ASCM. The numerical results show that with comparable accuracy MASCM has 50% reduction in run time compared with ASCM.