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

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 10times 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 100times speeds up.

[1]  P. Ghanta,et al.  A Framework for Statistical Timing Analysis using Non-Linear Delay and Slew Models , 2006, 2006 IEEE/ACM International Conference on Computer Aided Design.

[2]  Athanasios Papoulis,et al.  Probability, Random Variables and Stochastic Processes , 1965 .

[3]  Dongbin Xiu,et al.  The Wiener-Askey Polynomial Chaos for Stochastic Differential Equations , 2002, SIAM J. Sci. Comput..

[4]  C. E. Clark.  The Greatest of a Finite Set of Random Variables , 1961 .

[5]  Ankur Srivastava,et al.  A general framework for accurate statistical timing analysis considering correlations , 2005, Proceedings. 42nd Design Automation Conference, 2005..

[6]  K. Ritter,et al.  Simple Cubature Formulas with High Polynomial Exactness , 1999 .

[7]  Jun Li,et al.  A probabilistic collocation method based statistical gate delay model considering process variations and multiple input switching , 2005, Design, Automation and Test in Europe.

[8]  Yehea I. Ismail,et al.  Statistical static timing analysis: how simple can we get? , 2005, Proceedings. 42nd Design Automation Conference, 2005..

[9]  Yu Hen Hu,et al.  Correlation-preserved non-Gaussian statistical timing analysis with quadratic timing model , 2005, Proceedings. 42nd Design Automation Conference, 2005..

[10]  Bruno O. Shubert,et al.  Random variables and stochastic processes , 1979 .

[11]  Vladimir Zolotov,et al.  Parameterized block-based statistical timing analysis with non-Gaussian parameters, nonlinear delay functions , 2005, Proceedings. 42nd Design Automation Conference, 2005..

[12]  Gene H. Golub,et al.  Calculation of Gauss quadrature rules , 1967, Milestones in Matrix Computation.

[13]  Andrzej J. Strojwas,et al.  Correlation-aware statistical timing analysis with non-Gaussian delay distributions , 2005, Proceedings. 42nd Design Automation Conference, 2005..

[14]  Sachin S. Sapatnekar,et al.  Statistical timing analysis with correlated non-Gaussian parameters using independent component analysis , 2006, 2006 43rd ACM/IEEE Design Automation Conference.

[15]  Charlie Chung-Ping Chen,et al.  Non-gaussian statistical parameter modeling for SSTA with confidence interval analysis , 2006, ISPD '06.

[16]  Dongbin Xiu,et al.  High-Order Collocation Methods for Differential Equations with Random Inputs , 2005, SIAM J. Sci. Comput..

[17]  Xuan Zeng,et al.  A sparse grid based spectral stochastic collocation method for variations-aware capacitance extraction of interconnects under nanometer process technology , 2007 .

[18]  K. Ravindran,et al.  First-Order Incremental Block-Based Statistical Timing Analysis , 2004, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.