Hyperbolic Polynomial Chaos Expansion (HPCE) and its Application to Statistical Analysis of Nonlinear Circuits

Transcription

1 Hyperbolic Polynomial Chaos Expansion HPCE and its Application to Statistical Analysis of Nonlinear Circuits Majid Ahadi, Aditi Krishna Prasad, Sourajeet Roy High Speed System Simulations Laboratory Department of Electrical & Computer Engineering Colorado State University 1

2 Agenda Background of Polynomial Chaos PC methods Proposed Hyperbolic Polynomial Chaos Expansion HPCE approach Numerical example Summary 2

3 Motivation Importance of stochastic simulation: Manufacturing process variations at sub-90nm technology is random by nature Unpredictable operating conditions temperature variability, EM environment Propagation of device/structure level uncertainty to system level response can affect chip, package, board level characteristics 3

4 Generalized Polynomial Chaos gpc theory in recent years Limitations: Motivation Most methods scale at a polynomial fashion with respect to number of random variables curse of dimensionality. Each simulation of distributed networks scales as ON α ; where N - size of the network and 3 4 4

5 Generalized Polynomial Chaos Expansion Polynomial chaos expansion: t, λ t 0 0 λ 1 t 1 λ... P t P λ Orthogonal polynomials from Weiner-Askey family with respect to distributions P + 1 = m+n!, m being the common maximum degree of the m!n! univariate polynomials, n is number of random variables λ [ λ, λ 1 2,.., λ ] t n 5

6 Mean: Generalized Polynomial Chaos Expansion Variance: E x t, λ 0 t Var x t, λ t t... t P Coefficients can be found by intrusive approaches like Galerkin or non-intrusive methods such as stochastic collocation, pseudo spectral collocation and linear regression. 6

7 Non-intrusive Use of commercial software Scalability and Parallelization Intrusive Higher accuracy Non-intrusive vs. Intrusive The proposed approach applies to both- Focused on non-intrusive 7

8 Agenda Background of Polynomial Chaos PC methods Proposed Hyperbolic Polynomial Chaos Expansion HPCE approach Numerical example Summary 8

9 Polynomial Chaos in general nonlinear networks The stochastic modified nodal analysis MNA equation PC approximates circuit response as: Multidimensional bases: d = [d 1, d 2,, d n ] 9,,, t t dt t d t B λ F λ C λ λ λ G P 0 k k k t t, λ λ n i i d k i 1

10 Classical PC Linear constraint d 1 d d Proposed HPCE scheme d n m Proposed HPC hyperbolic constraint, u is the hyperbolic factor d u u u u 1/ u d d d m n λ 2 λ 2 d 1 m d u m; u<1 λ 1 λ 1 10

11 Adaptive determination of the hyperbolic factor λ 2 u = 1, HPCE turns to the full-blown PCE. u 0, HPCE turns to 1D bases. Main challenge: tuning the hyperbolic factor Increasing u λ 1 11

12 Adaptive determination of the hyperbolic factor 12 A E A: Information matrix made from 1-D polynomial bases. : Unknown coefficients. E: SPICE simulation results. K: Number of 1-D bases. ; ; K K K K K K λ λ E I λ I λ I λ I λ A

13 Adaptive determination of the hyperbolic factor Augmentation of A, and E. Step j : q bases more than step j-1. Matrix A = A + q new columns + q new rows using new bases. Vector = + q new coefficients of additional bases. Vector E = E + q new SPICE simulation results. The equation is solved again to find all members. Previous nodes and simulation results are reused. A E 13

14 Adaptive determination of the hyperbolic factor Augmentation continues until the desired accuracy is achieved. A E Z. Zhang, T. A. El-Moselhy, I. M. Elfadel and L. Daniel, Stochastic testing method of transistor level uncertainty quantification based on generalized polynomial chaos, IEEE Trans. Computer Aided Design, vol. 32, no. 10, pp , Oct Ahadi, Majid, and Sourajeet Roy. "Sparse Linear Regression SPLINER Approach for Efficient Multidimensional Uncertainty Quantification of High-Speed Circuits., IEEE Trans. on Computer-Aided Design of Integrated Circuits and Systems, 2015, Available as early access 14

15 Finding enrichment added to variance Enrichment per added base: residual error from previous step 15 d = Number of bases at step j c = Number of bases at step j-1 d c = q, number of added bases q t t t t En d c k k j c k k j d c k k j *

16 Program flow Yes No 16

17 Agenda Background of Polynomial Chaos PC methods Proposed Hyperbolic Polynomial Chaos Expansion HPCE approach Numerical example Summary 17

18 Numerical example Objective: Accuracy of the proposed method compared to traditional PC Random Variables Mean % Relative SD d µm d µm d 3 70 µm W µm W µm W µm h 1 30 µm h 2 20 µm 30 % Uniform Distribution h 3 40 µm C 1 C 2 C 3 1 pf 0.5 pf 1.5 pf TL Conductivity 5.800E+07 Ω -1 /m 18

19 Enrichment per added base Number of added bases * u < u < ** 0.79 u < u = 1 Average En.t of variance at N E E E E-05 Average En.t of variance at N E E E E-05 *When 0.5 u < 0.69: only 53 1-D bases. **573 bases are used for this example. 19

20 Statistical Results Mean +/- 3σ 20

21 Statistical Results PDF at maximum standard deviation of N 1 and N 2 outputs. 21

22 Scaling Results Number of RVs Random Variables CPU Time for Proposed HPC s CPU Time for Full-blown PCs 5 d 1, d 2, d 3, w 1, w d 1, d 2, d 3, w 1, w 2, w d 1, d 2, d 3, w 1, w 2, w 3, h d 1, d 2, d 3, w 1, w 2, w 3, h 1, h d 1, d 2, d 3, w 1, w 2, w 3, h 1, h 2, h 3, d 1, d 2, d 3, w 1, w 2, w 3, h 1, h 2, h 3, C d 1, d 2, d 3, w 1, w 2, w 3, h 1, h 2, h 3, C 1, C 2 12d 1, d 2, d 3, w 1, w 2, w 3, h 1, h 2, h 3, C 1, C 2, C 3 13d 1, d 2, d 3, w 1, w 2, w 3, h 1, h 2, h 3, C 1, C 2, C 3, TLC

23 Agenda Background of Polynomial Chaos PC methods Proposed Hyperbolic Polynomial Chaos Expansion HPCE approach Numerical example Summary 23

24 Summary A novel improvement to the polynomial chaos approach for the uncertainty analysis of high speed circuits is presented. The development of an alternative hyperbolic truncation scheme to replace the conventional linear truncation scheme. Results in a sparse PC expansion for marginal loss of accuracy. A greedy adaptive methodology to determine the number of basis terms. The accuracy and scalability is evaluated via a numerical example. 24

25 Thank You 25