EE6361 : Importance Sampling

Transcription

1 EE6361 : Importance Sampling Electrical Engineering Department IIT Madras Janakiraman Viraraghavan

2 Agenda Yield analysis in memories Statistical Compact Model Traditional Monte-Carlo Limitations of Monte-Carlo Variance Reduction Importance Sampling Statistical Blockade 2

3 Yield Analysis in Memories WL WL BLt BLc 6T SRAM Bit Cell Smallest possible transistors used Prone to large variability Very high yield expected ( %) Foundry has to guarantee 2

4 Variability in Fab (Idsat, Vtsat, Vtlin, Idlin) Statistical Modeling Focus (Local and Global variations to be captured) P (Nominal, USL, LSL) (Idsat, Vtsat, Vtlin, Idlin) Statistical compact model available to designers for use 2

5 Statistical Compact Model VD VG Idlin Maps measured variations of voltages and currents to process variations Back propagation of Variance 5

6 Monte Carlo Analysis VT Le SPI CE MODEL Tox SPICE measurements Generate random samples from the distribution Simulate with each set of values 6

7 Monte Carlo Analysis Trial # X1, X2.. XK Y 1 x11, x21.. xk1 Y1 2 x12, x22.. xk2 Y2 x1n, x2n.. xkn YN ~ N Mean Sample Mean Ŷ1N Variance 7

8 Monte Carlo Analysis Monte Carlo Run 1 Monte Carlo Run 1 Monte Carlo Run 1 Trial # Monte Carlo Run M Trial # # 1 Trial Trial # ~ 22 ~ N ~~ N N Mean N Mean Mean Mean Variance Variance Variance Variance Run # Mean 1 Ŷ1N 2 Ŷ1N ~ M ŶMN ŶMN 8

9 Monte Carlo Analysis CLT - Gaussian Run # Mean 1 Ŷ11 2 Ŷ12 ~ M Ŷ1M 9

10 Monte Carlo Analysis Variance Reduction As N increases variance decreases By increasing N, you can get arbitrarily close 10

11 Monte Carlo Analysis Estimating N Standard Normal RV 11

12 Rare Event Estimation Y=1 Y=0 Low probability 12

13 Monte Carlo Analysis Estimating N Y=1 Y=0 Rare event Unlikely to be sampled by Monte-Carlo 13

14 Monte Carlo Estimation P(Z > 1.5) Z Standard Normal Random Variable Different trials give different answers! D The difference reduces with large N

15 Monte Carlo Estimation P(Z > 6) Z Standard Normal Random Variable No sample generated in region of interest

16 Monte Carlo Analysis Variance Reduction As N increases - variance decreases Can you reduce the variance in any other way? 16

17 Monte Carlo Analysis Variance Reduction As N increases - variance decreases Can you reduce the variance in any other way? 17

18 Monte Carlo Analysis Variance Reduction 18

19 Monte Carlo Analysis Variance Reduction 19

20 Importance Sampling Define a new function Z where gx(x) is a new distribution Z should have the same mean as Y by lower variance 20

21 Importance Sampling - Mean Y - X is drawn from f() Z X is drawn from g() 21

22 Importance Sampling - Variance Y - X is drawn from f() Z X is drawn from g() 22

23 Importance Sampling - Variance Desired! 23

24 Importance Sampling - Variance Desired! Choose g(x) such that this ratio less than 1 for all values of x! 24

25 Importance Sampling Choice of g(x) How can a function be less than 1 for all values of x But have mean of 1? 25

26 Importance Sampling Variance Choice of g(x) How can a function be less than 1 for all values of x But have mean of 1? g(x) is also a PDF Has to integrate to unity! 26

27 Importance Sampling Choice of g(x) Ratio will be greater than 1 for some values of x If h(x) is strictly zero in those regions? 27

28 Failure Probability Estimation E[h(X)] = 1*p(X>= a) + 0*p(X<a) Low probability 28

29 Failure Probability Estimation Region of importance 29

30 Example: P(Z>6) Z Standard Normal Random Variable 2 1 f X ( x)= e 2 π x g X (x)= e 2 π 2 (x 2) 2 30

31 Estimation with Importance Sampling P(Z > 6) Z Standard Normal Random Variable Convergent estimate obtained

32 Failure Probability Estimation - Where does it fail? Prob ~ 0.4 Region of importance Centroid Beware when estimating large probabilities! 32

33 References Kanj, R.; Joshi, R.; Nassif, S., "Mixture importance sampling and its application to the analysis of SRAM designs in the presence of rare failure events," in Design Automation Conference, rd ACM/IEEE, vol., no., pp.69-72, doi: /DAC Probability Models second edition by Sheldon Ross Introduction to Probability and Statistics by Sheldon Ross 33