1. Chapter 4 Stochastic Modeling Prof. Lei He Electrical Engineering Department University of California, Los Angeles URL: eda.ee.ucla.edu Email: lhe@ee.ucla.edu μxμx σxσx x f (x)

2. Outline 1. Monte Carlo Simulation 2. SSTA

3. Monte Carlo Simulation Problem Formulation Given a set of random variables X=(X 1, X 2, … X n ) T and a function of X, Y=f(X), estimate the distribution of the Y Method Generate N samples of X=(X 1, X 2, … X n ) T For each sample of X, calculate the correspondent sample of Y=f(X) Obtain the distribution of Y from the samples of Y

4. Advantage and Disadvantage of MC simulation Advantage n Accurate – – Error→0 when N→∞ n Flexible – Works for any arbitrary distribution of X – Works for any arbitrary function of f n Simple – Easy to implement n Usually used as golden case in statistical analysis Disadvantage n Not efficient – Need large N to obtain high accuracy – Need to run large number of iterations n Not suitable for statistical optimization

5. Example Given X 1 and X 2 are independent standard Gaussian RVs, estimate the distribution of max(X 1, X 2 )

6. Quasi Monte Carlo Simulation Basic idea n Use deterministic samples instead of pure random samples n Select deterministic samples to cover the whole sample space evenly

7. Discrepancy Definition n N is total number of samples, A(B, P) is the number of points in bounding box B, λ s (B) is the volume of B Discrepancy measures how evenly the samples are in the sample place

8. Low Discrepancy Sequence Sample sequence with low discrepancy Low discrepancy array generation algorithms n Faure sequence n Neiderreiter sequence n Sobol sequence n Halton Sequence

9. Example: Halton Sequence Basic idea n Choose a prime number as base (let's say 2) n Write natural number sequence 1, 2, 3,... in base n Reverse the digits, including the decimal sign n Convert back to base 10: – 1 = 1.0 => 0.1 = 1/2 – 2 = 10.0 => 0.01 = 1/4 – 3 = 11.0 => 0.11 = 3/4 – 4 = 100.0 => 0.001 = 1/8 – 5 = 101.0 => 0.101 = 5/8 – 6 = 110.0 => 0.011 = 3/8 – 7 = 111.0 => 0.111 = 7/8 High dimensional array n Use different base for different dimension n Example 2-d array, X-base 2, y-base 3 – 1 => x=1/2 y=1/3 – 2 => x=1/4 y=2/3 – 3 => x=3/4 y=1/9 – 4 => x=1/8 y=4/9 – 5 => x=5/8 y=7/9 – 6 => x=3/8 y=2/9 – 7 => x=7/8 7=5/9

10. Advantage and Disadvantage of QMC Simulation Advantage n Efficient – Use fewer sample than random Monte Carlo simulation Disadvantage n Only works in low dimension cases n Very slow when number of random variations become large n Not very common in statistical analysis

11. Comparison of MC and QMC QMC converges faster than MC

12. Reference Singhee, A., Rutenbar, R. “From Finance to Flip Flops: A sstudy of Fast Quasi-Monte Carlo Methods from Computational Finance Applied to Statistical Circuit Analysis.” (see related student presentation at http://eda.ee.ucla.edu/EE201C/index.php?n=StudentPresentation09 S.StudentPresentation09S)

13. Homework 4 Yield Estimation using Monte Carlo Method n Consider “access time failure” : the time that voltage difference between BL_B and BL becomes larger than certain value. n The schematic are shown as below Initial Value: BL_B=1; Q_B=0; Q=1; BL=1; Variation Source: 1.V th (threshold voltage) of Mn1 and Mn2 2.L eff of Mn1 and Mn2 Device Model: Use BSIM3 model for all MOSFETs

14. netlist Netlist for 6-T cell SRAM * SRAM netlist Vdd dd 0 5 Mn1 3 2 0 0 nmos Mn2 3 5 4 4 nmos Mn3 2 3 0 0 nmos Mn4 2 5 1 1 nmos Mp5 3 2 dd dd pmos Mp6 2 3 dd dd pmos all MOSFETs should use BSIM3 model

15. Detailed Steps Performance Constraint: The voltage difference between BL_B and BL should be larger than ∆v at the time-step t thresh. Use Monte-Carlo and Quasi-Monte Carlo to calculate the yield Y, which is the percentage of circuits with satisfied performance. Steps: – (1) Use MC and QMC to generate random sequences for two variable parameters with Matlab code. – (2) Perform transient simulations with these sequences, and compare the variable performance with constraint. – (3) Calculate the yield rate with definition. Nominal Values, Performance Constraint and Matlab code will be provided soon Due May 8th