1. EE201C Final Project Adeel Mazhar Charwak Apte

2. Problem Statement Need to consider reading and writing failure – Pick design point which minimizes likelihood of failure – (Diagrams taken from problem statement PDF file – Credited to Fang Gong.)

3. Exhaustive Search Done in 2 phases – First phase, 10 steps per design variable, 100 QMC samples per design point – Second phase, 5 steps per design variable, 15000 QMC samples per design point – Yield was given priority over power and area – Multiple points were found with the same yield. – Sort by area, then pick lowest power @ min area

4. Many Points with same yield M2vthM5vthM2leffM5leffYieldPowerArea 0.4250.20.095 0.9997999873.19E-051.148813 0.4250.20.09750.0950.9997999873.18E-051.14975 0.4250.20.10.0950.9997999873.18E-051.150688 0.4250.20.10250.0950.9997999873.18E-051.151625 0.4250.20.1050.0950.9997999873.17E-051.152563 0.50.20.095 0.9997999873.18E-051.148813 0.50.20.09750.0950.9997999873.18E-051.14975 0.50.20.10.0950.9997999873.17E-051.150688 0.50.20.10250.0950.9997999873.17E-051.151625 0.50.20.1050.0950.9997999873.16E-051.152563

5. Results of Exhaustive Search M2 Length=95nm M5 Length=95nm M2 Vth= 0.5V M5 Vth= 0.2V Yield= 0.999799987 Power= 3.18E-05 (nom=3.2130e-5) Area=1.148813 (nom=1.1516)

6. Analytical study We wanted to see the effect of changing each parameter from nominal on the yield We found that certain parameters had a very strong and consistent effect on the yield We used this to allow us to reduce the solution space.

7. Analytical Study Parameter Effect of setting to max Effect of setting to min Setpoint M2 Len Read164.2199mV164.3353mV 0.095 um Write584.4249mV581.1114mV M5 Len Read164.2677mV 0.095 um Write624.6716mV539.8206mV M2 Vth Read163.4407mV165.1866mV 0.5V Write594.2763mV580.3916mV M5 Vth Read164.2677mV 0.2V Write931.9230mV511.0882mV ReadRd=164.2677mVLess than 164.3mV163.4004mV WriteWr=582.8680mVLess than 629mV482.6054mV

8. Finding Optimum Design Point Start at a point, and use gradient descent – Explicitly optimize for a cost function – Minimal number of wasted samples Leverage WLHS to quickly determine the yield of a design point The solution space is really only two variables, since M5 can be fixed to an optimum value, based on the yield criteria given. Method has to be problem specific – Gradient descent is bad if lots of local minima – Have not encountered any here

9. Gradient Descent Pick a starting point For each parameter – Simulate at starting point +/- that parameter – Pick best as new starting point – Repeat – If difference between old point and current point is small, exit

10. Initial Results Used QMC for each design point 500 points per design point 123 spice runs – 45min – Exhaustive is 10,000 spice runs – 4-7 hours Yield=0.9880 M2len = 0.0941 (need to fix constraint code) M5len = 0.100 M2vth = 0.3500 M5vth = 0.2206

11. Sampling Methods tried 1.Monte Carlo 2.Quasi Monte Carlo 3.Latin Hypercube Sampling 4.Weighed Latin Hypercube Sampling 5.Importance Sampling We have talked about MC and QMC, Next few slides look at LHS, WLHS and IS.

12. Latin Hypercube Sampling CDF is used to divide variable span into equi- probable partitions. The Gaussian distribution of the variable is preserved. LHS Samples for L1, L2, Vt1, Vt2 are obtained and randomly permuted. Only as good as QMC.

13. LHS vs QMC 1.Clusters and Voids can be identified in LHS and stratified LHS. 2.Convergence rate and accuracy are lower than QMC.

14. Weighted LHS Amplification of samples in the predicted failure zone. Assigning appropriate weight to the failures. The critical point where we start, oversampling affects the accuracy of weighted LHS.

15. Importance Sampling PDF is actually shifted to generate more samples in the predicted failure zone. Accuracy depends on choice of the shift vector and nature of the solution space. The failures are weighed: P(x)| old / P(x)| new.

16. Results TechniqueNumber of Samples YieldSpice RuntimeSpice Runtime Degradation Wrapper Runtime Monte Carlo 10 5 1.0 81x1.0 Quasi-MC 4.3x10 4 1.00.4335x1.2x LHS 4.7 x 1041.00.4738x1.42x Weighted LHS 122599.985%0.012251.01.8x Importance Sampling 140099.981%0.014001.142x1.7x

17. Convergence Rates 1 100 1000 10000 100000 QMC LHS MC IS WLHS

18. Conclusions Tested 5 different Sampling Algorithms. WLHS and IS are the most promising. We will leverage WLHS to find the optimum design point. Gradient Descent works for this problem due to the lack of local minima in the solution space. Monte Carlo is ~81x slower as compared to WLHS, and 70x slower as compared to IS.