1. Statistical Sampling-Based Parametric Analysis of Power Grids Dr. Peng Li Presented by Xueqian Zhao EE5970 Seminar

2. Outline — Motivation — Prior works — Importance sampling technique — Sampling-based localized sensitivity analysis — Sampling-based 2 nd order parametric analysis — Conclusion

3. Motivation Power/Ground integrity becomes a serious challenge for modern chip design IR drops reduce noise margin and increase circuit delay —10% supply voltage fluctuations may translate in more than 10% timing variation Technology scaling worsens the P/G integrity —Reduction of power supply voltage

4. Analysis Challenge Modern P/G networks routinely reach multi-million node complexity —Full grid analysis becomes very expensive Need to consider significant variation in power consumption —Active power is mode dependent —Process and temperature variability impact significantly the leakage power Power grids are also subject to parametric variations due to fabrication fluctuations Multiplicity of variations in power grids make the analysis even more difficult

5. Prior Works P/G can be modeled as a linear system: Direct methods: —LU, Cholesky decomposition Iterative methods: —Conjugate gradient (CG), preconditioned CG —Multigrid method Since G is a sparse matrix, the complexity of above methods is around O(n 2 )

6. Random Walks Convert an electrical network to a random walk Circuit response estimated locally via mean estimation —Average over a set of statistical samples Locality exploited naturally without solving the complete network

7. Random Walks (cont.) Transition probabilities between states are obtained from the electrical network The random walk can be described as Markov chain The complexity can be reduced to O(n), compared to prior works.

8. Locally solve for selected circuit nodes Compute the nominal node voltage (IR drop) for each node —Achievable through random walks Sensitivities with respect to multiple process/loading variations?? High order parametric dependencies?? Localized Analysis

9. Adjoint Sensitivity Analysis Classical adjoint sensitivity analysis Requires two complete linear solutions —Obscure the possibility of locality Localized sensitivity analysis??

10. Intuition Original circuit differs from the perturbed circuit in circuit element values Want to solve both circuits simultaneously by only sampling in the original circuit Need to scale each sample to correct the sampling bias: D’ k  (P’ k /P k )D’ k P k1 P k2 P k3 node n i V dd … Original Circuit (A) P’ k1 P’ k2 P’ k3 node n i V dd … Perturbed Circuit (B) Value: D k Prob: P k = P k1 P k2 Value: D’ k Prob: P’ k = P’ k1 P’ k2 …

11. Importance Sampling View random walks algorithms as a Monte Carlo method Circuit response is estimated via mean estimation Importance sampling allows us to estimate the mean of a statistical distribution while sampling according to another distribution Ratio estimate

12. Localized Sensitivity Analysis Design / process parameters Perform sampling only in the nominal circuit Estimate the response in any parametric circuit Need to propagate the first order sensitivities while sampling in the nominal circuit

13. Localized Sensitivity Analysis Propagate circuit element parametric sensitivities Perform a few scalar arithmetic operations —Additions, subtractions, multiplications and divisions

14. Localized Parametric Analysis Flow Pick a new move according to the nominal ckt Compute the prob. of this move and update the path prob. P path for the param. ckt Accumulate the cost incurred by the move Weight the gain of the complete walk by W path = P path /P path (nom) … + Mean estimate: Sum up and normalize Pick a new move according to the nominal ckt Compute the prob. of this move and update the path prob. P path for the param. ckt Accumulate the cost incurred by the move Weight the gain of the complete walk by W path = P path /P path (nom) …

15. Second Order Analysis 2 nd order analysis gives more accurate results for larger perturbations Straightforward implementation is prohibitively expensive —276 coefficients needed for 22 variables ! Can exploit the inherent spatial locality in the algorithm formulation Adopt two 2 nd order parametric forms: —Voltage response estimate /cost incurred due to current sources —State-transition/path probabilities

16. Exploring Spatial Locality Naïve 2nd order analysis not feasible for a large number of inter-/intra die variations Model variations sources using a hierarchical model —Global, semi-global and local variations Local data types impacted only by a small set of variations —Represented in a SPARSE 2nd order form Global Local Semi-Global

17. Exploring Spatial Locality Data interactions —Local + local: efficiently computable —Global + global: only happen at end of each random walk —Global + local: many counts / dominant cost! Exploring sparsity Dominant cost: O(N G N L ), N L << N G Node of Interest

18. Importance Sampling Estimators Importance sampling Integration estimator Ratio estimator

19. Importance Sampling Estimators Regression estimator

20. Results Comparison of estimators —40k-node grid —Solve the nominal ckt and the perturbed circuit simultaneously IR drop estimation in the perturbed circuit

21. Results Localized sensitivity analysis —Simultaneously solve for sensitivities —Compared with direct sensitivity

22. Results A 250K node grid Resistance variation  Average: 12.3%  Max: 55% 22 variation sources  1 st order analysis: 23 coefficients  2 nd order analysis: 276 coefficients Runtime  1 st order: 3.1s  2 nd order: 22s 1000 samples: 1 st order errors 1000 samples: 2 nd order errors

23. Results A 1.1 million node grid Resistance variation  Average: 13%  Max: 55% Loading variation  Average: 19%  Max: 164% 22 variation sources  1 st order analysis: 23 coefficients  2 nd order analysis: 276 coefficients Runtime  1 st order: 4s  2 nd order: 28s 1000 samples: 1 st order errors1000 samples: 2 nd order errors

24. Results Runtime as a function of number of variations —Near-linear complexity achieved by exploring spatial locality 2 nd order analysis runtime

25. Conclusion Power/ground network verification is becoming increasingly difficult due to large problem complexity The analysis complexity exacerbates as we address process variations and current loading uncertainties Efficient parametric analysis is proposed to analyze large power grids locally —Adopt importance sampling in Monte Carlo method —Lends itself naturally to a localized version of the classical sensitivity analysis 2 nd order analysis improves the accuracy for larger loading and process variations —Explore the spatial locality of the algorithm formulation to achieve near-linear complexity