1. SMM: Scalable Analysis of Power Delivery Networks by Stochastic Moment Matching Andrew B. Kahng, Bao Liu, Sheldon X.-D. Tan* UC San Diego, *UC Riverside

2. Outline Background Problem Formulation Random Walk Moment Computation in an RLC Tree SMM Theory Experiments Conclusion

3. P/G Supply Voltage Integrity Analysis Increasing Power/Ground supply voltage degradation in latest technologies  IR drop (DC/AC)  L dI/dt drop Effects:  Malfunction  Performance degradation P/G supply networks are special interconnects  Complex topology, numerous nodes, IOs Scalability improvement schemes  Top-down: multigrid-like, hierarchical, partition  Bottom-up: random walk

4. Random Walk A stochastic process which gives voltage of a specific P/G node Advantages:  Localization  Parallelism Limitations:  DC analysis  Transient analysis Our contribution:  Frequency domain analysis

5. Outline Background Problem Formulation Random Walk Moment Computation in an RLC Tree SMM Theory Experiments Conclusion

6. Problem Formulation Given  an RLC P/G supply network  power pads  supply current sources Find  P/G node voltages Challenges  Scalability  Accuracy

7. Kirchoff’s current law: A random wanderer pays for lodging every night, and has a probability to go to a neighboring location, until he reaches home A Monte Carlo method to a boundary value problem of partial differential equations Random Walk IqIq

8. Input: resistive network N, nodes B with known voltages Output: voltage of node s Start walking from a node s While (not reaching a node b  B) Pay A(q) at node q Walk to an adjacent node p with Pr(p, q) Gain V b the voltage of the boundary node b  B V s = net gain of the walk Random Walk in a Resistive Network

9. Moment Computation in an RLC Tree Current through Rpq charges all downstream capacitors Expanding the voltages in moments p q R pq

10. Input: RLC tree T, input nodes voltage moments Output: Output node voltage moments For each moment order j Depth-first traversal of the tree T In pre-order, compute m i-1 (p) for each node p In post-order, compute S k  Tp C k m i-1 (k) for each T p Moment Computation in an RLC Tree

11. Expanding moment computation in a tree to a general structure network Stochastic Moment Matching (SMM) IqIq CqCq q

12. A random walk process  Pr(p, q) transition probability  A(q) lodging cost Stochastic Moment Matching (SMM)

13. Input: RLC P/G network N, nodes B with known voltages, current sources S Output: P/G node voltages 1.For each current source s  S 2. Walk from s to a power pad with Pr(p, q) 3. For each node q in the path 4. For each moment order j 5. Compute m j (q) 6.Collect node moments 7.Compute poles and residues by moment matching 8.Output time domain waveforms and voltage drops SMM Algorithm

14. Numerical Stabilities Compute moments of all orders of a node based on the same random walk process  See algorithm Reduce number of random walks by reducing the number of node voltage moments needed  MMM vs. SMM Filtering out numerically instable solutions  Unvisited nodes, positive poles, etc. Take average

15. Runtime Number of moments M Average path length P (dominant)  = average distance from the node to a power pad  Independent to P/G network size Number of poles/residues for moment matching Time domain binary search for delay

16. Outline Background Problem Formulation Random Walk Moment Computation in an RLC Tree SMM Theory Experiments Conclusion

17. Convergence I.Solid curve: Random walk I II.Dashed curve: Random walk II III.Dotted curve: Liebmann’s method

18. Accuracy Randomly generated 100x100 power mesh of R=100W~1KW, C=0.1pF~1.0pF, L=0.1pH~1.0pH, Tr=0.5ns~2.5ns, Ip=0.5mA~2.0mA 1000 random walks vs. SPICE

19. Scalability Power mesh of R=1KW, C=1pF, Tr=1ns, Ip=1mA N/G1234 CPUVdopCPUVdropCPUVdropCPUVdrop 100.140.040.071.090.041.100.041.12 200.480.950.211.040.091.100.061.11 505.540.851.860.980.441.030.261.03 10023.080.917.790.931.970.971.151.02

20. SMM vs. Transient Random Walk I.SMM: 100 random walks II.TRW: 100 random walks for each time step, each of 5ps 1234567 ICPU12.87.39.512.84.44.66.9 Vdrop1.050.970.941.040.970.961.03 IICPU142.1141.5139.3135.0192.6107.6100.3 Vdrop1.121.151.091.211.321.090.94

21. Summary We extend random walk to frequency domain analysis by computing moments for RLC P/G networks Much better efficiency/accuracy than transient analysis random walk Advantages of random walk: locality, runtime which depends on average distance to a power pad, parallelism More stable moment computation in a bunch of stochastic processes

22. Thank you !