SMM: Scalable Analysis of Power Delivery Networks by Stochastic Moment Matching

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

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

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 P/G Supply Voltage Integrity Analysis

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 Random Walk

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

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 Iq

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 beB) Pay A(q) at node q Walk to an adjacent node p with Pr(p, q) Gain Vb the voltage of the boundary node b e B Vs = net gain of the walk Random Walk in a Resistive Network

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

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 mi-1(p) for each node p In post-order, compute Sk e TpCk mi-1(k) for each Tp Moment Computation in an RLC Tree

Expanding moment computation in a tree to a general structure network Stochastic Moment Matching (SMM) q Cq Iq

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

Input: RLC P/G network N, nodes B with known voltages, current sources S Output: P/G node voltages For each current source s e S Walk from s to a power pad with Pr(p, q) For each node q in the path For each moment orderj Compute mj(q) Collect node moments Compute poles and residues by moment matching Output time domain waveforms and voltage drops SMM Algorithm

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

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

Convergence • Solid curve: Random walk I • Dashed curve: Random walk II • Dotted curve: Liebmann’s method

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

Scalability • Power mesh of R=1KW, C=1pF, Tr=1ns, Ip=1mA

SMM vs. Transient Random Walk • SMM: 100 random walks • TRW: 100 random walks for each time step, each of 5ps

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

Thank you !

SMM: Scalable Analysis of Power Delivery Networks by Stochastic Moment Matching