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

Andrew B. Kahng, Bao Liu, Sheldon X.-D. Tan*

UC San Diego, *UC Riverside

OutlineOutline

Background

Problem Formulation

Random Walk

Moment Computation in an RLC Tree

SMM Theory

Experiments

Conclusion

P/G Supply Voltage Integrity AnalysisP/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

Random WalkRandom 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

OutlineOutline

Background

Problem Formulation

Random Walk

Moment Computation in an RLC Tree

SMM Theory

Experiments

Conclusion

Problem FormulationProblem Formulation Given

an RLC P/G supply network power pads supply current sources

Find P/G node voltages

Challenges Scalability Accuracy

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 WalkRandom Walk

I G V V

V

G V I

G

q pq p qp q E

q

pq pp q E

q

pqp q E

( )( , )

( , )

( , )

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 b B)

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 B

Vs = net gain of the walk

Random Walk in a Resistive NetworkRandom Walk in a Resistive Network

Moment Computation in an RLC TreeMoment Computation in an RLC Tree Current through Rpq charges all downstream

capacitors

Expanding the voltages in moments

V V R sC V

M q M p R C m k

q p pq k kk Tp

i i pq k ik Tp

( ) ( ) ( )1

pq 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 Tp Ck mi-1(k) for each Tp

Moment Computation in an RLC TreeMoment Computation in an RLC Tree

Expanding moment computation in a tree to a general structure network

Stochastic Moment Matching (SMM)Stochastic Moment Matching (SMM)

V

R sL

V

R sLsC V Iq

pq pqp q E

p

pq pqp q Eq q q

( , ) ( , )

IqCq

q

A random walk process Pr(p, q) transition probability A(q) lodging cost

Stochastic Moment Matching (SMM)Stochastic Moment Matching (SMM)

m q p q m p A q

p qG

G

A qC m q m I

G

j j

pq

pqp q E

q j j q

pqp q E

( ) P r( , ) ( ) ( )

P r( , )

( )( ) ( )

( , )

( , )

1

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 S2. 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 mj(q)

6. Collect node moments

7. Compute poles and residues by moment matching

8. Output time domain waveforms and voltage drops

SMM AlgorithmSMM Algorithm

Numerical StabilitiesNumerical 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

RuntimeRuntime

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

OutlineOutline

Background

Problem Formulation

Random Walk

Moment Computation in an RLC Tree

SMM Theory

Experiments

Conclusion

ConvergenceConvergence

I. Solid curve: Random walk I

II. Dashed curve: Random walk II

III. Dotted curve: Liebmann’s method

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

N/G 1 2 3 4

CPU Vdop CPU Vdrop CPU Vdrop CPU Vdrop

10 0.14 0.04 0.07 1.09 0.04 1.10 0.04 1.12

20 0.48 0.95 0.21 1.04 0.09 1.10 0.06 1.11

50 5.54 0.85 1.86 0.98 0.44 1.03 0.26 1.03

100 23.08 0.91 7.79 0.93 1.97 0.97 1.15 1.02

SMM vs. Transient Random Walk SMM vs. Transient Random Walk I. SMM: 100 random walks

II. TRW: 100 random walks for each time step, each of 5ps

1 2 3 4 5 6 7

I CPU 12.8 7.3 9.5 12.8 4.4 4.6 6.9

Vdrop 1.05 0.97 0.94 1.04 0.97 0.96 1.03

II CPU 142.1 141.5 139.3 135.0 192.6 107.6 100.3

Vdrop 1.12 1.15 1.09 1.21 1.32 1.09 0.94

SummarySummary 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 !Thank you !