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 !
Top Related