Statistical Sampling-Based Parametric Analysis of Power Grids

Dr. Peng Li

Presented by Xueqian Zhao

EE5970 Seminar

Outline

— Motivation

— Prior works

— Importance sampling technique

— Sampling-based localized sensitivity analysis

— Sampling-based 2nd order parametric analysis

— Conclusion

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

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

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(n2)

1, , ,n n nGv b G R v b R

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

1122

33

44

I1

G1 G3

G212

3

4

C1P12

P14P13

1122

33

44

C1P12

P14P13

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.

VDD

VDD

VDD

VDD

, ,1

m

a k a ii

P P

, , ,1

m

a k a i a ii

V P D

• 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

VDD

VDD

Target Node ni

??)( inV ??)(

j

i

p

nV

??),,()( 21 ppfnV i

Adjoint Sensitivity Analysis

• Classical adjoint sensitivity analysis

• Requires two complete linear solutions — Obscure the possibility of locality

• Localized sensitivity analysis??

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/Pk)D’k

Pk1 Pk2 Pk3node ni

Vdd…

Original Circuit (A)

P’k1 P’k2 P’k3

node niVdd…

Perturbed Circuit (B)

Value: Dk Prob: Pk = Pk1Pk2

Value: D’k Prob: P’k = P’k1P’k2…

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

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

Ts ],,,[ 21

0

Localized Sensitivity Analysis

• Propagate circuit element parametric sensitivities

• Perform a few scalar arithmetic operations— Additions, subtractions, multiplications and

divisions

Localized Parametric Analysis Flow

Pick a new move according to the

nominal ckt

Compute the prob. of this move and update the path prob. Ppath for the param.

ckt

Accumulate the cost incurred by the move

Weight the gain of the complete walk by Wpath = Ppath/Ppath(nom)

… …… …

+Mean estimate:

Sum up andnormalize

Pick a new move according to the

nominal ckt

Compute the prob. of this move and update the path prob. Ppath for the param.

ckt

Accumulate the cost incurred by the move

Weight the gain of the complete walk by Wpath = Ppath/Ppath(nom)

… …

Second Order Analysis

• 2nd 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 2nd order parametric forms: — Voltage response estimate /cost incurred due to current sources

— State-transition/path probabilities

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

+ +

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(NGNL), NL << NG

Node of Interest

X

Importance Sampling Estimators

• Importance sampling

• Integration estimator

• Ratio estimator

Importance Sampling Estimators

• Regression estimator

Results

• Comparison of estimators— 40k-node grid— Solve the nominal ckt and the perturbed circuit simultaneously

IR drop estimation in the perturbed circuit

Results

• Localized sensitivity analysis— Simultaneously solve for sensitivities— Compared with direct sensitivity

# Nodes Direct T. Int. T Int. Err. Ratio T. Ratio Err. Reg. T. Reg. Err 40K 83s 1.38s 45.1% 1.42s 7.00% 1.59s 0.35% 90K 356s 1.93s 9.01% 2.03s 2.49% 2.24s 1.31% 250K 40m30s 2.21s 0.14% 2.29s 0.14% 2.34s 0.06% 1.1M N/A 2.25s N/A 2.28s N/A 2.33s N/A 1.4M N/A 4.81s N/A 4.85s N/A 4.87s N/A

Results A 250K node grid Resistance variation

Average: 12.3% Max: 55%

22 variation sources 1st order analysis: 23 coefficients 2nd order analysis: 276 coefficients

Runtime 1st order: 3.1s 2nd order: 22s

1000 samples: 1st order errors 1000 samples: 2nd order errors

Results A 1.1 million node grid Resistance variation

Average: 13% Max: 55%

Loading variation Average: 19% Max: 164%

22 variation sources 1st order analysis: 23 coefficients 2nd order analysis: 276 coefficients

Runtime 1st order: 4s 2nd order: 28s

1000 samples: 1st order errors 1000 samples: 2nd order errors

Results

• Runtime as a function of number of variations — Near-linear complexity achieved by exploring spatial

locality

2nd order analysis runtime

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

• 2nd order analysis improves the accuracy for larger loading and process variations— Explore the spatial locality of the algorithm formulation to achieve near-linear

complexity