PiCAP: A Parallel and Incremental Capacitance Extraction Considering Stochastic Process Variation

Fang Gong1, Hao Yu2, and Lei He1

1Electrical Engineering Dept., UCLA

2Berkeley Design AutomationPresented by Fang GongPresented by Fang Gong

Outline

Background and Motivation

Algorithms

Experimental Results

Conclusion and Future Work

Outline

Background and Motivation

Algorithms

Experimental Results

Conclusion and Future Work

Process Variation and Cap Extraction

Process variation leads to capacitance variation

OPC lithography and CMP polishing

Capacitance variation affects circuit performance

Delay variation and analog mismatch

0.00%10.00%20.00%30.00%40.00%50.00%60.00%70.00%

-2~0

%0~

2%2~

4%4~

6%6~

8%

8~10

%

10~1

2%

Capacitance Variation (%)

% o

f Seg

men

ts

From [Kang and Gupta]

Background of BEM Based Cap Extraction

Capacitance extraction in FastCap Procedures

1. Discretize metal surface into panels

2. Form linear system by collocation

3. Results in dense potential coeffs

4. Solve by iterative GMRES

Fast Multipole method (FMM) to evaluate Matrix Vector Product (MVP) Preconditioned GMRES iteration with guided convergence

1 1

| |ij ipanel ii i j

P daa r r

P q v

Difficulties for stochastic capacitance extractionHow to consider variations in FMM?How to consider different variations in precondition?

Source Panel j

Observe Panel i

Motivation of Our Work

Existing works Stochastic integral by low-rank approximation

Zhu, Z. and White, J. “FastSies: a fast stochastic integral equation solver for modeling the rough surface effect”. In Proceedings of IEEE/ACM ICCAD 2005.

Pros: Rigorous formulationCons: Random integral is slow for full-chip extraction

Stochastic orthogonal polynomial (SOP) expansionCui, J., and etc. “Variational capacitance modeling using orthogonal

polynomial method”. In Proceedings of the 18th ACM Great Lakes Symposium on VLSI 2008.

Pros: An efficient non-Monte-Carlo approach Cons: SOP expansion results in an augmented and dense linear

system

Objective of our work Fast multi-pole method (FMM) with nearly early O(n) performance

with a further parallel improvement.parallel improvement. Pre-conditioner should be updated incrementally for different

variation.

Outline

Background and Motivation

Algorithms

Experimental Results

Conclusion and Future Work

Flow of piCAP

1. Represent Pij with stochastic geometric moments

2. Use parallel FMM to evaluate MVP of Pxq

3. Obtain capacitance (mean and variance) with incrementally preconditioned GMRES

Potential Coefficient

0 1 0 1( , )ij d wP M d d w w

Solve with GMRES

Build Spectrum preconditioner

Evaluate the MVP (Pxq) with FMM in parallel

Calculate Cij with the charge distribution.

Geometric Moments

0 1 0 1( , )d wM d d w w

Incrementally update

preconditioner

Geometry Info

Process Variation

0 0( , )d w

( , )d w

Stochastic Geometric Moment

Consider two independent variation sources: panel distance (d) and panel width (w)

Multipole expansion along x-y-z coordinates: multipole moments and local moments Mi and Li show an explicit dependence on geometry parameters, and are

called geometric moments.

source-cube

observer cube

source-panel

d0

r00

0

,x y z

x y z

r r x r y r z

d d x d y d z

0 0 0 00

1 0 1 030

22 0 2 0 05

0

1( ) , ( ) 1

( ) , ( )

3 1( ) , ( ) (3 )

6

kk

k lk l kl

l d m rd

dl d m r r

d

d dl d m r r r r

d

0 0 0 00 0 00 0 0

1 ( 1) 1( ) ( ) ( ) ( )

| | !

p

p p ppp p ppp

r r M l d m rr d p d

Stochastic Potential Coefficient Expansion

Stochastic Potential Coefficient Relate geometric parameters to random variables

Let be random variable for panel width w, and be random variable for panel distance d.

Geometric moments Mp and Lp are:

Now, the potential coefficient is

w d

0 1 0 1

0 1 0 1

ˆ ( , ) ( , )

ˆ ( , ) ( , )

p w d p w d

p w d p w d

M M w w d d

L L w w d d

0 1 0 1 0 1 0 1ˆ ˆ( , ) ( , ) ( , )w d w d w dP P w w d d M w w d d

Stochastic Potential Coefficient Expansion

Stochastic Potential Coefficient Relate geometric parameters to random variables

Let be random variable for panel width w, and be random variable for panel distance d.

Geometric moments Mp and Lp are:

Now, the potential coefficient is

w d

n-order stochastic orthogonal polynomial expansion of P

Accordingly, m-th order (m = 2n + n(n − 1)) expansion of charge is:

0 1 0 1 0 1 0 1ˆ ˆ( , ) ( , ) ( , )w d w d w dP P w w d d M w w d d

0 0 1 10

ˆ( ) ( ) ( ) ( ) ( )n

n n i ii

P P P P P

0

ˆ( ) ( )m

j jj

q q

0 1 0 1

0 1 0 1

ˆ ( , ) ( , )

ˆ ( , ) ( , )

p w d p w d

p w d p w d

M M w w d d

L L w w d d

Potential Coefficient

0 1 0 1( , )ij d wP M d d w w

Solve with GMRES

Build Spectrum preconditioner

Evaluate the MVP (Pxq) with FMM in parallel

Calculate Cij with the charge distribution.

Geometric Moments

0 1 0 1( , )d wM d d w w

Incrementally update

preconditioner

Geometry Info

Process Variation

0 0( , )d w

( , )d w

Augmented System

Recap: SOP expansion leads to a large and dense system equation

(0) (1)0

(1) (0) (1)1

(1) (0)2

0

2 0

0 2 0

P P b

P P P

P P

(0) (1)0 1

ˆ( ) ( ) ( )d dP P d P d

0

( ) ( )d j j di

q

ˆ( ) ( ) , ( ) 0d d j dP q b

Parallel Fast Multipole Method--upward

Overview of Parallel fast-multipole method (FMM) group panels in cubes, and build hierarchical tree for cubes

We use 8-degree trees in implementation, but use 2-degree trees for illustration here.

A parallel FMM distributes cubes to different processors

Upward PassLevel 0

Level 1

Level 2

Level 3

• starting from bottom level, it calculates stochastic geometric moments• Update parent’s moments by summing the moments of its children—called M2M operation

M2M

• M2M operations can be performed in parallel at different nodes

Parallel Fast Multipole Method--Downwards

Downward Pass

Level 0

Level 1

Level 2

Level 3

•Sum L2L results with near-field potential for all panels at bottom level and return Pxq• At the top level, calculation of potential between two cubes—called M2L operation.• potential is further distributed down to children from their parent in parallel—called L2L operation.

M2L

L2L

• Calculate near-field potential directly in parallel

M2M

M2M, L2L are local operations, while M2L is global operation.

How to reduce communication traffic due to global operation?

Reduction of traffic between processors

Global data dependence exists in M2L operation at the top level Pre-fetch moments: distributes its moments to all cubes on its

dependency list before the calculations. As such, it can hide communication time.

Cube 1

Cube under calculation

Cube 0

Dependency ListCube 0

Cube 1

…

Cube k

…

Cube k

Flow of piCAP

1. Use spectrum pre-conditioner to accelerate convergence

2. Incrementally update the pre-conditioner for different variation.

Potential Coefficient

0 1 0 1( , )ij d wP M d d w w

Solve with GMRES

Build Spectrum preconditioner

Evaluate the MVP (Pxq) with FMM in parallel

Calculate Cij with the charge distribution.

Geometric Moments

0 1 0 1( , )d wM d d w w

Incrementally update

preconditioner

Geometry Info

Process Variation

0 0( , )d w

( , )d w

Deflated Spectral Iteration

Why need spectral preconditioner GMRES needs too many iterations to achieve convergence. Spectral preconditioner shifts the spectrum of system matrix to

improve the iteration convergence

Deflated spectral iteration k (k=1 power iteration) partial eigen-pairs

Spectrum preconditioner

Why need incremental precondition Variation can significantly change spectral distribution Building each pre-conditioner for different variations is

expensive Simultaneously considering all variations increases the

complexity of our model.

1 1[ ,..., ], [ ,..., ]K K K KV v v D diag

1( ), TK K KW I V D V shifting value

Incremental Precondition

For updated system , the update for the i-th eigen vector is:

is the subspace composed of is the updated spectrum

Updated pre-conditioner W’ is

1 0P P P ( 1,..., )iv k K

1 Ti i i i iv V B V Pv

iV 1[ ,..., ,... ] ( , , 1,..., )j Kv v v j i and i j K iB

1[ ,..., ,..., ], ( , , 1,..., )i i i j i KB diag j i and i j K

1

1 1

1 1

( ( ) ( ) )

( )

( )

( )

TK K K

K K K K K K

T T TK K K K K K K

T T TK K K K K K K

W I V D V W W

W E V D F D V

where E V D V V D V

F V V D V V D

Inverse operation only involves

diagonal matrix DK

Consider different variations by updating the nominal preconditioner partially.

Outline

Background and Motivation

Algorithm

Experimental Results

Conclusion and Future Work

Accuracy Comparison Setup: two panels with random variation for distance d and width w

Result: Stochastic Geometric Moments have high accuracy with average error of 1.8%, and can be up to ~1000X faster than MC

2 panels, d0 = 7.07μm, w0 = 1μm, d1 = 20%d0

MC (3000) piCAP

Cij (fF) -0.3113 -0.3056

Time(s) 10.786037 0.008486

2 panels, d0 = 11.31μm, w0 = 1μm, d1 = 10%d0

MC (3000) piCAP

Cij (fF) -0.3861 -0.3824

Time(s) 10.7763 0.007764

2 panels, d0 = 4.24μm, w0 = 1μm, d1 = 20%d0, w1 = 20%h0

MC (3000) piCAP

Cij (fF) -0.2498 -0.2514

Time(s) 11.17167 0.008684

Runtime for parallel FMM Setup

Two-layer example with 20 conductors.Other: 40, 80, 160 conductorsEvaluate Pxq (MVP) with 10% perturbation

on panel distance

ResultAll examples can have about 3X speedup with 4 processors

#wire 20 40 80 160

#panels 12360 10320 11040 12480

1 proc. 0.737515/1.0 0.541515/1.0 0.605635/1.0 0.96831/1.0

2 proc. 0.440821/1.7X 0.426389/1.4X 0.352113/1.7X 0.572964/1.7X

3 proc. 0.36704/2.0X 0.274881/2.0X 0.301311/2.0X 0.489045/2.0X

4 proc. 0.273408/2.7X 0.19012/2.9X 0.204606/3.0X 0.340954/2.8X

Efficiency of spectral preconditioner

Setup: Three test structures: single plate, 2x2 bus, cubic

ResultCompare diagonal precondition with spectrum preconditionSpectrum precondition accelerates convergence of GMRES (3X).

# panel # variable diagonal prec. spectral prec.

# iter Time(s) # iter Time(s)

plate 256 768 29 24.59 11 8.625

cubic 864 2592 32 49.59 11 19.394

bus 1272 3816 41 72.58 15 29.21

Speedup by Incremental Precondition

Setup Test on two-layer 20 conductor example Incremental update of nominal pre-conditioner for different

variation sources Compare with non-incremental one

discretizationw-t-l

#panel #variable Total Runtime (s)

Non-incremental incremental

3x3x7 2040 6120 419.438 81.375

3x3x15 3960 11880 3375.205 208.266

3x3x24 6120 18360 - 504.202

3x3x50 12360 37080 - 3637.391

Result: Up to 15X speedup over non-incremental results, and only incremental one can finish all large examples.

Conclusion and Future Work

Introduce stochastic geometric moments

Develop a parallel FMM to evaluate the matrix-vector product with process variation

Develop a spectral pre-conditioner incrementally to consider different variations

Future Work: extend our parallel and incremental solver to solve other IC-variation related stochastic analysis

ThanksPiCAP: A Parallel and Incremental Capacitance

Extraction Considering Stochastic Process Variation

Fang Gong, Hao Yu and Lei He