1

Stochastic Integral Equation Solver for Efficient Variation-

Aware Interconnect Extraction

Tarek A. El-Moselhy

and Luca Daniel

2

Motivation: On/Off-chip Variations Irregular geometries: On-chip Rough-surfaces: On-package

and on-board

Irregularities change in impedance

Irregularities are random but current extraction tools are deterministic

[Courtesy of IBM and Cadence]

[Braunisch06]

3

Definition of Stochastic Solver

Stochastic Field Solver

Geometry of interconnect structure

Distribution describing the geometrical variations

Statistics of interconnect input impedance

1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Input Impedance

PD

F

-3 -2 -1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Width

PD

F

width

input impedance

PD

F

PD

F

4

Magneto-Quasistatic MPIE

Current conservation

0 rJ

AB

J

J J

J

rrdrrrGjrV

''', 3JJ

mmT IMLjRM

Mesh matrix M

Piecewise constant basis functions + Galerkin testing

Stochastic

m kV V km

mk rdrdAA

rrGL 33 '

',mV m

mm rdA

R 3

Vm Vk

A x b

5

Linear System Abstraction

The system matrix elements are functions of the random variables describing the geometry

Vector represents n (Gaussian) correlated random variables

The objective is to find the distribution of the unknown vector

bHxHA

bHAHx 1

C

HCHHP

H

T

5.0

1

2

5.0exp

H

HAijH

Single matrix element depends on a small subset of the physical parameters

6

Outline

Motivation and Problem Definition

Previous Work and Standard Techniques

Contribution New Theorem for orthogonal projection New simulation technique

Results

7

Sampling-Based Techniques

Monte Carlo, stochastic collocation method [H.Zhu06]

Solve the system Mc times for Mc different realizations of

Compute required statistics from the ensemble

Advantages: Highly parallelizable, very simple

Disadvantages: requires solving the system Mc times which means complexity is

ci MiH ,1,

bHAHx ii1

3NM c

bHxHA

8

Neumann Expansion

Computing the statistics is very expensive

Complexity: O(N4)

bAHAHEAbAHEAbAHxE AAA10

10

10

10

10

10

=0 10

10

AvecHHE

HAHvecE

AA

AA

convergence criterion 1max 1

0 HA A

bHxHA

bAHAHAbAHAbAHx

bHAHx

bHAHx

AAA

A

10

10

10

10

10

10

10

1

2D capacitance [Z.Zhu04], 3D inductance [Moselhy07], on-chip capacitance [Jiang05]

9

Stochastic Galerkin Method [Ghanem91]

expand in terms of orthogonal polynomials

write as a summation of same polynomials

substitute and assemble linear system to compute

the unknowns

K

jjjxx

0

bHxHA

K

kkkAA

0

LH

1L

need to decouple random variables

Step 2

Step 1

Step 3

Step 4

10

Stochastic Galerkin Method (Con’t)

Problem 1: Very expensive multi-dimensional integral

Step 2. Polynomial Chaos Expansion

Use multivariate Hermite polynomials

M-dimensional

For a typical interconnect structure M > 100

1 2

21

2

21

2

5.0exp,,,

,

M

MM

T

kMij

kijij

k

dddA

AA

K

kkkAA

0

MMMM

MMMMM

AA

AAAAAA

1,1212,1

2,

211,1110 11

11

Stochastic Galerkin Method (Con’t)

Step 4. System Assembly

Problem 2: Very large linear system O(KN)

Use Galerkin Testing to obtain a deterministic linear system of equations

K+1 unknowns each of length N

K

jjjxx

0

m

K

k

K

jmjkjk bxA

,,0 0

0

01

0

001

00

01

011

001

00

010

000

b

x

x

x

AAA

AAA

AAA

KK

kkKKk

K

kKkk

K

kKkk

K

kkKk

K

kkk

K

kkk

K

kkKk

K

kkk

K

kkk

kjm

K

kkkAA

0

bxA

bxAK

k

K

jjkjk

0 0

12

Outline

Motivation and Problem Definition

Previous Work and Standard Techniques

Contribution New Theorem for orthogonal projection New simulation technique

Results

13

Solution of Problem 1: Efficient Multi-Dimensional Projection

Current techniques include: Monte Carlo integration Quasi-Monte Carlo integration Sparse grid integration

We propose to solve the problem by reducing the dimension of the integral.

1 2

21

2

21

2

5.0exp,,,

,

M

MM

T

kMij

kijij

k

dddA

AA

14

Solution of Problem 1: Efficient Multi-Dimensional Projection (con’t)

for a second order expansion

2

1

i

i

1 LAHA ijij

H is a small subset of the

vector containing the

physical parameters

H

1 2

21

2

21

2

5.0exp,,,

,

M

MM

T

kMij

kijij

k

dddA

AA

15

Corollary

100-D Integral

8-D Integral

Matrix elements depend on a small subset of the physical random variables

Second order expansion

1 6 1 2

216121612161 ,,,,,,,

,

H H

kij

kijij

k

dddHdHHHPHHA

HAA

1 2

21

2

21

2

5.0exp,,,

,

M

MM

T

kMij

kijij

k

dddA

AA

Original Polynomial Chaos ExpansionOriginal Polynomial Chaos Expansion

New TheoremNew Theorem

16

Theorem Given the matrix elements

the coefficients of the Hermite expansion ( ) are given by:

where is the subset of parameters on which the matrix element depends and is the subset of random variables on which the polynomial depends

If dimension of then the above formula is more efficient than the traditional approach

1 LAHA ijij

dHdHPHAHAA kijkijijk ,,

H

H

CH

HCC

HV

C

VCVHP

T

TH

H

T

,

,~,,

2

~5.0exp

,5.0

1

K

kkkAA

0

17

Solution of Problem 2: Efficient Stochastic Solver Use Neumann expansion to reduce system size

Use Polynomial Chaos expansion to simplify computation of the statistics:

Rearranging above expansion we obtain the required expansion of the output:

bAAAbAAbAxK

kkk

K

kkk

K

kkk

10

1

10

1

10

10

1

10

10

K

kkkA

1

K

kj

K

jkj

Tk

K

kkk

To

TT YQQxxbxby1 11

0

bAx 100 0xQ kk jj QAY 1

0

bAHAHAbAHAbAHx AAA10

10

10

10

10

10

18

Efficient Stochastic Solver

Obtain directly an expansion of the output in terms of some orthogonal polynomials

Complexity is transformed into a large number of vector matrix products

Highly parallelizable

Requires independent system solves (same system matrix), currently implemented using direct system solvers and re-using the LU factorization

Efficiency can be even further enhanced using block iterative solvers

K

kj

K

jkj

Tk

K

kkk

To

T YQQxxby1 11

0

bAx 100 0xQ kk jj QAY 1

0

19

Outline

Motivation and Problem Definition

Previous Work and Standard Techniques

Contribution New Theorem for orthogonal projection New simulation technique

Results

20

Definition of Stochastic Solver

Stochastic Field Solver

Geometry of interconnect structure

Distribution describing the geometrical variations

Statistics of interconnect input impedance

Rough surface with Gaussian profile and

correlation

21

Results: Accuracy Validation

Microstrip line W=50um, L=0.5mm, H=15um

sigma=3um, correlation length=50um

mean: 0.0122, std (MC, SGM) = 0.001, std (New algorithm)= 0.00097

SGM +

22

Results: Complexity Validation

Example Technique Properties for 5% accuracy Memory Time

Long Microstrip line

DC Only

400 unknowns

Monte Carlo

Neumann*

SGM

New Algorithm

10, 000

2nd order

96 iid, 4753 o.p.

96 iid, 4753 o.p.

1.2 MB

1.2 MB

(72 GB)

1.2 MB

2.4 hours

0.25 hours

-

0.5 hours

Transmission Line

10 freq. points

800 unknowns

Monte Carlo

Neumann*

SGM

New Algorithm

10, 000

2nd order

105 iid, 5671 o.p.

105 iid, 5671 o.p.

10 MB

10 MB

(300 TB)

10 MB

16 hours

24 hours

-

7 hours

Two-turn Inductor

10 freq. points

2750 unknowns

Monte Carlo

Neumann*

SGM

New Algorithm

10, 000

2nd order

400 iid, 20604*

400 iid, 20604*

121 MB

121 MB

(800 PB)

121 MB

(150 hours) X 4p

(828 hours) X 4p

-

8 hours X 4p

23

Results: Large Example

Two-turn inductor

Simulation at 1GHz for different rough surface profiles

Input resistance is 9.8%, 11.3% larger than that of smooth surface for correlation lengths 5um, 50um, respectively

Variance increases proportional to the correlation length

Inductance is decreased by about 5%

Quality factor decreases

0.23 0.235 0.24 0.245 0.25 0.255 0.260

100

200

300

400

500

600

700

800

Re(Impedance) in

Pro

bab

ility

De

nsity

Fu

nctio

n

0.23 0.235 0.24 0.245 0.25 0.255 0.260

20

40

60

80

100

120

140

Re(Impedance) in

Pro

bab

ility

De

nsity

Fu

nctio

n

correlation length = 5um

correlation length = 50um

24

Conclusion

Developed a new theorem: efficient Hermite polynomial expansion new inner product many orders of magnitude reduction in computation time suitable for any algorithm that relies on polynomial expansion

Developed new simulation algorithm: merged both Neumann and polynomial expansion does not require the solution of a large linear system easy to compute the statistics parallelizable.

Verified our algorithm on a variety of large examples that were not solvable before.

25

Thank You

26

Inductor Example

27

Proof The main step is to prove the orthogonality of the polynomial using

the modified inner product definition

Consequently,

Pji

ji

ji

jiHPji

dP

ddHHP

dHdHP

,

,

,,,

k

kkaHa

kk

kk

Haa

,

,

28

Alternative Point of View The same theorem can be proved by doing a variable transformation and

making use of Mercer Theorem:

Remember from Mercer Theorem:

VCV

DV

TT

r

1

2212221

1111211~~~~

~~~~1

1

rrrr

rrrr

V

MM

MM

kkk ~

r

CH

HDD

D

DDDD

DC

T

TT

Trr

Tr

Tr

TTr

T

r

,

,

H

C T