stochastic integral equation solver for efficient variation-aware interconnect extraction
DESCRIPTION
Stochastic Integral Equation Solver for Efficient Variation-Aware Interconnect Extraction. Tarek A. El-Moselhy and Luca Daniel. Motivation: On/Off-chip Variations. Rough-surfaces: On-package and on-board. Irregular geometries: On-chip. [Courtesy of IBM and Cadence]. [Braunisch06]. - PowerPoint PPT PresentationTRANSCRIPT
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