Pattern Based Method For P/G Grid Analysis

Jin Shi, Yici Cai, Xianlong Hong Sheldon X.-D. Tan

EDA Lab. CS Department, Tsinghua University

EE Department, University of California Riverside

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Outline• Practical Computation Problems• Overview of Existing Methods• Acceleration Tech• Some Observations• Pattern Based Method• Conclusion

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Outline• Practical Computation Problems• Overview of Existing Methods• Acceleration Tech• Some Observations• Pattern Based Method• Conclusion

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Practical Problems #1• Linear computation complexity does not mean linear

increasing time cost due to Cache Miss

3( )O n

-1

0

1

23

4

5

6

0 1 2 3 4 5

log Problem Size

log

cycl

es/f

lop

T = N4.7

Size 2000 took 5 days

12000 would take1095 years

matrix multiply operation algorithm looks like 5( )O n

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Practical Problems #1• Summary of Cache Miss

– Linear computation complexity is not enough

– Optimize algorithms together with cache performance

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Practical Problems #2• Iterative efficiency

– Preconditioner’s performance decreases as the matrix size increases

1 2 3 4 5 6 7 8 9 10 110

50

100

150

200

250

300

350

400

2 log(size)

itera

tion

times

Performance of Different Preconditioner

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Practical Problems #3• Memory efficiency limitation• When the Design is too large …

– Hard to load it from DB– Impossible to build matrix– Vector malloc run out of memory– Too many years to get results

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Our Contribution

• Alleviate Cache Miss • Reduce the overall memory usage• Present more efficient preconditioner under a

partition framework• Constant preconditioner performance

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Outline• Practical Computation Problems• Overview of Existing Methods• Acceleration Tech• Some Observations• Pattern Based Method• Conclusion

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Summary of Existing Method• Computation Complexity <

– LU – PCG p slightly larger than 1– M-G with small coefficient– R-W with large coefficient– ADI with small coefficient

• Memory Efficiency– LU – PCG – M-G – R-W – ADI

• Trade off between speed and accuracy

2( )O n

( )pO n

2( )O n

( )O n( )O n

( )O n

( )O n

2( )O n

( )O n( )O n

( )O n

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Summary• Direct Methods vs Iterative Methods

– Time cost of LU:

– Time cost of PCG:

– Which is faster depends on the fill in ratio and performance of preconditioner

1 11 2

11

( ( )) ( ( ) ( ) )

( ) (1 ) ( ) :

t d A N t L v t U v

t L v f nnz A f fill in ratio

− −

−

+ × +

∝ + ⋅

( ( ) ) ( ( ) )( ) ( )

t d A N n t A vt A v n n z A

+ × ⋅∝

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Outline• Practical Computation Problems• Overview of Existing Methods• Acceleration Tech• Some Observations• Pattern Based Method• Conclusion

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Acceleration Tech• Model Order Reduction

– S domain Based before 2000– Electrical Equivalent Circuit Based 2002– Topological Partition Based 2004

• Among these methods, topological partition method is the most powerful one to simulate large P/G grid

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Partition Benefits• For direct method

– Decrease decomposition time

– Decrease fill in ratio – Decrease Cache Miss

• For iterative method– Decrease preconditioner

construction time– Increase preconditioner

performance– Decrease iteration time– Decrease Cache Miss

2 2( )n x n x⋅ <

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Outline• Practical Computation Problems• Overview of Existing Methods• Acceleration Tech• Some Observations• Pattern Based Method• Conclusion

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Observation #1• Too many elements share the same value

– Extract R L C with BEM Solver– All most all elements are extracted from M1 and M2– More than 80% elements in M1 and M2 have the same value

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Observation #2• P/G grid topology is

self-similar– One layer contains many

routed metal in the same direction

– Metal rails share the same width and pitch

• Possible to transform topology similarity to matrix similarity ?– Yes

• How about irregular P/G grid ?– Do Local regularization to

make elements in local area share the same value

• Continuous in topology domain– Local regularization will not

introduce obvious computation errors

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Observation #3• When matrix size is medium, PCG method is

faster than any other method– PCG can converge within 10 times iteration– LU usually has fill in factor larger than 30– R-W is inefficient for small case– M-G needs auxiliary computation structure and time to

construct them• Traditional preconditioner can be improved

– Use topology similarity property– Use element value similarity property

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Outline• Practical Computation Problems• Overview of Existing Methods• Acceleration Tech• Some Observations• Pattern Based Method• Conclusion

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Pattern Based Method• Regular P/G Grid in 3D

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Pattern Base Method• Self Similarity: Global Similarity (Global Pattern)

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Pattern Base Method• Self Similarity: Local Similarity (Local Pattern)

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Pattern Base Method• Similarity Summary

– Patterns are elements similar to each other– Patterns exist not only in global area but also in local

area• Strategy

– Partition global area to blocks, perform relaxation iteration between global blocks

– Reuse local pattern to perform local simulation

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Pattern Based Method• Local Matrix Generation

– Element fill in under NA method – Node order is important

g

g g g

g g

⎡ ⎤⎢ ⎥−⎢ ⎥⎢ ⎥⎢ ⎥−⎢ ⎥⎢ ⎥⎣ ⎦

O L L L L

L L L

L L O L L

L L L

L L L L O

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Pattern Based Method• Local Matrix Generation

– Ordering node number according to the routing direction of each layer

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Pattern Based Method• Diagonal Block

– transform topology similarity to matrix similarity– one tridiagonal matrix need to be stored

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Pattern Based Method• Via Block

– All the vias share the same value between two adjacent layers

– Fill in of via element is regular : size , pitch– No need to store sub matrixes caused by vias

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Pattern Based Method• Drawback of traditional preconditioner

– Performance of ILU or incomplete choleskeydecomposition are restricted to memory usage

– More fill in cause faster convergence speed

1748.411e-44410.961e-3932.791e-22211.51e-1

Avg Iter timesFill in ratioDropthreshold

No odering residual=1e-6

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Pattern Based Method• Preconditioner Generation: Two Metal Layer Case

• Schur-alike Decomposition to get approximate inverse

mod( , )

0ij via

ijTij

a b j i c gA CA a B b C c

C B other ca b

α β⎡ ⎤ ⎡ ⎤ = =⎧⎡ ⎤ ⎪⎢ ⎥ ⎢ ⎥= = = ⎨⎢ ⎥ ⎢ ⎥ ⎢ ⎥ =⎣ ⎦ ⎪⎩⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

1

1

1

1 1 1

11

'

'

'

T T

T

T T

A C I A I A CC B C A I B I

B B C A C

A C II A C AC B C A II B

−

−

−

− − −

−−

⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤= ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦= −

⎡ ⎤ ⎡ ⎤−⎡ ⎤ ⎡ ⎤= ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

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Pattern Based Method• Preconditioner Generation

– Perform Precondition in CG Algorithm1 1 1

11

1 1 11 11

1 12 1 1 2 21

1 12

2 1 1 22 112

3

'

1:

2:''

3:

T T

T T T

A C II A C AP r r r

C B C A II B

Ir r rstep r r p A r

C A Ir C A r r C p r

pr Astep r r

B rr B

step r

− − −

−−

−− −

−

−−

⎡ ⎤ ⎡ ⎤−⎡ ⎤ ⎡ ⎤⋅ = = ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤= = = = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥− − + − +⎣ ⎦⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= = =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦⎣ ⎦

1 1 212 2

2 22

r A CrI A Cr

rI

−− ⎡ ⎤⎡ ⎤ −−= = ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

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Pattern Based Method• Preconditioner Generation

– Inverse of diagonal block

– Other matrix vector dot operation1 2

2 1T T

A C A r C rr

C B B r C r

aA r a r

a

⎡ ⎤+⎡ ⎤= ⎢ ⎥⎢ ⎥ +⎣ ⎦ ⎣ ⎦

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

1

1 1

1

aA r a r

a

−

− −

−

⎡ ⎤⎢ ⎥= ⎢ ⎥⎢ ⎥⎣ ⎦

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Pattern Based Method• Benefit of Using Pattern Structure

– Memory cost is reduced from to less than

– Reuse of sub matrixes and vectors can reduce Cache Miss dramatically

– Better preconditioner can be constructed to accelerate convergence speed

0.5( )O n( )O n

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Pattern Based Method• Algorithm Flow

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Pattern Based Method• Experimental Performance: Speed

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Pattern Based Method• Experimental Performance: Memory

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Conclusion• New pattern based method is presented• Combine advantages of direct method and

iterative method• Decrease the Cache Miss dramatically • Reduce the memory efficiency dramatically• New preconditioner is faster than traditional

preconditioner• Fast linear PCG can be achieved even the grid

size is huge

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Thank you !