Pattern Based Method For P/G Grid Analysis Jin Shi, Yici Cai, - - PowerPoint PPT Presentation

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

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( ) O n

• 1

1 2 3 4 5 6 1 2 3 4 5 log Problem Size log cycles/flop

T = N4.7

Size 2000 took 5 days 12000 would take 1095 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 11 50 100 150 200 250 300 350 400 2 log(size) iteration 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

( )

p

O 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 1 1 2 1 1

( ( )) ( ( ) ( ) ) ( ) (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 v t 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 choleskey decomposition are restricted to memory usage – More fill in cause faster convergence speed

17 48.41 1e-4 44 10.96 1e-3 93 2.79 1e-2 221 1.5 1e-1 Avg Iter times Fill in ratio Drop threshold 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( , )

ij via ij T ij

a b j i c g A C A a B b C c C B

• ther

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

1 1 1 1 1 1 1 1

' ' '

T T T T T

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

− − − − − − − −

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

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Pattern Based Method

• Preconditioner Generation

– Perform Precondition in CG Algorithm

1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 2 1 1 1 2 2 1 1 2 2 1 1 2 3

' 1: 2: ' ' 3:

T T T T T

A C I I A C A P r r r C B C A I I B I r r r step r r p A r C A I r C A r r C p r p r A step r r B r r B step r

− − − − − − − − − − −

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

1 1 2 1 2 2 2 2 2

r A Cr I A C r r I

− −

⎡ ⎤ ⎡ ⎤ − − = = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦

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Pattern Based Method

• Preconditioner Generation

– Inverse of diagonal block – Other matrix vector dot operation

1 2 2 1 T T

A C A r C r r C B B r C r a A r a r a ⎡ ⎤ + ⎡ ⎤ = ⎢ ⎥ ⎢ ⎥ + ⎣ ⎦ ⎣ ⎦ ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

1 1 1 1

a A 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 !