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

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PiCAP A Parallel and Incremental Capacitance
Extraction Considering Stochastic Process
Variation
Fang Gong1, Hao Yu2, and Lei He1 1Electrical
Engineering Dept., UCLA 2Berkeley Design
Automation Presented by Fang Gong
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Outline

• Background and Motivation
• Algorithms
• Experimental Results
• Conclusion and Future Work

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Outline

• Background and Motivation
• Algorithms
• Experimental Results
• Conclusion and Future Work

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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

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Background of BEM Based Cap Extraction
Source Panel j

• Capacitance extraction in FastCap
• Procedures
• Discretize metal surface into panels
• Form linear system by collocation
• Results in dense potential coeffs
• Solve by iterative GMRES
• Fast Multipole method (FMM) to evaluate Matrix
  Vector Product (MVP)
• Preconditioned GMRES iteration with guided
  convergence

Observe Panel i

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

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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 formulation
• Cons Random integral is slow for full-chip
  extraction
• Stochastic orthogonal polynomial (SOP) expansion
• Cui, 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 O(n)
  performance with a further parallel improvement.
• Pre-conditioner should be updated incrementally
  for different variation.

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Outline

• Background and Motivation
• Algorithms
• Experimental Results
• Conclusion and Future Work

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Flow of piCAP
Incrementally update preconditioner
Potential Coefficient

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

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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.

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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

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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

• n-order stochastic orthogonal polynomial
  expansion of P
• Accordingly, m-th order (m 2n n(n - 1))
  expansion of charge is

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Augmented System

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

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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 Pass

Level 0
Level 1
Level 2
M2M
Level 3

• Update parents moments by summing the moments
  of its childrencalled M2M operation

• starting from bottom level, it calculates
  stochastic geometric moments

• M2M operations can be performed in parallel at
  different nodes

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Parallel Fast Multipole Method--Downwards

• Downward Pass

M2L
Level 0
L2L
Level 1
Level 2
M2M
Level 3
M2M, L2L are local operations, while M2L is
global operation. How to reduce communication
traffic due to global operation?

• 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 cubescalled M2L operation.

• potential is further distributed down to
  children from their parent in parallelcalled L2L
  operation.

• Calculate near-field potential directly in
  parallel

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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.

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Flow of piCAP

1. Use spectrum pre-conditioner to accelerate
   convergence
2. Incrementally update the pre-conditioner for
   different variation.

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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 (k1 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.

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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

Inverse operation only involves diagonal matrix DK
Consider different variations by updating the
nominal preconditioner partially.
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Outline

• Background and Motivation
• Algorithm
• Experimental Results
• Conclusion and Future Work

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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

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Runtime for parallel FMM

• Setup
• Two-layer example with 20 conductors.
• Other 40, 80, 160 conductors
• Evaluate Pxq (MVP) with 10 perturbationon panel
  distance
• Result
• All examples can have about 3X speedup with 4
  processors

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Efficiency of spectral preconditioner

• Setup Three test structures single plate, 2x2
  bus, cubic
• Result
• Compare diagonal precondition with spectrum
  precondition
• Spectrum precondition accelerates convergence of
  GMRES (3X).

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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

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

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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

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Thanks
PiCAP A Parallel and Incremental Capacitance
Extraction Considering Stochastic Process
VariationFang Gong, Hao Yu and Lei He