ISPD

1
ISPD2005Fast IntervalValued StatisticalInterco
nnect Modeling And Reduction

• James D. Ma and Rob A. Rutenbar
• Dept of ECE, Carnegie Mellon University
  jdma, rutenbar_at_ece.cmu.edu
• Funded in part by C2S2, the MARCO Focus Center
  for Circuit System Solutions

2
New Battlefield Manufacturing Variations

• CMOS scaling
• Good for speed
• Good for density

• Bad for variation
• Bad for manufacturability
• Bad for predictability

• No longer realistic to regard device or
  interconnect as deterministic
• Continuous random distribution with complex
  correlations

3
New Problem Statistical Analysis

• Statistical static timing analysis
• Propagate correlated normal distribution
• A limited number of operators sum and maximum

• Statistical interconnect timing analysis
• Require a richer palette of computations
• Not easy to represent statistics and push them
  through model reduction algorithms

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Approaches to Statistical Interconnect Analysis

• Straight-forward Monte Carlo simulation
• Repeat model reduction algorithms at the
  outermost loop
• General, accurate, but computationally expensive
• Control-theoretic (model order reduction)
• Based on perturbation theory Liu-et al, DAC99
• Multi-parameter moment matching Daniel-et al,
  TCAD04
• Circuit performance evaluation
• Low-order analytical delay formula Agarwal-et
  al, DAC04
• Asymptotic non-normal probability extraction
  Li-et al, ICCAD04
• Classical interval delay analysis
  Harkness-Lopresti, TCAD92

5
New Interval Ideas

• Affine interval
• Define central point and radius
• Keep source of uncertainties
• Handles correlations by uncertainty sharing

• Classical interval
• Define two end-points
• No inside information
• Unable to consider correlations

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Affine Arithmetic An Overview
Results are still affine (accurate or
conservative)
Replace second-order terms with one new
uncertainty term ?

• Develop a library for most affine arithmetic
  operations
• More accurate or efficient approximations are
  also available

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From Intervals to Statistics

• Statistical assumption for the uncertainty
  symbols?

• Uniform distribution?
• Keep conservative bounds
• Not realistic for modeling manufacturing
  variations

• Choose normal distribution
• µ 0, s2 1 for each symbol ei
• Probability not equal in the interval
• Model the central mass of the infinite,
  continuous distribution

• Essential assumption
• Mechanics of calculation for finite affine
  intervals are a reasonably good approximation of
  how statistics move through the same computations

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Putting Altogether From Intervals to Algorithms

• Scalar-valued linear solve
• Backward substitution

• Classical interval-valued linear solve
• Backward substitution
• Classical interval arithmetic

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From Intervals to Algorithms (Contd)

• Affine interval-valued linear solve

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Our New Approach Affine Interval-Valued
Statistical Interconnect Model Reduction

• Ma-Rutenbar, ICCAD2004

• Represent variational RLC elements as correlated
  intervals

• Replace scalar computation with interval-valued
  computation by pushing intervals through chain of
  model reduction

Interval computation
Reduced set of intervals

• Stop, and repeatedly sample a reduced set of
  intervals

Sampling
Scalar computation

• Continue with scalar-valued computation

• Obtain delay distribution

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Interval Modeling of Interconnect Parameters

• Global variations inter-die
• Affect all the device and interconnect, in a
  similar way
• Local variations intra-die
• Affect device and interconnect close to each
  other, in a similar way
• Linearized combination of global and local
  variations

Affine forms
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Interval-Valued AWE 1st Generation

• Interval-valued MNA and LU for model reduction
• Interval-valued pole/residue analysis
• Mostly fundamental affine operations
• Compare intervals based on their central values
• Obtain a reduced, small set of interval poles and
  residues
• Sample and continue scalar transient analysis
• Monte Carlo sampling over this reduced model is
  very fast
• Similar approach for interval-valued PRIMA

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Interval-Valued AWE 2nd Generation

• 1st improvement
• Replace MNA formulation LU decomposition with
  path-tracing for tree-structured circuits to
  compute interval-valued moments much more
  efficiently

• 2nd improvement
• Stop interval-valued computation at moments, not
  poles/residues
• Then switch to sampling and scalar-valued
  computation

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1st Improvement Interval LU vs. Path-Tracing

• Path-tracing DC analyses for moments via
  depth-first search
• Tree topology does not change DFS only once
• Tracing order can be stored and remembered

• Interval estimation errors
• Like floating-point errors, but more macroscopic,
  not so easy to ignore
• The longer the chain of computation, the more
  errors

• Replace interval LU with interval path-tracing
• Reduce number of approximate affine operations
  significantly
• Improve greatly both efficiency and accuracy

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Interval-Valued AWE 2nd Generation

• A reduced, small set of interval moments via
  interval-valued path-tracing
• Sample over moment intervals to produce a set of
  scalar moments
• Continue scalar computation, just like a standard
  AWE
• Monte Carlo sampling over the reduced model is
  very fast
• Similar approach for interval path-tracing-based
  PRIMA

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2nd Improvement AWE Interval/Scalar Tradeoff

• 2nd generation
• Hybrid interval/scalar strategy

• 1st generation
• Pervasive interval computation

• Interval computation for large-scale near-linear
  model reduction
• Scalar sampling small-scale nonlinear root
  finding

• Similar tradeoff for 2nd generation of
  interval-valued PRIMA

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Benchmarks

• 3 tree-structured RC(L) interconnects
• From 120 to 2400 elements
• Deterministic unit step input
• 6 21 variation symbols
• One global, shared by all RLCs
• Others local, shared by a
  cluster of nearby RLCs
• Relative s of global / local vars
• 20 / 10, 10 / 20, 5 / 30

• Able to accommodate
• Any number of uncertainties, from most types of
  variation sources
• Any reasonable combinations of global / local
  variations

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2nd Generation Implementation

• Interval arithmetic library and AWE/PRIMA in
  C/C
• Compare distribution of 50 delay
• 2nd generation (statAWE/statPRIMA) vs. RICE 4/5
  used in a simple Monte Carlo loop (RMC)
• Determine proper number of Monte Carlo samples
  using standard confidence interval techniques
  Burch-et al, TVLSI93
• Specify accuracy within 1, with 99 confidence
  level
• 3000 samples for each design combination

vs.
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Pole Distribution

• At the end of 2nd generation interval AWE/PRIMA,
  an interval-valued reduced model is obtained
• How well do the reduced interval model produce
  scalar poles?

• design0, 123 RLCs, 5 global variation, 30
  local variation, 6 variation terms, 8th order
  AWE, distributions of 4 dominant poles on complex
  plane

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Accuracy Efficiency
Mean Delay Err Stdev Err Speed-up
statAWE 1.7 1.8 11
statPRIMA 2.5 2.6 10

• CPU time 1 interval analysis 300
  deterministic runs

• Delay PDFs ex 1275 RCs, 5 global, 30 local,
  4th order models

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Interval/Scalar Tradeoff

• Compare 4 AWE interval strategies

Interval Path-tracing MNA LU
Moments I (2nd gen.) II
Poles/residues III IV (1st gen.)

• If 510 error is OK, one can still use
  intervals pervasively
• 1st ? 2nd generation 10X less CPU, 34X
  less error

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Conclusions and Ongoing Work

• Affine interval model statistical
  interpretation allow us to
• Represent the essential mass of a random
  distribution
• Preserve 1st-order correlations among
  uncertainties
• Retarget classical model reduction to
  interval-valued computations
• Improved 2nd generation
• Smarter interval linear solves and
  interval/scalar tradeoffs
• 10X faster, and 34X less error
• Whats next?
• Works well for interconnect reduction but how
  general is the idea?
• Can we bring statistics into arbitrary CAD tools
  efficiently?
• In progress interval-valued physics-based
  TCAD/DFM modeling