ISPD’2005 Fast IntervalValued Statistical Interconnect Modeling And Reduction James D. Ma and Rob A. Rutenbar Dept of ECE, Carnegie Mellon University {jdma, [email protected]} Funded in part by C2S2, the MARCO Focus Center for Circuit & System Solutions
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ISPD’2005 Fast IntervalValued Statistical Interconnect Modeling And Reduction
ISPD’2005 Fast IntervalValued Statistical Interconnect Modeling And Reduction. James D. Ma and Rob A. Rutenbar Dept of ECE, Carnegie Mellon University {jdma, [email protected]} Funded in part by C2S2, the MARCO Focus Center for Circuit & System Solutions. - PowerPoint PPT Presentation
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ISPD’2005
Fast Interval Valued StatisticalInterconnect Modeling And Reduction
ISPD’2005
Fast Interval Valued StatisticalInterconnect Modeling And Reduction
Our New Approach: Affine Interval-Valued Statistical Interconnect Model Reduction
Represent variational RLC elements as correlated intervals
Interval computationInterval computation
Sam
plin
g Reduced set of intervalsReduced set of intervals
Scalar computationScalar computation
delay
Replace scalar computation with interval-valued computation by pushing intervals through chain of model reduction
Stop, and repeatedly sample a reduced set of intervals
Continue with scalar-valued computation
Obtain delay distribution
[Ma-Rutenbar, ICCAD’2004]
Slide 11
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
One variation source may contribute to multiple RLC’s & lead to correlation
Any variation can have positive or negative impact on RLC
R = R0 + ∑(∆Ri i ) + ∑(∆Rj j) + ∑(∆Rk k)
C = C0 + ∑(∆Ci i ) + ∑(∆Cj j) – ∑(∆Ck k)
L = L0 + ∑(∆Li i ) – ∑(∆Lj j) – ∑(∆Lk k)Affi
ne fo
rms
Slide 12
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
LU decompositionLU decomposition
Solve for residuesSolve for residues
Delay distributionDelay distribution
Solve for polesSolve for poles
MNA formulationMNA formulation
Sam
plin
g Poles/residuesPoles/residues
Transient analysisTransient analysis
Hankel matrix & vectorHankel matrix & vector
Vandemonde matrixVandemonde matrixIn
terv
als
Inte
rval
sSc
alar
sSc
alar
s
Slide 13
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
1 2
3 4
5 6
0
0 1 2 3
5
4
6
C
C
C
C
R
R
R
R
R
2nd improvement Stop interval-valued computation at moments, not poles/residues Then switch to sampling and scalar-valued computation
Slide 14
1st Improvement: Interval LU vs. Path-Tracing
Interval estimation errors Like floating-point errors, but more macroscopic, not so easy to ignore The longer the chain of computation, the more errors
LU decomposition
rang
e
Replace interval LU with interval path-tracing Reduce number of approximate affine operations significantly Improve greatly both efficiency and accuracy
Path-tracing
rang
e
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”
Slide 15
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
Transient analysis Transient analysis & delay distribution& delay distribution
Tree & path-tracingTree & path-tracing
Scal
ars
Scal
ars
Slide 16
2nd Improvement: AWE Interval/Scalar Tradeoff
Interval MNA & LUInterval MNA & LU
Interval momentsInterval moments
Interval root findingInterval root finding
Interval poles/residuesInterval poles/residues
Scalar delayScalar delaySam
plin
g
Inte
rval
sIn
terv
als
Scal
ars
Scal
ars
Inte
rval
sIn
terv
als
Scal
ars
Scal
ars
Interval tree Interval tree & path-tracing& path-tracing
Interval momentsInterval moments
Scalar root findingScalar root finding
Scalar poles/residuesScalar poles/residues
Scalar delayScalar delay
Sam
plin
g
1st generation Pervasive interval computation
2nd generation Hybrid interval/scalar strategy
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
Slide 17
Benchmarks
ε1
ε2
ε3
ε4
ε5
3 tree-structured RC(L) interconnects From 120 to 2400 elements Deterministic unit step input
6 — 21 variation symbols One global, shared by all RLC’s Others local, shared by a
cluster of “nearby” RLC’s
Relative σ 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
Slide 18
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, TVLSI’93]
Specify accuracy within 1%, with 99% confidence level ~ 3000 samples for each design combination
CPU time: 1 interval analysis ≈ 300 deterministic runs
Slide 21
Interval/Scalar Tradeoff
Interval Path-tracing MNA & LU
Moments I (2nd gen.) II
Poles/residues III IV (1st gen.)
Compare 4 AWE interval strategies
0
0.5
1
1.5
2
2.5
3
3.5
0% 2% 4% 6% 8% 10%mean error
log
(ru
n t
ime
)
IIV
II
III
Run time vs. mean error
0
0.5
1
1.5
2
2.5
3
3.5
0% 2% 4% 6% 8%std error
log
(ru
n t
ime
)
III
IVII
I
Run time vs. std error
If ~5–10% error is OK, one can still use intervals pervasively 1st 2nd generation: ~10X less CPU, ~3–4X less %error
Slide 22
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 ~3–4X less %error
What’s 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