speeding up monte carlo simulation using an emulator · introduction: the objective...
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Where innovation starts
Speeding Up Monte Carlo SimulationUsing An Emulator
A.K. Tyagi, W.H.A. Schilders, X. Jonsson, T.G.J. Beelen
Center for Analysis, Scientific Computingand Applications (CASA),
Mentor Graphics R©, Grenoble (FR)
November 04, 2015
1/22
/
Outline
1 Introduction
2 Project Startup
3 Motivation
4 An Emulator Technique
5 Results
6 Conclusion and Further Steps
1/22
/
Outline
1 Introduction
2 Project Startup
3 Motivation
4 An Emulator Technique
5 Results
6 Conclusion and Further Steps
2/22
/
Introduction: The Project
ASIVA14: Analog Simulation and Variability Analysis for 14nm design
http://www.itn-asiva14.eu
3/22
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Introduction: The Project
ASIVA14 Objectives
1) Speed up time-domain simulations fornearly-periodic circuits from 1 week to 2hours (70x speedup) (Giovanni De Luca)
2) Speed up time-domain simulations in fullchip verification by orders of magnitude(10x speedup) (Nhunh Dang)
3) Variability Analysis: Reduce the number ofMonte Carlo runs for variability and uncertaintyanalysis in ICs (1010x speedup /) (Anuj Tyagi)
4/22
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Introduction: The Project
Promotor:
Prof. Wil SchildersTU Eindhoven, Netherlands
email: w.h.a.schilders@tue.nl
My Supervisors:
Dr. Xavier JonssonMentor Graphics, Grenoble
email: xavier_jonsson@mentor.com
Dr. Theo BeelenTU Eindhoven, Netherlandsemail: t.g.j.beelen@tue.nl
5/22
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Introduction: The Project
How we work
• Regular Meetings
5/22
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Introduction: The Project
How we work
• Reports
• Weekly reports
• Technical reports
6/22
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Introduction: Variability Analysis
• With each new technology, device sizes shrink
• Biggest challenge: uncertainty in circuit performance
• Uncertainty leads to circuit failure that affects the production yield
6/22
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Introduction: Variability Analysis
• With each new technology, device sizes shrink
• Biggest challenge: uncertainty in circuit performance
• Uncertainty leads to circuit failure that affects the production yield
6/22
/
Introduction: Variability Analysis
• With each new technology, device sizes shrink
• Biggest challenge: uncertainty in circuit performance
• Uncertainty leads to circuit failure that affects the production yield
7/22
/
Introduction: The objective
• Estimate the circuit failure probability
• Uncertainty and variability of the probability estimator
• Estimation shall be possible within required computational time
7/22
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Introduction: The objective
• Estimate the circuit failure probability
• Uncertainty and variability of the probability estimator
• Estimation shall be possible within required computational time
7/22
/
Introduction: The objective
• Estimate the circuit failure probability
• Uncertainty and variability of the probability estimator
• Estimation shall be possible within required computational time
8/22
/
Introduction: State of the Art
• Monte Carlo (MC) Simulation
Prb(h(x) ≤ µ) =1N
N∑i=1
1h(Xi)
where Xi are N i.i.d. random numbers generated from the pdf(x) and
1h(x) =
{1 if h(x) ≤ µ0 otherwise
,
8/22
/
Introduction: State of the Art
• Monte Carlo (MC) Simulation
Prb(h(x) ≤ µ) =1N
N∑i=1
1h(Xi)
where Xi are N i.i.d. random numbers generated from the pdf(x) and
1h(x) =
{1 if h(x) ≤ µ0 otherwise
,
9/22
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Introduction: State of the Art
• Monte Carlo (MC) Simulation
Benefits:
• Robustness• Doesn’t care about the number of parameters• Easy to implement
Drawbacks:
• Computationally expensive
9/22
/
Outline
1 Introduction
2 Project Startup
3 Motivation
4 An Emulator Technique
5 Results
6 Conclusion and Further Steps
10/22
/
Project Startup
• Sensitivity Analysis
• Given, input x→ IC Model → output h(x)
• Find∂h∂x
and get most dominant parameters
• We do this to use Uncertainty Quantification
Drawback: UQ limited up to few parameters
10/22
/
Project Startup
• Sensitivity Analysis
• Given, input x→ IC Model → output h(x)
• Find∂h∂x
and get most dominant parameters
• We do this to use Uncertainty Quantification
Drawback: UQ limited up to few parameters
10/22
/
Project Startup
• Sensitivity Analysis
• Given, input x→ IC Model → output h(x)
• Find∂h∂x
and get most dominant parameters
• We do this to use Uncertainty Quantification
Drawback: UQ limited up to few parameters
10/22
/
Outline
1 Introduction
2 Project Startup
3 Motivation
4 An Emulator Technique
5 Results
6 Conclusion and Further Steps
11/22
/
Motivation
• Large number of statistical parameters
• Monte Carlo doesn’t care about the number of parameters
• Keep this method and Speed it up
• An Emulator approach can be used
11/22
/
Outline
1 Introduction
2 Project Startup
3 Motivation
4 An Emulator Technique
5 Results
6 Conclusion and Further Steps
12/22
/
An Emulator Technique
• Given IC model
IC Modelθ(x, α1, · · · , αk)
inputx = (x1, · · · , xp) output h(x)
12/22
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An Emulator Technique
• Given IC model
IC Modelθ(x, α1, · · · , αk)
inputx = (x1, · · · , xp) output h(x)
• Take m uniformly distributed random vectors Xi and get a training set (X,H)
IC Modelθ(x, α1, · · · , αk)X =
X1X2...
Xm
h(X1)h(X2)...
h(Xm)
= H
12/22
/
An Emulator Technique
• Given IC model
IC Modelθ(x, α1, · · · , αk)
inputx = (x1, · · · , xp) output h(x)
• Fit a regression model y = f (x) such that h(x) = y(x) + δ(x)
12/22
/
An Emulator Technique
• Given IC model
IC Modelθ(x, α1, · · · , αk)
inputx = (x1, · · · , xp) output h(x)
• Fit a regression model y = f (x) such that h(x) = y(x) + δ(x)
12/22
/
An Emulator Technique
• Given IC model
IC Modelθ(x, α1, · · · , αk)
inputx = (x1, · · · , xp) output h(x)
• Fit a regression model y = f (x) such that h(x) = y(x) + δ(x)
12/22
/
An Emulator Technique
• Given IC model
IC Modelθ(x, α1, · · · , αk)
inputx = (x1, · · · , xp) output h(x)
• Fit a regression model y = f (x) such that h(x) = y(x) + δ(x)
13/22
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An Emulator Technique
• Compute outputs y(X∗) for X∗ ∼ N(µ,Σ) using regression model
RegressionModel y = f (x)X∗ =
X∗1X∗2...
X∗N∗
y(X∗1)y(X∗2)...
y(X∗N∗ )
= Y∗
13/22
/
An Emulator Technique
• Compute outputs y(X∗) for X∗ ∼ N(µ,Σ) using regression model
RegressionModel y = f (x)X∗ =
X∗1X∗2...
X∗N∗
y(X∗1)y(X∗2)...
y(X∗N∗ )
= Y∗
• An Emulator model
13/22
/
An Emulator Technique
• Compute outputs y(X∗) for X∗ ∼ N(µ,Σ) using regression model
RegressionModel y = f (x)X∗ =
X∗1X∗2...
X∗N∗
y(X∗1)y(X∗2)...
y(X∗N∗ )
= Y∗
• An Emulator model
13/22
/
An Emulator Technique
• Compute outputs y(X∗) for X∗ ∼ N(µ,Σ) using regression model
RegressionModel y = f (x)X∗ =
X∗1X∗2...
X∗N∗
y(X∗1)y(X∗2)...
y(X∗N∗ )
= Y∗
• An Emulator model
he(x) =
{h(x) if |y(x) − µ| ≤ zαδ(x)y(x) otherwise
where zα is such that Φ(zα) = α, Φ(x) is cdf of standard normaldistribution.
14/22
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An Emulator Technique
• The failure probability is given by:
Prb(h(x) ≤ µ) ≈ Prb(he(x) ≤ µ) =1
N∗
N∗∑i=1
1he (X∗
i )
Note:
• We can reduce the rejections by having a ‘best’ regression model
• The ‘best’ model predict a good approximation in the importantregion
• An adaptive sampling might be used
14/22
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An Emulator Technique
• The failure probability is given by:
Prb(h(x) ≤ µ) ≈ Prb(he(x) ≤ µ) =1
N∗
N∗∑i=1
1he (X∗
i )
Note:
• We can reduce the rejections by having a ‘best’ regression model
• The ‘best’ model predict a good approximation in the importantregion
• An adaptive sampling might be used
15/22
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An Emulator Technique: Adaptive Sampling
15/22
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Outline
1 Introduction
2 Project Startup
3 Motivation
4 An Emulator Technique
5 Results
6 Conclusion and Further Steps
16/22
/
Results
Example:
• Input:x = (x1, x2) ∈ R2
∼ N(0, I)
• Output:
h1(x) = x21 − 8x2 + 16,
h2(x) = −16x1 + x22 + 32
h(x) = max(h1(x), h2(x))
• Evaluate Prb(h ≤ 20)
• We used Matlab Kriging toolbox to fit Gaussian regression model y(x)
17/22
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Results
Figure: α = 0.01 i.e., 99% confidence bound for accept model
Prb_err = 1.0%
18/22
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Results
Figure: α = 0.001 i.e., 99.9% confidence bound for accept model
Prb_err = 0.9%
19/22
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Results
Figure: Number of rejections w.r.t. adaptive runs for α = 0.01 and α = 0.001
20/22
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Results: Check Validity of the Emulator
• Get the failure probability Prmc using Monte Carlo simulator
• Evaluate failure probability PrEm through Emulator
• Repeat the processor few number of times
• Evaluate Root Mean Squared Error (RMSE) for probability estimator
RMSE =
√√√1
Nrep
Nrep∑i=1
(PrEmi − Prmc)2
20/22
/
Results: Check Validity of the Emulator
• Get the failure probability Prmc using Monte Carlo simulator
• Evaluate failure probability PrEm through Emulator
• Repeat the processor few number of times
• Evaluate Root Mean Squared Error (RMSE) for probability estimator
RMSE =
√√√1
Nrep
Nrep∑i=1
(PrEmi − Prmc)2
20/22
/
Results: Check Validity of the Emulator
• Get the failure probability Prmc using Monte Carlo simulator
• Evaluate failure probability PrEm through Emulator
• Repeat the processor few number of times
• Evaluate Root Mean Squared Error (RMSE) for probability estimator
RMSE =
√√√1
Nrep
Nrep∑i=1
(PrEmi − Prmc)2
20/22
/
Results: Check Validity of the Emulator
• Get the failure probability Prmc using Monte Carlo simulator
• Evaluate failure probability PrEm through Emulator
• Repeat the processor few number of times
• Evaluate Root Mean Squared Error (RMSE) for probability estimator
RMSE =
√√√1
Nrep
Nrep∑i=1
(PrEmi − Prmc)2
21/22
/
Results: Check Validity of the Emulator
Figure: Distribution of the probability P(h ≤ 20)
21/22
/
Outline
1 Introduction
2 Project Startup
3 Motivation
4 An Emulator Technique
5 Results
6 Conclusion and Further Steps
22/22
/
Conclusion and Further Steps
Conclusion
• The Emulator technique is easy to implement
• We assume N > 10(m + N∗rej) and get a speedup ofN
m + N∗rej> 10x, where
N∗rej is the number of rejections by the Emulator
Further Steps
• The Gaussian process doesn’t work well [4] for the dim > 10 therefore ouraim is to use dimension reduction methods
• Find alternatives (independent of dimensions) of Gaussian Process.
22/22
/
Conclusion and Further Steps
Conclusion
• The Emulator technique is easy to implement
• We assume N > 10(m + N∗rej) and get a speedup ofN
m + N∗rej> 10x, where
N∗rej is the number of rejections by the Emulator
Further Steps
• The Gaussian process doesn’t work well [4] for the dim > 10 therefore ouraim is to use dimension reduction methods
• Find alternatives (independent of dimensions) of Gaussian Process.
22/22
/
References
C. E. Rasmussen and C. K. I. Williams: Gaussian Processes forMachine Learning, the MIT Press, 2006.
V. Dubourg, Adaptive surrogate models for reliability analysisand reliability-based design optimization. PhD Thesis,BlaisePascal University - Clermont II, 2011.
S. N. Lophaven, H. B. Nielsen and J. SondergaardL: DACE, AMATLAB Kriging Toolbox Version 2.0, Technical University ofDenmark , August 1, 2002.
Marrel, A., Iooss, B., Van Dorpe, F. and Volkova, E.: An efficientmethodology for modeling complex computer codes withGaussian processes. Comput. Statist. Data Anal (2008).
22/22
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1/2
/
Gaussian Process
• A generalization of a multivariate Gaussian distribution.
• Fully specified by a mean function m(x) and covariance function k(x, x′):
f (x) ∼ GP(m(x), k(x, x′))
• We choose
m(x) = 0
k(x, x′) = exp(−
12l|x − x′|2
).
where, l is a scaled factor called hyperparameter.
1/2
/
Gaussian Process
• A generalization of a multivariate Gaussian distribution.
• Fully specified by a mean function m(x) and covariance function k(x, x′):
f (x) ∼ GP(m(x), k(x, x′))
• We choose
m(x) = 0
k(x, x′) = exp(−
12l|x − x′|2
).
where, l is a scaled factor called hyperparameter.
1/2
/
Gaussian Process
• A generalization of a multivariate Gaussian distribution.
• Fully specified by a mean function m(x) and covariance function k(x, x′):
f (x) ∼ GP(m(x), k(x, x′))
• We choose
m(x) = 0
k(x, x′) = exp(−
12l|x − x′|2
).
where, l is a scaled factor called hyperparameter.
2/2
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Gaussian Process
• data: x, h (training set)
• model: h = y(x) + ε
Gaussian likelihood:h|x, y(x) ∼ N(y, σ2I)
Gaussian process prior:y(x) ∼ GP(0, k(x, x′))
Gaussian process posterior:
y(x)|x,h ∼ GP(mpost(x) = k(x, x)T[K(x, x) + σ2I]−1y,
kpost(x, x′) = k(x, x′) − k(x, x)[K(x, x) + σ2I]−1k(x, x′)).
Gaussian predictive distribution:
y∗|x∗, x, y ∼ N(k(x∗, x)T[K+σ2I]−1y, k(x∗, x∗)+σ2−k(x∗, x)T[K+σ2I]−1k(x∗, x)). (1)
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