DOWNLOAD FROM GP TECHNOLOGIES, INC., www.ghiocel-tech.com 1 HCF Conference, New Orleans, LO, March 8 HCF Conference, New Orleans, LO, March 8 - - 11, 2005 11, 2005 Stochastic Response Surface Approximation Stochastic Response Surface Approximation Using (Bayesian and Fuzzy) Hierarchical Models Using (Bayesian and Fuzzy) Hierarchical Models Dr. Dan M. Ghiocel Dr. Dan M. Ghiocel Email: dan.ghiocel@ghiocel Email: dan.ghiocel@ghiocel - - tech.com tech.com Phone: 585 Phone: 585 - - 641 641 - - 0379 0379 Ghiocel Predictive Technologies Inc. Ghiocel Predictive Technologies Inc.
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HCF Conference, New Orleans, LO, March 8HCF Conference, New Orleans, LO, March 8--11, 200511, 2005
Stochastic Response Surface Approximation Stochastic Response Surface Approximation Using (Bayesian and Fuzzy) Hierarchical ModelsUsing (Bayesian and Fuzzy) Hierarchical Models
Dr. Dan M. GhiocelDr. Dan M. GhiocelEmail: dan.ghiocel@ghiocelEmail: [email protected]
Phone: 585Phone: 585--641641--03790379
Ghiocel Predictive Technologies Inc.Ghiocel Predictive Technologies Inc.
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To discuss some of advanced stochastic response approximation tools that can useful for engine applications.
The proposed hierarchical models are capable of accurately approximating complex stochastic response problems.
The use of hierarchical approximation models is very efficient when employed in conjunction with (i) stochastic ROM and for (ii) RBDO analysis.
Objective of this PresentationObjective of this Presentation
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Two Major Stochastic Modeling Aspects:Two Major Stochastic Modeling Aspects:
1. Develop Accurate Stochastic Approximation Models for High-Complexity Behavior Given Sample Dataset (Data-based Stochastic ROM) - Global and Local Accuracy in Statistical Data Space – ROM in Data Space- For both System Inputs and Outputs
2. Develop Fast Stochastic Simulation Models Given the Physics (PDE) and Stochastic Inputs (Physics-based Stochastic ROM) - Global and Local Accuracy in Physical Space – ROM in Physical Space- For System Outputs
3. Combine 1) and 2) to build efficient Stochastic ROMs for stress predictions
Implicit Formulation: Using joint PDF estimation of
y
jx −
y
jx −
==)(f),y(f)y(f
xxx
T],y[z x=
dy)y(yf]y[E xx ∫∞
∞−
=
--- Least-square fitting ( is explicit)
--- Stochastic Neural Networks ( is implicit)Decomposes overall complex JPDF in localized simple JPDFs.
y
y
Limited
Refined
)()(ry xxx ε+=
Solution is obtained by stochastic interpolation
Explicit Formulation: Using function approximation via nonlinear regression
Convergence: Minimizing Mean Square Error (in Mean Square sense)Causal relationship. Implicit assumption of Gaussian variations.
)(r]y[E xx =
Convergence: Using Maximumum Likelihood Function (in Probability sense)Non causal relationship. No implicit assumption of Gaussian variations.
SecondSecond--Order (SO) Approximation of Stochastic FieldsOrder (SO) Approximation of Stochastic Fields
HighHigh--Order (HO) Approximation of Stochastic FieldsOrder (HO) Approximation of Stochastic Fields
Defined by stochastic vector or field
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))(z)(u(g)(z)(u),(ui
iii
ii ∑∑ θ=θ=θ xxx
SO Stochastic Field ExpansionSO Stochastic Field ExpansionA Non-Gaussian (translation) stochastic field can be expanded:
(1) Original Space Expansiondx ),x(u )x(u)(z
Dii ∫ θ=θCompute Non-Gaussian Variables:
(2) Transformed Space Expansiona. Transform Original Field in A Gaussian Imageb. Perform Expansion in Gaussian Image Space c. Back-Transform to Non-Gaussian Original Space
Non-Gaussian GaussianStaticMappingCase (1) Case (2)
)(z)(u)(z)(),(u i
n
0iiii
n
0ii θ=θΦλ=θ ∑∑
==
xxx
Example: KL/PCA Expansion KL/PCAEigenDecompositionREDUCED STOCHASTIC SPACE
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1L denoted one computational layer ( stochastic FEA)
1L 1L 1L
Symbolic Representation
Input Output
∑=k
kkji )h(fw)x,x(f
jxix1h
2h
jxix1h
2h
∑=k
kklji )h(fw)x,x,x(f3D
2DLocal JPDF ModelsLocal JPDF Models
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Comparison between RBF and Local PPCA ExpansionsComparison between RBF and Local PPCA Expansions
Local PPCA ExpansionLocal PPCA ExpansionRBF ExpansionRBF Expansion
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Best Fitted Fitted
Best Fitted, Simpler Best Fitted, More Refined
Model Fitting (Estimation Problem)Model Fitting (Estimation Problem)
Model Selection (Evidence Problem)Model Selection (Evidence Problem)
Optimize ModelParameters Given Data
Select Most Plausible ModelGiven Data
Stochastic Model Fitting and Selection Stochastic Model Fitting and Selection
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-1-0.5
00.5
11.5
-1
0
1
20
100
200
300
400
X1
Mean Surface actual
X2
F(X
1,X
2)
-1
0
1
2 -1-0.5
00.5
11.5
0
100
200
300
400
500
600
X2
2D Surface at 90% Probability after smoothing
X1
F(X
1,X
2)
-1-0.5
00.5
1
0100
200300
400500
0
2
4
6
X1
F(Y) at a given X=0
Y
F(Y
)
50% Probability Surface50% Probability Surface
IntegrationIntegration)G(f x
),G(f x
90% Probability Surface90% Probability Surface
Decomposition in Local Stochastic ModelsDecomposition in Local Stochastic Models Joint and Conditional Joint and Conditional PDFsPDFs ComputationComputation
Computation of ProbabilityComputation of Probability--Level Response SurfacesLevel Response Surfaces
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Stochastic vs. Fuzzy Approximation Stochastic vs. Fuzzy Approximation
Stochastic ApproximationStochastic Approximation
Priors are assumed to be independent standard Gaussian PDF
Stochastic 2L HM Approximation
∑∑∏
∏∑
=
= =
=
=
==M
1jM
1ji
ji
n
1ii
ij
i
n
1ii
j
M
1jjj
)x(fa
)x(fay)x(by)x(f
Fuzzy BF ApproximationSingleton fuzzifier, Gaussian membership functions, center average defuzzifier and product-inference rule
Fuzzy ApproximationFuzzy Approximation
∑∑∏
∏∑
=
=
=
==M
1jM
1iiii
i
jjii
j
M
1jjj
)s(f)sx(f
)s(f)sx(f)(y)(h)(yy[E xxxx]
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Smoothed Response SurfaceSmoothed Response Surface Summation of Summation of UnweightedUnweighted Local Local JPDFsJPDFs for Xfor XFuzzy ClusteringFuzzy Clustering--Based Local BF ModelsBased Local BF Models
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EuristicEuristic Stochastic Field Interpolation SchemesStochastic Field Interpolation Schemes
Weighted Average Constant Interpolation (WACI)Weighted Average Constant Interpolation (WACI)
[ ] )(yyE xx = =
∑
∑
=
=NC
1iii
ii
NC
1ii
)s(p)s(f
)s(p)s(f]y[E
x
x=
∑
∑
=
=NC
1iiii
iii
NC
1ii
p),(f
p),(fy
xx
xx
Weighted Average Linear Interpolation (WALI)Weighted Average Linear Interpolation (WALI)
[ ] )(yyE xx = =
∑
∑
=
=NC
1iiii
iii
NC
1ii
p),(f
p),(f)(y
xx
xxx=
∑
∑
=
=+
NC
1iiii
iii
NC
1ii
Ti
p),(f
p),(f)b(
xx
xxxa
[ ]Tp,i2,i1,i1p,i
xiy
ii ,...,,11
φφφφ
−=φ
−=+
Φa
iTiy
ii
1b zΦφ
=
ia
ib
= slopes= slopes
= intercept= intercept
[ ] )(yyE xx =∑∑
∑∑
= =
= =
µ
µn
1i
m
1jj,ij,i
n
1ij,ij,i
m
1jj,i
)(f
)(f]y[E
x
xx= )(hy j,i
n
1i
m
1jj,i x∑∑
= =
[ ] )(yyE xx =∑∑
∑∑
= =
= =
µ
µn
1i
m
1jj,ij,i
n
1ij,ij,i
m
1jj,i
)(f
)(f]y[E
x
xx= )(h)b( j,i
n
1i
m
1jj,i
Tj,i xxa∑∑
= =+
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01
23
45
0
2
45
-10
0
10
20
30
40
X1X2
F(X
1,X
2)
2D Nelson Plot
NelsonNelsonSurfaceSurface
10 Local JPDF with WACI10 Local JPDF with WACI 10 Local JPDF with WALI10 Local JPDF with WALI
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13 Local JPDFs plus WALI 7 Local JPDFs plus WALI
8 Local JPDFs plus WALI 14 Local JPDFs plus WALI
SubstractiveSubstractive Clustering with WALIClustering with WALINelsonNelson
RosenbrockRosenbrock
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2L HM2L HM
TwoTwo--Level Hierarchical Model Versus Level Hierarchical Model Versus KriggingKrigging with 10% Noisewith 10% Noise50% Probability Surface50% Probability Surface
KriggingKrigging95% Probability Surface 95% Probability Surface
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Hamiltonian MCMCHamiltonian MCMC
Dynamic MC for Stochastic RS ApproximationDynamic MC for Stochastic RS Approximation
No particle inertiaNo particle inertia With particle inertiaWith particle inertia
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MCMCMCMC--Based Response Surface ApproximationBased Response Surface ApproximationSimulated Conditional Mean SurfacesSimulated Conditional Mean SurfacesData and Local ModelsData and Local Models
Particles Particles with inertiawith inertia
Remark:Remark: Confidence Intervals Depend on Local Data Density and ParticleConfidence Intervals Depend on Local Data Density and Particle Inertia! Inertia!
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4. Concluding Remarks4. Concluding Remarks
1.1. Hierarchical stochastic approximation models, such as 2L and 3L Hierarchical stochastic approximation models, such as 2L and 3L HMsHMs,,are accurate tools for response surface modeling for complex stoare accurate tools for response surface modeling for complex stochasticchasticproblems. problems.
2. Using 2. Using HMsHMs, probability, probability--level response surfaces can be easily computed.level response surfaces can be easily computed.They save large computational efforts in RBDO analyses.They save large computational efforts in RBDO analyses.
3. The best stochastic approximation results were obtained using3. The best stochastic approximation results were obtained using thetheproposed 3L HM that combines a pair of two 2L proposed 3L HM that combines a pair of two 2L HMsHMs. .
4. The current practice approaches based using quadratic regress4. The current practice approaches based using quadratic regression withion withDOE sampling rules can be inadequate for complex stochastic respDOE sampling rules can be inadequate for complex stochastic responses,onses,as illustrated herein (slide 20).as illustrated herein (slide 20).