impec07_crestaux.pdf

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Polynomial Chaos expansionUncertainty and sensitivity analysis by PC

Application : Advection-dispersion

Polynomial Chaos Expansion for UncertaintiesQuantication and Sensitivity Analysis

Thierry Crestaux1, Jean-Marc Martinez1,Olivier Le Maitre2, Olivier Lafitte 1,3

1Commissariat lnergie Atomique, Centre dtudes de Saclay, France ;2Universit dvry, France ; 3Universit Paris Nord, France

Thierry Crestaux SAMO, JUNE 2007

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Introduction

Uncertainties quantification in numerical simulation by PolynomialChaos expension is a technic which has been used recently fornumerous problems.This method can also be used in global sensitivity analysis by theapproximation of sensitivity indices.


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Plan

1 Polynomial Chaos expansionPolynomial ChaosIntrusive method : Galerkin projectionNon-intrusive methods

Least square approximationNon Intrusive Spectral Projection

2 Uncertainty and sensitivity analysis by PCUncertainty analysisSensitivity Analysis

Sobol decomposition of the PC surrogate modelSensitivity indicesExamples

3 Application : Advection-dispersion


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Polynomial ChaosIntrusive method : Galerkin projectionNon-intrusive methods

Polynomial Chaos expansion


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Polynomial Chaos

Polynomial Chaos (PC) expansions of (2nd order) stochastic processes :

y(x , t , ) =

k=0k (x , t)k (()) (Wiener 1938).

Application to uncertainty quantication by Ghanem and Spanos. = (1, 2, . . . , d ) a set of d independent second order randomvariables with given joint density p() =

pi (i).

(k ())kN multidimensional orthogonal polynomials with regardto the inner product (mathematical expectation)< k ,l >

k ()l ()p()d = kl ||k ||2.


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Polynomial Chaos

y(x , t , ) =

k=0k (x , t)k (),

where k (x , t) are the PC coefcients or stochastic modes of y . Knowledge of the k fully characterizes the process y .For practical use, truncature at polynomial order no :

P + 1 = (d + no)!d !no! y(x , t , ) P

k=0k (x , t)k ().

Fast increase of the basis dimension P according to no.Need for numerical procedure to compute k .


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Intrusive method : Galerkin projection

Galerkin projectionA two steps procedure to solve spectral problems :

The introduction of the truncated spectral expansions into modelequations.Determination of the PC coefficients such that the residual isorthogonal to the basis.

M(y ; D()) = 0M(

ii i (()); D()),k (())

= 0 k .

Comments :? A set of P + 1 coupled spectral problems.? Require rewriting / adaptation of existing codes.


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Non-intrusive methods

Construction of a sample set {(i)} of and corresponding set ofdeterministic solutions {y (i) = y(x , t , (i))}.Use the solution set to estimate/compute the PC coefficients k .

Comments : Solve a (large) number of deterministic problems. Transparent to non linearities. Convergence with the sample set dimension and error estimation.

Currently we use two different non-intrusive methods :Least square approximation of the k .Non Intrusive Spectral Projection (NISP).


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Least square approximationLeast square problem for a sample sets B = ((i)) and y = (y (i)).

R(B) = (Z T Z )1Z T ywhere Z T Z is the Fisher matrix :

Z =

1 1((1)) . . . P((1))1 1((2)) . . . P((2))...

... . . ....

1 1((n)) . . . P((n))

Open questions :

Selection of the sample set ?Design Optimal Experiment, active learning ?Error estimation ?Model selection ?


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Non Intrusive Spectral Projection : NISP

Exploit orthogonality of the PC basis :

k =y(),k ()

2k , y(),k =

y()k ()pdf ()d.

Numerical integration :

y()k ()pdf ()d N

i=1y((i))k ((i))w (i) = k

2k,

with (i) and w (i) are integration quadrature points / weights. Independent computation of the PC coefficients. Curse of dimension (cubature formula, adpative construction,Monte-Carlo, . . .)


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Uncertainty analysisSensitivity Analysis

Uncertainty and sensitivityanalysis by PC


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Uncertainty analysis

Uncertainty analysis from PC coefficients is immediate :

The expectation and the variance of the process are given byE{y(x , t)} = 0(x , t) andE{(y(x , t) E{y(x , t)})2} = k=1 2k (x , t)||k ||2.Higher moments too.Fractiles and density estimation can be calculated byMonte-Carlo simulations of the PC surrogate model

y(x , t , ) P

k=0k (x , t)k ()

(only polynomials to be computed : not the full model).


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Global Sensitivity Analysis

The computation of sensitivity indices from PC coefficients is alsoimmediate.

Indeed we know exactly the Sobol decomposition of the PCs.

So thanks to orthogonality of the basis and linearity of the PCexpansion one can immediately deduce the Sobol decomposition ofthe PC expansion.


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Sobol decomposition of the PC surrogate modelFor each integrable function f , there is a unique decomposition :

f () =

u{1,2,...,d}fu(u), (Sobol1993)

with f = f0.The Sobol decomposition of a troncated PC expansion y is,

y() =

u{1,2,...,d}yu(u) =

Pk=0

k k ()

The terms of the decomposition are

yu(u) =kKu

k k ()

with K = {0, 1, ...,P}, Ku := {k K |k () = k ( = u)}and y = 00


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Sensitivity indices

Sensitivity indices are calculated with the formula

Su =2u2y

Where 2y is2y =

u{1,2,...,d}\

2u

and 2u are explicits for PC expansions

2u =

y2u ()p()d =

kKu

2k ||k ||2


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Example : Homma-Saltellif () = sin(1) + 7sin2(2) + 0.143sin(1) .

1e-05

1e-04

0.001

0.01

0.1

1

1 10 100 1000 10000

L-1

erro

r on

sens

itivi

ty in

dice

s

Sample set dimension

cub S1cub S2cub S5MC S1MC S2MC S3MC S4MC S5MC S6MC S7

FIG.: L-1 error sensitivity indicescomputed by PC coefficients andMonte-Carlo simulation vs. thesample set dimension

k computed by NISP usingSmolyak cubature.

The figure shows the expectationof the error on the computation byMonte-Carlo over 100simulations.


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Example : Saltelli-Sobol, non smooth functiong() =

Qpi=1(|4i 2|+ ai )/(1 + ai ), ai = (i 1)/2, p = 5 .

1e-04

0.001

0.01

0.1

1

10

1 10 100 1000 10000 100000

L-1

erro

r on

sens

itivi

ty in

dice

s

Sample set dimension

cub S1cub S2cub S3cub S4cub S5MC S1MC S2MC S3MC S4MC S5MC S6MC S7MC S8MC S9

MC S10MC S11MC S12MC S13MC S14MC S15

FIG.: L-1 error sensitivity indicescomputed by PC coefficients andMonte-Carlo vs. the sample setdimension

k computed by NISP usingSmolyak cubature.

The figure shows the expectationof the error on the computation byMonte-Carlo over 100simulations.


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Application :Advection-dispersion in a porousmedia


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Equation of advection-dispersion

(1 + R) Ct

(z, t) = z

(qC(z, t) (D0 + |q|)C

z(z, t)

),

(+ Initial and boundary conditions).

Deterministic inputR 0 decay rate,q Darcy velocity, ]0, 1] porosity,D0 mol. diffusivity.

Input uncertainties hydrodynamic dispersion coefficient :

= ab,

where a and b random

log(a) U([104, 102]), b U([3.5,1]).


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Application : Advection-dispersiont = 5h. t = 8h. t = 10h.

0.01

0.1

1

10

100

1000

0 0.2 0.4 0.6 0.8 1

pdf(C

)

C

GalerkinNISP

0.01

0.1

1

10

100

1000

0 0.2 0.4 0.6 0.8 1

pdf(C

)

C

GalerkinNISP

0.01

0.1

1

10

100

0 0.2 0.4 0.6 0.8 1

pdf(C

)

C

GalerkinNISP

t = 12h. t = 13h. t = 14h

0.01

0.1

1

10

100

0 0.2 0.4 0.6 0.8 1

pdf(C

)

C

GalerkinNISP

0.1

1

10

100

0 0.2 0.4 0.6 0.8 1

pdf(C

)

C

GalerkinNISP

0.01

0.1

1

10

0 0.2 0.4 0.6 0.8 1

pdf(C

)

C

GalerkinNISP

t = 15h. t = 18h. 20h.

0.1

1

10

100

0 0.2 0.4 0.6 0.8 1

pdf(C

)

C

GalerkinNISP

0.1

1

10

100

0 0.2 0.4 0.6 0.8 1

pdf(C

)

C

GalerkinNISP

0.001

0.01

0.1

1

10

100

1000

0 0.2 0.4 0.6 0.8 1

pdf(C

)

C

GalerkinNISP

FIG.: Comparison between pdf of the concentration at x = 0.5 for differenttimes obtained by Galerkin and NISP (no = 6).


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 100 200 300 400 500 600

indi

ces d

e se

nsib

ilit

pas de temps

S_aS_b

S_ab

FIG.: Sensitivity indices computed thanks to the PC coefficients computed vs.time


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Conclusion

SummaryAlternative techniques (intrusive / non-intrusive) available forpracticle determination of PC coefficients ;PC expansion contains a great deal of information in aconvenient compact format ;Global sensitivity analysis proceeds immediately from PCexpansion ;Limited to low-moderate dimensionality of the input uncertainty ;Issues in application to non-smooth processes (remedy : usenon-smooth basis).


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Conclusion

PerspectivesImprovement of non-intrusive methods (development of efficientadaptive quadrature techniques, automatic enrichment of samplesets using active learning techniques) ;Reduced basis approximation ;Application to industrial problems ;Application to identification and optimization problems.


Polynomial Chaos expansionPolynomial ChaosIntrusive method: Galerkin projectionNon-intrusive methods

Uncertainty and sensitivity analysis by PCUncertainty analysisSensitivity Analysis

Application: Advection-dispersion

impec07_crestaux.pdf

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