Variance-based sensitivity indices for models with ... - Blue Sky eLearn

Variance-based sensitivity indices for models with correlated inputs using polynomial chaos expansions B. Sudret1 , Y. Caniou2,3 1 Laboratoire

2 Phimeca 3

Navier, IFSTTAR/Ecole des Ponts ParisTech, 6-8 Avenue Blaise Pascal, 77465 Marne-la-Vall´ ee Cedex 2

Engineering, Centre d’Affaires du Z´ enith, 63800 Cournon, France

Clermont Universit´ e, IFMA, EA 3867, Laboratoire de M´ ecanique et Ing´ enieries, BP 10448, 63000 Clermont-Ferrand, France

Introduction Polynomial chaos expansions PC-based sensitivity analysis Case of dependent variables Application examples

Motivation : robust design of multi-scale systems Modern engineering makes an intensive use of computer simulation (e.g. finite element models) in order to optimize and assess the design of systems.

The design assessment often refers to various scales of analysis (multiscale models) and various physics (multiphysics analysis)

Bruno Sudret (Laboratoire Navier)

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Issues for imbricated multiphysics models X1 X1.1

X2

M1

X1.2

Y1 = X 3.1

...

X3.2

X1.n1 = X2.1

...

X2.2 ...

X2.n2

M2

M3

Y3

Y2 = X 3.n3

X2

The computational workflow consists in an imbrication of models (multiscale/multiphysics analysis) : Models of different levels have common parameters (e.g. geometrical dimensions for thermal, mechanical, acoustics problem). The output of micro-scale models is the input of the macro-scale models. Bruno Sudret (Laboratoire Navier)

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Sensitivity analysis Specifics of multi-level imbricated models

X1 X1.1

X2

M1

X1.2

Y1 = X 3.1

...

X3.2

X1.n1 = X2.1

...

X2.2 ...

X2.n2

M2

Y2 = X 3.n3

X2

M3

Y3

The PDF of the parameters of the intermediate levels (margins + dependence structure (correlations)) is implicitely defined by uncertainty propagation : semi-parametric representation + copula theory.

Need for sensitivity indices usable for dependent variables In industrial applications, need for low computational cost (e.g. a few hundred runs of the computational models) : use of meta-models Polynomial chaos expansions


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Sensitivity analysis with dependent parameters Some bibliography Saltelli & Tarantola (J. Amer. Statist. Assoc., 2002) : brute force Monte Carlo to estimate variances of conditional expectations Xu & Gertner (Reliab. Eng. Sys. Safe, 2008) : regression-based approach to identify a correlated (resp. uncorrelated) contribution of parameters. Extension in Mara & Tarantola (Reliab. Eng. Sys. Safe, in press) Li, Rabitz et al. (J. Phys. Chem., 2010) : covariance decomposition, leading to structural and correlative contributions Kucherenko et al. (Comput. Phys. Comm., 2012) : generalization of Sobol’ indices. No distinction between corr./uncorr. contributions Chastaings, Gamboa & Prieur (2012) : generalized Hoeffding decomposition and estimation by local polynomial regression Borgonovo, Tarantola et al. (2007,2008, 2011) : distribution-based “δ” sensitivity indices that are also well defined in the dependent case. This contribution is related to mixing polynomial chaos expansions and the covariance decomposition Bruno Sudret (Laboratoire Navier)

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Outline 1

Introduction

2

Polynomial chaos expansions Mathematical setting Computing the coefficients

3

PC-based sensitivity analysis

4

Case of dependent variables

5

Application examples Linear model g-Sobol’ function Mechanical example : composite beam


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Mathematical setting Computing the coefficients

Spectral representation of a random response

Soize & Ghanem, 2004

Principle Consider a computational model M : D ⊂ RM 7→ R and a probabilistic model for the uncertainty in the input parameters, say QM X ∼ fX (x) = i=1 fXi (xi ).

Assuming that E M(X)2 < ∞ one can represent the random response Y = M(X) in a suitable Hilbert space. There exists a countable orthonormal basis {ψj , j ∈ N} such that : Y =

∞ X

yj Ψj (X)

j=0

where : yj : coefficients to be computed (coordinates) Ψj : basis functions e.g. multivariate orthonormal polynomials Bruno Sudret (Laboratoire Navier)

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Polynomial chaos basis Univariate orthogonal polynomials For each marginal distribution fXi (xi ) one can define a functional inner product : Z hφ1 , φ2 ii =

φ1 (x) φ2 (x) fXi (xi ) dxi Di

and a family of orthogonal polynomials {Pki , k ∈ N} such that : Pji , Pki =

Z

Pji (x) Pki (x) fXi (x) dx = aji δjk

Classical families Type of variable

Weight function

Orthogonal polynomials

Uniform

1]−1,1[ (x)/2 2 √1 e−x /2 2π x a e−x 1 + (x) R (1−x)a (1+x)b 1]−1,1[ (x) B(a) B(b)

Legendre Pk (x)

Gaussian Gamma Beta


Hermite He (x) k

Hilbertian basis ψk (x) 1 Pk (x)/ 2k+1 √ He (x)/ k! k

p

p

Γ(k+a+1) Laguerre La (x) La (x)/ k! k k a,b a,b Jacobi J (x) J (x)/Ja,b,k k k a+b+1 Γ(k+a+1)Γ(k+b+1) J2 = 2 2k+a+b+1 Γ(k+a+b+1)Γ(k+1) a,b,k

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Polynomial chaos basis Multivariate polynomials Let us define the multi-indices (tuples) α = {α1 , . . . , αM }, of degree |α| =

M X

αi . The associated multivariate polynomial reads :

i=1

Ψα (x) =

M Y

Ψiαi (xi )

i=1

The set of multivariate polynomials {Ψα , α ∈ NM } forms a basis of the space X of second order variables : Y = yα Ψα (X) α∈NM

Truncated series A truncature scheme is selected and the associated finite set of multi-indices is generated, e.g. : M +p A = {α ∈ NM : |α| ≤ p} card A ≡ P = p Bruno Sudret (Laboratoire Navier)

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Regression method

Berveiller & Sudret, 2006

Principle One considers the PC expansion as the sum of a truncated series and a residual : P−1

Y = M(X) =

X

yj Ψj (X) + εP ≡ YT Ψ(X) + εP

j=0

where :

Y = {y0 , . . . , yP−1 } Ψ(x) = {Ψ0 (x), . . . , ΨP−1 (x)}

The coefficients gathered into a vector Y are obtained by solving a least-square minimization problem : ˆ = arg min E Y


h

YT Ψ(X) − M(X)

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2 i

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Least-square minimization in practice An experimental design is selected, say X = {x (i) ∈ DX , i = 1, . . . , n}, e.g. LHS design or Quasi Monte Carlo sequences. The set of response values is computed by running the model for each x (i) , say : Y = {M(x (i) ), i = 1, . . . , n} The experimental matrix is computed : Aij = Ψj (x (i) )

i = 1, . . . , N ; j = 0, . . . , P − 1

The least-square minimization problem is solved : ˆ reg = AT A Y

−1


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Error estimation and sparse expansions Ph.D thesis Blatman, 2009

The truncation error may be quantified by a cross validation technique known as leave-one-out, which leads to the Q 2 error estimator : ELOO =

n 2 1X MPC \i (x (i) ) − M(x (i) ) n i=1

Q2 = 1 −

ELOO ˆ V[Y]

By means of such an error estimator and adaptive algorithms, sparse PC expansions may be obtained : Blatman & Sudret, Prob. Eng. Mech. (2010) –, Reliab. Eng. Sys. Safe (2010)

- forward construction

- model selection using the Least Angle Regression algorithm by Efron et al. (2004) Blatman & Sudret, J. Comput. Phys., 2011 Sparse expansions allow one to address high dimensional problems up to M = 100 input parameters.


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Outline

1

Introduction

2

Polynomial chaos expansions

3


4


5

Application examples


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Sobol’ decomposition (functional ANOVA)

Sobol’, 1993 Saltelli et al. , 2000

Consider a model M : x ∈ [0, 1]M → M(x) ∈ R. The Sobol’ decomposition reads : M(x) = M0 +

M X

X

Mi (xi ) +

i=1

Mij (xi , xj ) + · · · + M12...M (x)

1≤i 0 ⇔ k ∈ (i1 , . . . , is )}


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PC-based Sobol’ indices Computation of the Sobol’ indices The partial variances Di1 ...is are obtained by summing up the square of selected PC coefficients. Di

=

X

2 yα

Ai = {α : αi > 0, αj6=i = 0}

α∈Ai

Di1 ...is

=

X

2 yα

Ai1 ...is = {α : αk > 0 ⇔ k ∈ (i1 , . . . , is )}

α∈Ai1 , ... ,is

The Sobol’ indices come after normalization : Si1 ...is =

Di1 ...is D

Once the PC expansion is available, the full set of Sobol’ indices are obtained for free !


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Outline

1

Introduction

2


3


4


5

Application examples


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Back to the PC expansion Consider a computational model M : D ⊂ RM 7→ R and a probabilistic model for the uncertainty in the input parameters, say X ∼ fX (x). The joint CDF/PDF may be represented by its copula C (·) (resp. copula density c(·)) : FX (x) = C (FX1 (x1 ), . . . , FXM (xM )) fX (x) = c (FX1 (x1 ), . . . , FXM (xM ))

M Y

fXi (xi )

i=1

A generalized spectral expansion may be derived (Sudret,2007) : Y =

X α∈NM

yα p

Ψα (X) c (FX1 (x1 ), . . . , FXM (xM ))

However, it cannot be cast anymore as a hierarchical decomposition.


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Covariance decomposition

Li, Rabitz et al. , 2010

Suppose a functional decomposition of the model exists, e.g. : M(x) =

X

Mu (x u )

u⊂{1, ... ,M}

The variance of the model response reads :

 Var [M(X)] = Var 

 X

Mu (X u )

u⊂{1, ... ,M}

X

=

Var [Mu (X u )] +

X

X

Cov [Mu (X u ); Mv (X v )]

u⊂{1, ... ,M} v(u

u⊂{1, ... ,M}

NB : in case of independence, the second term is zero.


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Covariance-based sensitivity indices

Li, Rabitz et al. , 2010

Total sensitivity index Su(T) = Cov [Mu (X u ), M(X)] /Var [Y ]

Structural sensitivity index Su(S) = Var M2u (X u ) /Var [Y ]

Correlative sensitivity index Su(C ) = Su(T) − Su(S) These indices are denoted respectively by Spj , Spaj and Spbj in Li, Rabitz et al. (2010). (C )

Su

= 0 on the case of independent parameters.


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Covariance-based indices using PC expansions Principle Functional decomposition A truncated polynomial expansion is computed assuming that the parameters are independent : M(x)≈

X

yα Ψα (x)

α∈A

The terms are grouped according to their input parameters : Mu (x u ) =

X

yα Ψα (x)

Au = {α : k ∈ u ⇔ αk > 0}

α∈Au


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Covariance-based indices using PC expansions Monte Carlo estimators

A sample set X = x (i) , i = 1, . . . , N the joint PDF fX (x).

of size N is drawn according to

Classical estimators are used : \ Var [Y ] =

N 2 1 X M(x (i) ) − µM N −1 i=1

d (T) =

Su

d (S) =

Su

1 N −1 1 N −1

N X

Mu (x (i) ) − µMu

\ M(x (i) ) − µM /Var [Y ]

Mu (x (i) ) − µMu

2

\ /Var [Y ]

i=1 N X i=1

d d d (C ) (T) (S) = Su − Su

Su

NB : all the functions are multivariate polynomials. N = 105 evaluations may be carried out in a matter of seconds. Bruno Sudret (Laboratoire Navier)

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Linear model g-Sobol’ function Mechanical example : composite beam

Outline

1

Introduction

2


3


4


5

Application examples Linear model g-Sobol’ function Mechanical example : composite beam


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Linear model with dependent input

Li, Rabitz et al. (2010)

Computational model Y = 5X1 + 4X2 + 3X3 + 2X4 + X5 where X is a Gaussian vector : 

µXi = 0.5

CXX

   =   

1 0.6 0.2 0 0

0.6 1 0.2 0 0

 0.2 0 0  0.2 0 0   1 0 0    0 1 0.2  0 0.2 1.0

Polynomial chaos expansion max. degree p = 1 , P = 6 terms. Obtained accuracy : Q 2 = 1.000 Bruno Sudret (Laboratoire Navier)

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Linear model with dependent input

Li, Rabitz et al. (2010)

Independent case : Sobol’ indices Parameters X1 X2 X3 X4 X5

S 0.455 0.291 0.164 0.073 0.018 1.001

P

ST 0.455 0.291 0.164 0.073 0.018 1.001

Dependent case Parameters X1 X2 X3 X4 X5

P

S (T) 0.442 0.336 0.159 0.047 0.015 1.000

S (S) 0.273 0.175 0.098 0.044 0.011 0.601

S (C ) 0.169 0.161 0.061 0.003 0.004 0.399

Results are identical to those obtained by Li, Rabitz et al. (2010). Bruno Sudret (Laboratoire Navier)

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g-Sobol function

Sobol’ (1993) Sudret (2008)

Independent variables Computational model (4 variables) Y =

4 Y |4 Xi − 2| + ai i=1

1 + ai

a = [1 , 2 , 5 , 10]

Case of independent variables :Xi ∼ U(0, 1) The partial variances and indices are obtained analytically : D=

q Y

(Di + 1) − 1,

i=1


Di =

1 3(1 + ai )2

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Si1 ...is =

s 1 Y Di D i=1

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g-Sobol function Dependent variables The joint PDF is defined by : Xi ∼ U (0, 1) a Gaussian copula parameterized by the Spearman’s correlation coefficients : C (u1 , . . . , un ) = Φn (Φ−1 (u1 ), . . . , Φ−1 (un ) ; R) where the correlation matrix may be linked to the Spearman’s rank correlation coefficients ρS of each pair of variables :

Rij = 2 sin


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π S ρij 6

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g-Sobol function Dependent variables Correlation matrix of the Gaussian copula  1  0.8 ρS =   0.8 0

0.8 1 0.8 0

0.8 0.8 1 0



0 0   0  1

Polynomial chaos expansion max. degree p = 10 P = 1001 terms. Obtained accuracy : Q 2 = 0.993


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g-Sobol’ function Case of independent variables : Sobol’ indices Parameters X1 X2 X3 X4

P

Si (anal.) 0.608 0.270 0.067 0.020 0.966

Si (chaos) 0.606 0.269 0.066 0.020 0.961

SiT (chaos) 0.639 0.299 0.079 0.027 1.044

S (S) 0.317 0.139 0.035 0.009 0.500

S (C ) 0.181 0.159 0.097 0.000 0.437

Case of dependent variables Parameters X1 X2 X3 X4

P


S (T) 0.498 0.298 0.132 0.009 0.937

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Simply supported composite beam Problem statement A composite beam of length L and section b × h is made of a fraction f of carbon fibers (Ef , ρf ) and a fraction (1 − f ) of epoxyde matrix (Ef , ρf ). The beam is loaded by its dead weight q = ρhom gh :

Of interest is the maximum midspan deflection v : v= Bruno Sudret (Laboratoire Navier)

5 qL4 384 Ehom I

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Simply supported composite beam Basic random variables and intermediate models Models Ehom

=

I

=

q

=

v

=

f Ef + (1 − f ) Em bh 3 12 [f ρr + (1 − f ) ρm ] g h 5 qL4 384 Ehom I

Basic random variables Parameter

Distribution

Mean

Coefficient of variation

L b h F Ef

Lognormal Lognormal Lognormal Lognormal Lognormal

1m 10 cm 1 cm 200 N 300 GPa

3% 3% 3% 20% 15%

Em ρf

Lognormal

ρm f

Lognormal Lognormal

10 GPa 1800 kgm−3 1200 kgm−3

15%

Lognormal


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Implementation PC expansions of the intermediate variables Ehom , q, I Ehom

=

X

eα Ψα (X)

α∈AE

q

=

X

qα Ψα (X)

α∈Aq

I

=

X

iα Ψα (X)

α∈AI

Non parametric estimation of the margins

bfI (i) Bruno Sudret (Laboratoire Navier)

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Implementation (2) Estimation of the dependence structure : Gaussian copula based on the Spearman’s rank correlation coefficients.

ρS (q, E) = 0.09

ρS (q, I ) = 0.79

Isoprobabilistic transform of L, I , q, Ehom into standard normal variates U PC expansion of v and computation of covariance-based indices (chaos degree p = 3, P = 35 terms, Q 2 = 0.999) Bruno Sudret (Laboratoire Navier)

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Results Ignoring the dependence (independent copula) Parameters q L E I

S 0.046 0.236 0.537 0.164 0.983

P

ST 0.048 0.246 0.551 0.172 1.017

Using the Gaussian copula Parameters q L E I

P


S (T) 0.043 0.248 0.531 0.160 0.982

S (S) 0.043 0.240 0.528 0.160 0.971

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S (C ) 0.000 0.008 0.003 0.000 0.011

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Conclusions Polynomial chaos expansions are a versatile tool for computing efficiently sensitivity indices. The built-in orthogonality of the terms of a PC expansion leads to a straightforward evaluation of the Sobol’ indices in the independent case. The functional decomposition obtained assuming the independence may be post-processed together with the original joint PDF of the parameters in order to compute the covariance-based indices. The latter do not provide a clear distinction between interaction terms and correlative contributions to first-order terms. Note that the generalized Sobol’ decomposition proposed by Kucherenko et al. (2011) also provide different results.

Thank you very much for your attention !


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