Adaptive method for indirect identification of the ...

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Adaptive method for indirect identification of thestatistical properties of random fields in a Bayesian

frameworkGuillaume Perrin, Christian Soize

To cite this version:Guillaume Perrin, Christian Soize. Adaptive method for indirect identification of the statistical prop-erties of random fields in a Bayesian framework. Computational Statistics, Springer Verlag, 2020, 35,pp.111-133. 10.1007/s00180-019-00936-5. hal-02373628

https://hal-upec-upem.archives-ouvertes.fr/hal-02373628

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Noname manus ript No.

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Adaptive method for indire t identi ation of the

statisti al properties of random elds in a Bayesian

framework

Guillaume Perrin · Christian Soize

Re eived: date / A epted: date

Abstra t This work onsiders the hallenging problem of identifying the sta-

tisti al properties of random elds from indire t observations. To this end,

a Bayesian approa h is introdu ed, whose key step is the nonparametri ap-

proximation of the likelihood fun tion from limited information. When the

likelihood fun tion is based on the evaluation of an expensive omputer ode,

this work also proposes a method to sele t iteratively new design points to re-

du e the un ertainties on the results that are due to the approximation of the

likelihood. Two appli ations are nally presented to illustrate the e ien y of

the proposed pro edure: a rst one based on analyti data, and a se ond one

dealing with the identi ation of the random elasti ity eld of an heteroge-

neous mi rostru ture.

Keywords Bayesian framework · un ertainty quanti ation · statisti alinferen e · sto hasti pro ess · kernel density estimation

1 Introdu tion

Random eld analysis has be ome a major tool in many s ienti elds, su h

as un ertainty quanti ation, material s ien e, biology, medi ine, signal pro-

essing, quantitative nan e, et . However, in most of these appli ations, the

knowledge of these random elds, whi h we write X, is limited. Numeri al

methods are therefore needed to identify the probability distribution of X

from the available information.

G. Perrin

CEA/DAM/DIF, F-91297, Arpajon, Fran e

E-mail: guillaume.perrin2 ea.fr

C. Soize

Université Paris-Est, MSME UMR 8208 CNRS, Marne-la-Vallée, Fran e

E-mail: hristian.soizeu-pem.fr

2 Guillaume Perrin, Christian Soize

When the information is onstituted of dire t measurements of the eld

to model, several te hniques have been proposed to perform su h an identi-

ation. For instan e, the AutoRegressive-Moving-Average (ARMA) models

[Whittle, 1951,Whittle, 1983,Box and Jenkins, 1970, allow the des ription of

Gaussian stationary random elds as a parameterized integral of a Gaus-

sian white noise. When onsidering a priori non-Gaussian and nonstation-

ary random elds, the identi ation is generally based on a two-step pro e-

dure. The rst step is the approximation of the random eld by its proje tion

on a redu ed number of deterministi fun tions [Ghanem and Spanos, 2003,

Le Maître and Knio, 2010, using for instan e the proper orthogonal de ompo-

sition [Atwell and King, 2001, the proper generalized de omposition [Nouy, 2010,

or the Karhunen-Loève expansion [Williams, 2011,Perrin et al., 2014,Perrin et al., 2013.

The se ond step is the identi ation of general sto hasti representations of the

proje tion oe ients in high sto hasti dimension [Soize, 2010,Soize, 2011,

Perrin et al., 2012,Nouy and Soize, 2014,Soize and Ghanem, 2016,Perrin et al., 2018.

The main spe i ity of this work omes from the fa t that only indi-

re t observations are available for the identi ation, in the sense that the

experimental data is made of the transformations of a limited number of

independent realizations of X through a bla k-box time- onsuming nonlin-

ear mapping, denoted by g. To make this identi ation tra table, we as-

sume that the random eld to identify belongs to a known parametri lass.

Thus, identifying the distribution of X amounts to identifying the values of

these parameters, whi h are gathered in the ve tor z. A Bayesian framework

is then onsidered [Marzouk and Najm, 2009,Stuart, 2010,Arnst et al., 2010,

Matthies et al., 2016,Emery et al., 2016: parameter z is supposed to be ran-

dom, and we sear h its posterior distribution given the available data.

Markov Chain Monte Carlo (MCMC) [Rubinstein and Kroese, 2008,Tian et al., 2016

is generally onsidered as a powerful tool to explore the posterior distribution

for these parameters. However it an be omputationally prohibitive when

ea h posterior evaluation requires evaluations of a omputationally expensive

ode, as it the ase here. To ir umvent this problem, a standard approa h is to

repla e the ode by a surrogate model, and to dire tly sample from the approxi-

mated posterior distribution asso iated with the modied likelihood using las-

si al MCMC pro edures. The surrogate model an be based on polynomial rep-

resentations [Marzouk and Najm, 2009,Marzouk and Xiu, 2009,Wan and Zabaras, 2011,

Li and Marzouk, 2014,Tsilis et al., 2017, Gaussian pro ess regression [Kennedy and O'Hagan, 2001,

Santner et al., 2003,Higdon et al., 2008,Bilionis and Zabaras, 2015,Sinsbe k and Nowak, 2017,

Damblin et al., 2013, or runs of the ode at dierent resolution levels [Higdon et al., 2003,

Chen and S hwab, 2015. Alternatively, the surrogate model an be used to

adapt the proposal distribution. In that ase, the number of expensive pos-

terior evaluations per MCMC step an be strongly redu ed, while sampling

asymptoti ally from the exa t posterior distribution (see [Rasmussen, 2003,

Fielding et al., 2011,Conrad et al., 2016,Conrad et al., 2018 for further details

about this approa h).

This work an be seen as an extension of these methods to the ase of

sto hasti odes. Indeed, for a given value of z, as X(z) is random, g(X(z))

Adaptive method for indire t identi ation 3

is also a random quantity. But if the distribution of X(z) is known on e

z is xed, the distribution of g(X(z)) is unknown, and its identi ation is

omputationally demanding. Therefore, instead of onstru ting a surrogate

model of the ode, we fo us on the approximation of the probability density

fun tion (PDF) of g(X(z)).To run a MCMC pro edure based on the asso iated approximated likeli-

hood in a reasonable omputational time, this approximation of the PDF of

g(X(z)) in any z has to be onstru ted from a xed number of already om-

puted ode evaluations. To this end, we rst propose to dire tly work on the

joint PDF of (g(X(z)), z). Then, we fo us on the Gaussian kernel density esti-

mation (G-KDE) [Wand and Jones, 1995,S ott and Sain, 2004,Perrin et al., 2018

for the PDF approximation. Indeed, this method is parti ularly interesting

for its ability to model non-Gaussian distributions with omplex dependen e

stru tures, but also be ause it allows an expli it derivation of the PDF of

g(X(z))|z on e the joint PDF is known. To onstru t relevant PDF ap-

proximations of this potentially high-dimensional random ve tor from a re-

du ed number of ode evaluations, we nally introdu e two adaptations of

the lassi al G-KDE formalism. First, an optimal partitioning of the om-

ponents of g(X(z)) is introdu ed, whi h onsists in de omposing the ran-

dom ve tor to model in well- hosen groups of omponents that an reason-

ably be onsidered as independent. Se ondly, a sequential strategy is proposed

to hoose the evaluations points on whi h the G-KDE relies. Starting from

a spa e-lling design, the obje tive is to sequentially add new ode evalu-

ations in the regions where the posterior distribution of the parameters is

high. We refer to [M Kay et al., 1979,Fang and Lin, 2003,Fang et al., 2006,

Dragulji¢ et al., 2012,Joseph et al., 2015 for the onstru tion of the initial

spa e-lling designs when the input spa es is an hyperre tangle, and to [Stinstra et al., 2003,

Stinstra et al., 2010,Auray et al., 2012,Dragulji¢ et al., 2012,Lekivetz and Jones, 2015,

Mak and Joseph, 2016,Perrin and Cannamela, 2017 for the general ase.

The outline of this work is as follows. Se tion 2 presents the theoreti al

framework of the proposed method. Se tion 3 rst illustrates the e ien y

of the method on an analyti al example, and then shows its potential for

the identi ation of the me hani al properties of an unknown heterogeneous

medium.

2 Indire t identi ation of the statisti al properties of random

elds

The obje tive of this se tion is to des ribe the adaptive pro edure we propose

for the identi ation of the statisti al properties of random elds when the

available information is a set of indire t observations.

2.1 Denitions and notations

Let (Ω,A,P) be a probability spa e. For dx, dy, dz ≥ 1,


P(X,Rdx) denotes the spa e of all the se ond-order random elds dened

on (Ω,A,P), with values in Rdx, indexed by a ompa t and onne ted spa e

X;

L2(X,Rdx) is the spa e of all the square-integrable fun tions dened on X

with values in Rdx;

g is a nonlinear measurable mapping whose omputational ost an be

high:

g :

L2(X,Rdx) → R

dy

h 7→ g(h); (1)

X (Rdz ,Rdx) refers to a parti ular lass of random elds in P(X,Rdx), whosestatisti al properties are parameterized by a deterministi ve tor z ∈ R

dz.

For instan e, X (Rdz ,Rdx) an orrespond to the set of Gaussian random

elds, whose mean and ovarian e fun tions are parameterized by the same

dz oe ients.

For all z in Rdz, X(z) is an element of X (Rdz ,Rdx).

Let X⋆be a parti ular element of P(X,Rdx), whi h an belong or not to

X (Rdz ,Rdx), and Y ⋆be its transformation by g. By onstru tion, Y ⋆

is a

dy-dimensional random ve tor. For ea h realization of X⋆, whi h we denote

by X⋆(θ) with θ ∈ Ω, Y ⋆(θ) := g(X⋆(θ)) denes a parti ular realization of

Y ⋆.

Given N independent realizations of Y ⋆, gathered in the set

S(N) := Y ⋆(θn)1≤n≤N , θn ∈ Ω,

the purpose of this work is to propose a Bayesian formalism for the identi-

ation of z⋆, su h that the probability distribution of X(z⋆) is the losest to

the one of X⋆.

Remarks

As mentioned in Introdu tion, it is important to noti e that for ea h z ∈R

dz, g(X(z)) is random. This strongly limits the possibility of repla ing

mapping z 7→ g(X(z)) by a surrogate model, as it is lassi ally done when

solving inverse problems that invoke omputationally expensive models.

In the following, for the sake of simpli ity, we assume thatX⋆ ∈ X (Rdz ,Rdx).If it was not the ase, it ould be ne essary to introdu e an error term to

model the dieren e betweenX⋆andX(z) [Kennedy and O'Hagan, 2001.

2.2 Bayesian formulation of the problem

In this work, z⋆is modeled by the random ve tor Z, to take into a ount

the fa t that its value is unknown. Let fZ be the probability density fun tion


(PDF) of Z, whi h is supposed to be known as a prior model. Hen e, identify-

ing z⋆amounts to sear hing the posterior PDF of Z | S(N), whi h we denote

by fZ|S(N). Using the Bayes theorem, it omes:

fZ|S(N)(z) =LS(N)(z)fZ(z)

E[LS(N)(Z)

] , z ∈ Rdz . (2)

There, E [·] is the mathemati al expe tation and LS(N) is the likelihood

fun tion. The elements of S(N) being statisti ally independent, it follows:

LS(N)(z) =N∏

n=1

fY (z)(Y⋆(θn)), z ∈ R

dz , (3)

in whi h fY (z) is the PDF of Y (z) := g(X(z)) for given z in Rdz, and is un-

known. To approximate fY (z), a rst possibility is to generateM independent

realizations of Y (z). Thus, based on this set, the value fY (z)(y) of fY (z) in

any point y in Rdy

an be approximated using any parametri or nonparamet-

ri statisti al learning te hnique. However, this means that fun tion g has to

be evaluated M ×Q times to evaluate fun tion LS(N) in Q points for z. This

qui kly be omes burdensome when the omputational ost for ea h evaluation

of g is relatively high (between several minutes to several hours CPU for the

onsidered appli ations). One possible approa h to ir umvent this problem is

to dire tly approximate the joint PDF of the (dy + dz)-dimensional random

ve tor (Y (Z),Z) [Soize and Ghanem, 2017. Indeed, M independent realiza-

tions of (Y (Z),Z) an be obtained from the following two-step pro edure:

we rst draw at randomM independent realizations of Z a ording to the

distribution fZ , whi h we denote by Z(ω1), . . ., Z(ωM ), where ω1, . . . , ωM

are in Ω;

for ea h value of z in Z(ω1), . . . ,Z(ωM ), we draw, at random and inde-

pendently the ones from the others, a parti ular realization of X(z), andwe dedu e a realization of Y (z) by evaluating g in this realization of X(z).

For the sake of simpli ity, we denote these realizations by Y (ωm), 1 ≤ m ≤M . Based on these realizations, the kernel estimator of fY ,Z is:

fY ,Z(y, z;H) :=det(H)−1/2

M

M∑

m=1

K(H−1/2 ((y, z)− (Y (ωm),Z(ωm)))

).

(4)

Here, det(·) is the determinant operator, K is any positive fun tion whose

integral over Rdy+dz

is one, and H is a ((dy + dz) × (dy + dz))-dimensional

positive-denite symmetri matrix, whi h is generally referred as the "band-

width matrix". In the following, we fo us on the ase where K is the Gaussian

multidimensional density, and where H is proportional to the empiri al esti-

mation of the ovarian e matrix of (Y (Z),Z), denoted by C:

H = h2C, h ∈ R. (5)


The main interest of this hypothesis omes from the fa t that it strongly

redu es the number of parameters that need to be identied for the onstru -

tion of H , while generally leading to very interesting results for the modeling

of multivariate PDFs (see [Perrin et al., 2018 for more details). Other parsi-

monious parameterizations ould be proposed for H, su h as diagonal repre-

sentations, but for su iently high values ofM , the inuen e of this hoi e on

the identi ation results is expe ted to be small.

Hen e, the PDF of (Y (Z),Z) is approximated by a mixture ofM Gaussian

PDFs, for whi h the means are the available realizations of (Y (Z),Z) and the

ovarian e matri es are all parameterized by a unique s alar h:

fY ,Z(y, z;h) =1

M

M∑

m=1

φ((y, z); (Y (ωm),Z(ωm)), h2C

). (6)

There, for any Rd-dimensional ve tor µ and for any (Rd×R

d)-dimensional

symmetri positive-denite matrixC, φ(·;µ,C) is the PDF of any Rd-dimensional

Gaussian random ve tor with mean µ and ovarian e matrix C:

φ (x;µ,C) :=exp

(− 1

2 (x− µ)T C−1 (x− µ))

(2π)d/2√det(C)

, x ∈ Rd. (7)

In addition, the blo k de omposition of C is written as:

C =

[CY Y CY Z

CT

Y Z CZZ

]. (8)

For all (y, z) ∈ Rdy ×R

dz, the kernel approximation of fY (z)(y), whi h we

denote by fY (z)(y;h), an therefore be written as follows (see Appendix for

more details about this expression):

fY (z)(y;h) =fY ,Z(y, z;h)∫

Rdy fY ,Z(v, z;h)dv

=

M∑

m=1

γm(z;h)∑M

m′=1 γm′(z;h)φ (y;µm(z),Cm(h)) ,

(9)

γm(z;h) := exp

(−

1

2h2(z −Z(ωm))

TC

−1

ZZ (z −Z(ωm))

), (10)

µm(z) := Y (ωm) + CY ZC−1

ZZ(z −Z(ωm)), (11)

Cm(h) := h2(CY Y − CY ZC

−1

ZZCT

Y Z

). (12)

It follows that the posterior PDF of Z is estimated for ea h z in Rdz

by:


fZ|S(N)(z) ≈LS(N)(z;h)fZ(z)

E

[LS(N)(Z)

] , LS(N)(z;h) :=N∏

n=1

fY (z)(Y⋆(θn);h). (13)

Remarks

One key step of these methods is the exploration of the whole spa e of the

input variables. To maximize this overing, it is generally worth hoosing

Z(ω1), . . . ,Z(ωM ) as a spa e lling design of experiments that preserves

good proje tion properties for ea h s alar input (see [Fang and Lin, 2003,

Fang et al., 2006,Perrin and Cannamela, 2017 for the onstru tion of su h

designs when prior density fZ is uniform or not).

Another ru ial aspe t of these Bayesian approa hes is the hoi e of prior

distribution fZ . Indeed, the more informative it is, the less measurements

we need to get a useful posterior distribution for Z. But if it is over on-

dent around values that are potentially biased, the un ertainty arried by

the posterior distribution may not be large enough to adequately apture

the true value of Z (see [Marin and Robert, 2007 for more details on the

onstru tion of this prior distribution).

In the standard ase, the M ode evaluations are generally used to on-

stru t a surrogate model of a omputationally expensive but determin-

isti ode. Hen e, depending on the dimension of the input spa e and

the regularity of the ode output with respe t to the inputs, interest-

ing approximations an be obtained using relatively small values of M

[Perrin et al., 2017. On the ontrary, in our ase, as z 7→ g(X(z)) is a

sto hasti simulator, the value of M is likely to be higher, as we want the

ode evaluations to allow a pre ise approximation of the dependen e stru -

ture between Y (Z) and Z in the onstru tion of their joint PDF. And the

higher dy + dz is, the higher value of M we may need. However, when on-

fronted to expensive simulators, the maximal number of ode evaluations

is generally limited (M must be less than 1000 for instan e). In that ase,

it is parti ularly important to work on methods that allow the most pre ise

identi ation of the parameters at the minimal ost. This is the obje tive of

the following se tions. In Se tion 2.3, we rst propose to de ompose Y (Z)in several groups to improve the relevan e of the nonparametri represen-

tation of PDF fY ,Z for a xed value of M . Then, sele tion riteria are

proposed in Se tion 2.5 to sequentially on entrate the ode evaluations in

the most likely regions for Z, and therefore redu e the un ertainties on its

posterior PDF fZ|S(N).

2.3 Optimal partitioning

As it is explained in [Perrin et al., 2018, when dy be omes high, separating

in dierent groups the omponents of Y (Z)|Z that ould reasonably be on-

sidered as independent an strongly improve the relevan e of fY (z) for a xed


number of ode evaluations. Let b = (b1, . . . , bdy) be a parti ular group de-

omposition of Y (Z)|Z in the sense that:

if bi = bj , Yi(Z)|Z and Yj(Z)|Z are supposed to be dependent and there-

fore belong to the same blo k,

if bi 6= bj, Yi(Z)|Z and Yj(Z)|Z are supposed to be independent and they

an belong to two dierent blo ks.

To avoid redundan ies in this blo k by blo k representation, ve tor b an

be hosen in the set:

B(dy) :=

b ∈ 1, . . . , dy

dy | b1 = 1, 1 ≤ bj ≤ 1 + max1≤i≤j−1

bi, 2 ≤ j ≤ dy

.

(14)

Hen e, for any b in B(dy), we an dene

Max(b) as the maximal value of b,

Y (ℓ)(z, b) as the random ve tor that gathers all the omponents of Y (Z)|Z =z with a blo k index equal to ℓ,

y(ℓ)(y, b) as the ve tor that gathers all the omponents of y with a blo k

index equal to ℓ.

For all b in B(dy), z in Rdz

and h := (h1, . . . , hMax(b)) in R

Max(b), if

fY (ℓ)(z,b)(y

(ℓ)(y, b);hℓ) is the kernel estimator of the PDF of Y (ℓ)(z, b), it omes:

fY (z)(y) ≈ fY (z)(y;h, b) :=

Max(b)∏

ℓ=1

fY (ℓ)(z,b)(y

(ℓ)(y, b);hℓ), y ∈ Rdy , (15)

leading to another approximation of fZ|S(N)(z) for ea h z in Rdz:

fZ|S(N)(z) ≈ fZ|S(N)(z) :=LS(N)(z;h, b)fZ(z)

E

[LS(N)(Z)

] , (16)

LS(N)(z;h, b) :=N∏

n=1

fY (z)(Y⋆(θn);h, b). (17)

2.4 Estimation of the kernel parameters

To evaluate LS(N), the values of h and b have to be identied. This an be

done by solving the following optimization problem:

(hAIC, bAIC) ≈ arg minh∈]0,+∞[Max(b), b∈B(dy)

AIC

LOO(h, b), (18)


AIC

LOO(h, b) := 2Max(b)−2 log

M∏

m=1

Max(b)∏

ℓ=1

f(−m)

Y (ℓ)(Z(ωm),b)(Y (ℓ)(Y (ωm), b);hℓ)

,

(19)

where f(−m)

Y (ℓ)(Z(ωm),b)is the kernel estimator of the PDF of Y (ℓ)(Z(ωm), b) that

is based on all the evaluations of g but the mth

one. Indeed, given Eq. (9), this

amounts to minimizing a "Leave-One-Out" version of the Akaike information

riterion (AIC) [Akaike, 1974 asso iated with the PDF of Y (Z)|Z (very lose

results would be obtained by onsidering another information riterion su h

as the Bayesian information riterion (BIC)). We refer to [Perrin et al., 2018

for more details about the solving of this optimization problem.

2.5 Adaptive strategy

By onstru tion, the pre ision of the estimation of z⋆depends on the num-

ber of experimental measurements, N , and the number of ode evaluations,

M . Classi ally, the value of N is xed, whereas it should be possible to im-

prove the a ura y of fY (z), whi h is dened by Eq. (15), by adding new

ode evaluations in the learning set. For instan e, Mnew

new points ould be

added to the learning set by evaluating the ode inMnew

independent realiza-

tions of Z|S(N) (we remind that no ode evaluations are required to hoose

these new points). However, as the kernel density estimator is based on the

post-pro essing of independent and identi ally distributed realizations of the

random ve tor to model, non onsistent results ould be obtained by mixing

realizations of Z|S(N) with realizations of Z. If su h a sele tion riterion was

hosen, this would mean that theM ode evaluations at the initial step should

not be used for the rening.

As an alternative, we propose to evaluate the fun tion

z 7→ f(z) := LS(N)(z;hAIC, bAIC)fZ(z)

in ea h value of Z(ω1), . . . ,Z(ωM ). For ea h 1 ≤ m ≤ M , let πm be the

following weights:

0 ≤ πm :=f(Z(ωm))

∑Mm′=1 f(Z(ωm′))

≤ 1. (20)

Without loss of generality, these weights are assumed to be sorted in de-

reasing order, π1 ≥ π2 ≥ . . . ≥ πM . Hen e, for 0 < α < 1, if we denote by Qα

the smallest integer su h that:

Qα−1∑

m=1

πm ≥ α, (21)


the domain Zα := z ∈ Rdz | f(z) ≥ f(Z(ωQα

) an be seen as a onservative

α- redible set for Z|S(N), in the sense that the probability for Z|S(N) to

be in Zα is likely to be higher than α. Therefore, adding new realizations of

Z|Z ∈ Zα seems a good mean to enri h the set of points on whi h the kernel

density estimator is based. Indeed, the most likely values of z at the former

step are kept in the adaptive pro edure, while a good exploration of the input

domain is expe ted if the value of α is hosen su iently high.

Finally, hoosing fZ|Z∈Zαinstead of fZ for the prior distribution of Z,

and repeating several times this pro edure, it is possible to iteratively redu e

the un ertainties about z⋆.

Remarks

By adding new ode evaluations, the obje tive is to make fY (z) be as lose

to fY (z) as possible, su h that the approximate posterior fZ|S(N) is as lose

to the true (but unknown) posterior fZ|S(N) as possible. Choosing a value

of α that is stri tly inferior to one only aims at limiting the number of new

ode evaluations that will be in the region where true posterior fZ|S(N) is

almost zero. However, this value has not to be hosen too small, as it would

arti ially redu e the un ertainty asso iated with the estimation of z⋆by

utting too mu h the tails of the true posterior. Hen e, in the appli ations

that will be presented in Se tion 3, α is hosen equal to 0.99. A ording to Eq. (21), we deliberately add one to the value of Qα to be

onservative for the estimation of the α- redible set. This is parti ularly

important for ases when after the rst iteration, π1 ≈ 1. Indeed, even if

one value of z appears to be mu h more relevant than the others, we do

not want to fo us too mu h around a single mode.

3 Appli ations

The purpose of this se tion is to illustrate the method proposed in Se tion 2

on two appli ations.

3.1 Analyti al appli ation

In this rst appli ation, X(z) refers to the Gaussian random elds whose

mean is equal to t 7→ sin(2πz3t + z4), and whose ovarian e fun tion is equal

to (t, t′) 7→ z21 exp(− (t−t′)2

2z22

).

This lass of random elds is therefore parameterized by four quantities:

two parameters for the mean value, denoted by z3 and z4, and two parameters

for the ovarian e fun tion, denoted by z1 and z2. We then introdu e U(X(z))as the image of X(z) by the following nonlinear mapping:


0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

PSfrag repla ements

t

U(t, ω1)

U(t, ω2)

U(t, ω3)

(a) Realizations of U(X(Z))

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

PSfrag repla ements

t

U⋆(t, θ1)

U⋆(t, θ2)

U⋆(t, θ3)

(b) Realizations of U(X(Z))|Z = z⋆

Fig. 1 Comparison of independent realizations U(X(Z)) and U(X(Z))|Z = z⋆.

U(X(z)) := X(t; z) sin(X(t; z)), t ∈ [0, 1] . (22)

The value of z⋆is hosen equal to (0.3, 0.2, 2, 1), and it is a priori modeled

by a uniformly distributed over [0.1, 1]× [0.05, 1]× [1, 3]× [0, 2] random ve tor,

denoted by Z. To identify z⋆, the available information is made of N = 10

independent realizations of U(X(Z))|Z = z⋆, denoted by U⋆(θ1), . . . , U

⋆(θN ).To solve the inferen e problem, M = 500 independent realizations of Z have

been drawn, whi h we write Z(ω1), . . . ,Z(ωM ). For ea h 1 ≤ m ≤ M , we

then draw at random one realization of U(X(Z(ωm))), and we denote it by

U(ωm) for the sake of simpli ity. As an illustration, several realizations of

U(X(Z)) and U(X(Z))|Z = z⋆are ompared in Figure 1.

In prin iple, the Bayesian formulation an be applied to any multi-variate

output ode. But in pra ti e, it is generally very onvenient to ondense (if

it is possible) the statisti al ontent of the ode output in a low-dimensional

ve tor [Perrin, ress. In our ontext, it is even more important, as a key step of

the proposed method is the identi ation of the joint distribution between the

parameters to be identied and the asso iated ode output, whose omplexity

strongly in reases with the dimension of the ode output. In that prospe t, we

introdu e ψp, p ≥ 1 as the solutions of the following eigenvalue problem:

∫ 1

0

M∑

m=1

U(t, ωm)U(t′, ωm)ψp(t′)dt′ = λpψp(t), (23)


λ1 ≥ λ2 ≥ · · · → 0,

∫ 1

0

ψp(t′)ψq(t

′)dt′ = δpq, (24)

where δpq is the Krone ker symbol that is equal to 1 if p = q and 0 otherwise. To

solve the inferen e problem, we nally introdu e Y (z) as the ve tor gatheringthe proje tion oe ients of U(X(Z)) on the former eigenfun tions asso iated

with the dy highest eigenvalues:

Y (Z) :=

(∫ 1

0

U(t;X(Z))ψ1(t)dt, . . . ,

∫ 1

0

U(t;X(Z))ψdy(t)dt

). (25)

The value of dy an then be hosen to guarantee a relevant representation

of the observations. To this end, we introdu e ε2 as the following quantity:

ε2(dy) :=

∑Nn=1

∫ 1

0

(U⋆(t, θn)− U⋆(t, θn; dy)

)2dt

∑Nn=1

∫ 1

0 (U⋆(t, θn))2dt

, (26)

U⋆(t, θn; dy) :=

dy∑

p=1

ψp(t)

(∫ 1

0

U⋆(t′, θn)ψp(t′)dt′

). (27)

As an illustration, Figure 2 shows the evolution of ε2(dy) with respe t to

dy, as well as the dieren e between U⋆(t, θ1) and U

⋆(t, θ1; dy) for three valuesof dy. For this appli ation, dy was hosen equal to 12, whi h orresponds to a

value of ε2 that is less than 1%.

Based on theseM realizations of (Y (Z),Z), and on these N realizations of

Y (z⋆) := Y (Z)|Z = z⋆, the adaptive Bayesian formalism presented in Se tion

2 is now applied. For this appli ation, the parameter α, whi h was introdu ed

in Se tion 2.5, is hosen equal to 0.99. At ea h iteration, new samples are

therefore added in the region where fZ|S(N) is not too small using a reje tion

approa h until we get a total of M points (in luding the points omputed at

the former iterations) in the α- redible set Zα, whose denition is also given

in Se tion 2.5. After 5 iterations, the total number of alls to the ode is

equal to 2300, whi h means that around 450 new points have been added at

ea h iteration. The results are summarized in Table 3.1 and Figures 3 and

4. As a rst omment, we verify that the identi ation of z⋆after only one

iteration is not very pre ise, in the sense that the predi tion un ertainties are

very high. This is not surprising, as we are trying to approximate the PDF

of a 16-dimensional random ve tor (dy = 12, dz = 4) on its whole denition

domain from only 500 realizations. Moving from M = 500 to M = 2300,that is to say spending the total budget at the rst iteration, does not really

help. Indeed, the results we get in terms of mean and varian e of Z|S(N)are approximatively the same. This is explained by the fa t that even if the

number of points is almost multiplied by ve, the overage of the denition


0 10 20 30 40 50 60 70 80−12−11−10

−9−8−7−6−5−4−3−2−1

0

PSfrag repla ements

log10(ε

2)

(

%

)

dy

(a) Error onvergen e

0 0.5 1−0.5

0

0.5

1

PSfrag repla ements

t

U⋆(t, θ1)

U⋆(t, θ1; dy = 15)

U⋆(t, θ1; dy = 5)

U⋆(t, θ1; dy = 10)

(b) Redu ed basis representation

Fig. 2 Evolution of the proje tion error with respe t to dy .

Z1 Z2 Z3 Z4

Referen e 0.3 0.2 2 1

E [Z|S(N)], M = 2300, i = 1 0.25 0.57 2.00 1.02

E [Z|S(N)], M = 500, i = 1 0.24 0.59 2.00 1.02

E [Z|S(N)], M = 500, i = 2 0.35 0.31 2.00 1.04

E [Z|S(N)], M = 500, i = 3 0.34 0.23 2.00 1.04

E [Z|S(N)], M = 500, i = 4 0.34 0.24 2.00 1.04

E [Z|S(N)], M = 500, i = 5 0.29 0.19 2.00 1.04

Table 1 Evolution of the posterior mean with respe t to the iteration number.

domain stays very sparse. On the ontrary, adding iteratively around 450 new ode evaluations in the most likely region, whose volume is mu h smaller than

the initial volume, allows E [Z|S(N)] to tend to z⋆, and strongly redu es the

redible intervals. This onvergen e is qui ker for the mean parameters than

for the ovarian e parameters, whi h was also expe ted, as the mean fun tion

is generally easier to identify than the ovarian e. Fo using on Figure 4, it is

also interesting to noti e that the referen e value does not need to be in the

99%- redible ellipse asso iated with Z|S(N) at the rst iteration to be in the

99%- redible ellipses at the next iterations.


1 2 3 4 50

1

2

3

PSfrag repla ements

V

a

l

u

e

s

o

f

Z1,Z2,Z3,Z4

Iteration

99%- redible interval of Z1|S(N)

Mean value of Z1|S(N)







z⋆1

z⋆2

z⋆3

z⋆4

Fig. 3 Evolution of the 99%- redible intervals and of the mean values of the omponents

of Z|S(N) with respe t to the iteration number.

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

PSfrag repla ements

z2

z1

i = 1

i = 2

i = 3

i = 4

i = 5

Referen e point

Fig. 4 Evolution of the 99%- redible ellipses of the two rst elements of Z|S(N) with

respe t to the iteration number.

3.2 Appli ation to the identi ation of the me hani al properties of an

unknown anisotropi material

The se ond appli ation deals with the identi ation of the me hani al proper-

ties of an heterogeneous mi ro-stru ture, whi h is modeled by a random elasti

medium. To this end, several experimental tests are performed on a series of

spe imens made of the same material. To be oherent with the notations in-

trodu ed in Se tion 2, we denote by X the elasti ity eld hara terizing the


Champ nu

SCAL

> 2.87E−01

< 3.15E−01

0.29

0.29

0.29

0.29

0.29

0.29

0.30

0.30

0.30

0.30

0.30

0.30

0.30

0.31

0.31

0.31

0.31

0.31

0.31

0.31

0.32

(a) Evolution of the Poisson ratio

Champ young

SCAL

> 2.04E+02

< 2.17E+02

2.05E+02

2.05E+02

2.06E+02

2.07E+02

2.07E+02

2.08E+02

2.08E+02

2.09E+02

2.10E+02

2.10E+02

2.11E+02

2.11E+02

2.12E+02

2.13E+02

2.13E+02

2.14E+02

2.14E+02

2.15E+02

2.16E+02

2.16E+02

2.17E+02

(b) Evolution of the Young modulus

Fig. 5 One potential spatial variation of the Young modulus and the Poisson ratio asso i-

ated with the sto hasti model X(z⋆).

Fig. 6 Representation of the studied me hani al phenomenon.


me hani al properties of the material that onstitutes the spe imens. Several

sto hasti models have been proposed in the framework of the heterogeneous

anisotropi linear elasti ity [Soize, 2006,Soize, 2008,Clouteau et al., 2013,Guilleminot and Soize, 2013.

It should be noted that the elasti ity eld is not a real-valued random eld, but

a tensor-valued random eld, and that the dierent omponents of this ran-

dom eld annot be identied separately due to algebrai onstraints. For this

appli ation, the sto hasti model for the elasti ity eld is based on the model

proposed in [Soize, 2006 and [Guilleminot and Soize, 2013 in 2D plan stresses

for the sake of simpli ity. Hen e, the distribution of X is non-Gaussian, and

it is parameterized by a 5-dimensional deterministi ve tor z = (z1, . . . , z5),where:

z1 is a positive dispersion oe ient that ontrols the level of u tuations,

z2, z3 are two spatial orrelation lengths,

z4 is the mean value of the Young Modulus (×109 Pa); z5 is the mean value of the Poisson ratio.

We then assume that N = 100 ubi spe imens are available, whose re-

spe tive me hani al properties are hara terized by one parti ular realization

of X(z⋆), with z⋆ = (2000, 0.1, 0.15, 210, 0.3). As an illustration, Figure 5

shows, for one parti ular spe imen, the evolution of the Young modulus and

the Poisson ratio in ea h point of [0, 1]2. The same pressure eld fS = −fSe2is then imposed on the top of ea h spe imen, and we only have a ess to the

indu ed displa ement eld on the boundaries of these spe imens (see Figure

6 for an illustration of the experimental proto ol). Let U⋆(θ1), . . . ,U⋆(θN ) be

these measured displa ements.

Based on this set of measurements, the method des ribed in Se tion 2 ould

dire tly be applied to the identi ation of z⋆. To speed up this identi ation,

following the works a hieved in [Nguyen et al., 2015, we propose an alterna-

tive method, whi h is based on a two-step pro edure. First, z⋆4 and z⋆5 will

be identied by onfronting the measured displa ements to the homogeneous

ase. On e z⋆4 and z⋆5 have been found, a Bayesian formalism will be proposed

for the identi ation of the three remaining omponents of z⋆.

Indeed, if the spe imens were made of a homogeneous material, hara -

terized by its young modulus E and its Poisson ratio ν, it is well known

[Lai et al., 2010 that the indu ed displa ement in ea h point s ∈ [0, 1]2 wouldbe equal to uhomo(s) = (as1, bs2), with

(a

b

)=

[λ λ+ 2µ

λ+ 2µ λ

]−1(0

−fS

), µ =

E

2(1 + ν), λ =

2µν

(1 − 2ν). (28)

Hen e, as we are onsidering a lass of stationary random pro esses, the

values of z⋆4 and z⋆5 an be identied as the arguments that minimize the L2

distan e between the N measured displa ements and the asso iated homoge-

neous displa ements. In this two-step approa h, the Bayesian identi ation is


no longer arried out in dimension 5, but in dimension 3. This strongly redu es

the number of ode evaluations that will be needed for a orre t identi ation

of (z⋆1 , z⋆2 , z

⋆3).

Thus, in the following, only z⋆1 , z⋆2 and z⋆3 are modeled by random quan-

tities. They are gathered in the ve tor Z, whose omponents are assumed

independent and distributed a ording to the following distributions:

log(Z1) ∼ U(4.6, 11.5), Z2 ∼ U(0.01, 0.3), Z3 ∼ U(0.01, 0.3), (29)

where for all a < b, U(a, b) is the uniform distribution over [a, b]. For a given

value of Z, it is possible to simulate independent realizations of X(Z), andto approximate (using the Finite Element Method) the displa ements indu ed

by the experimental for e eld, whi h we write U(X(Z)). Thus, for this se -ond appli ation, we rst hose at random M = 1000 values of Z a ording to

its prior distribution. For ea h of these values, a parti ular realization of the

elasti ity tensor was then generated over [0, 1]2, and the me hani al problem

that orresponds to the experimental proto ol was solved (using the software

Cast3M) to get the displa ements at the boundary of the ube. In the same

manner than in Se tion 3.1, we nally introdu e Y (Z) as the proje tion of

U(X(Z)) on the dy rst eigenfun tions asso iated with the empiri al estima-

tion of the ovarian e of U(X(Z)) based on the M ode evaluations. In the

same manner, we gather in S(N) the proje tion oe ients of ea h measured

displa ement U⋆(θn) on this redu ed basis. To hoose the value of dy, the

normalized error dened by Eq. (26) is on e again onsidered. For this appli-

ation, dy is hosen equal to 23 in order to orre tly represent most of the lo al

os illations of the displa ements. A ording to Figure 7, this orresponds to a

proje tion error that is less than 0.1%.

Following the framework proposed in Se tion 2, the PDF of Z|S(N) is

dedu ed from the kernel estimator of the PDF of (Y (Z),Z). An adaptive

pro edure (with α = 0.99) is moreover introdu ed to better on entrate the

distribution of Z|S(N) on the true value of z⋆. To be more pre ise, 900 new

ode evaluations were added between the two rst iterations, and 620 betweenthe two last iterations, leading to a total budget of 2520 ode evaluations. Therelevan e of this approa h is shown in Figure 8, where the blue ontinuous

lines orrespond to the 95%- redible ellipses asso iated with the distribution

of Z|S(N). After three iterations, the values of z⋆1 , z⋆2 and z⋆3 are indeed iden-

tied with a high pre ision. To emphasize the interest of the partitioning pre-

sented in Se tion 3.2, these results are ompared to the ase where there is no

optimization of the blo k stru ture (the ellipses in red dotted lines). Although

these two approa hes are based on the same information, there is no denying

that sear hing groups of independent omponents of Y (Z)|Z is really helpful.

This is espe ially true for the rst iteration, where 23 groups of independent

omponents were hosen, and for the se ond iteration, where 8 groups of in-

dependent omponents were hosen. For the third iteration, as only 4 groups


0 20 40 60 80 100−5

−4

−3

−2

−1

0

PSfrag repla ements

log10(ε

2)

dy

(a) Error onvergen e

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5x 10−4

PSfrag repla ements

U⋆ 1

s2

Measured data

Proje tion with dy = 10



(b) Redu ed basis representation of U⋆1 (1, s2; θ1)− uhomo(1, s2)

0 0.2 0.4 0.6 0.8 1−5

0

5x 10−5

PSfrag repla ements

U⋆ 2

s1

Measured data




( ) Redu ed basis representation of U⋆2 (s1, 1; θ1)− uhomo(0, 1)

Fig. 7 Inuen e of trun ation parameter dy on the representation of the measured data.


6 8 10

0.05

0.1

0.15

0.2

0.25

0.3

PSfrag repla ements

z1

z2

, blo k stru ture

, no blo k stru ture


Referen e point

(a) (z1, z2), i = 1

6 8 10

0.05

0.1

0.15

0.2

0.25

0.3

PSfrag repla ements

z1

z2

, blo k stru ture



Referen e point

(b) (z1, z2), i = 2

6 8 10

0.05

0.1

0.15

0.2

0.25

0.3

PSfrag repla ements

z1

z2

, blo k stru ture



Referen e point

( ) (z1, z2), i = 3

6 8 10

0.05

0.1

0.15

0.2

0.25

0.3

PSfrag repla ements

z1

z3

, blo k stru ture



Referen e point

(d) (z1, z3), i = 1

6 8 10

0.05

0.1

0.15

0.2

0.25

0.3

PSfrag repla ements

z1

z3

, blo k stru ture



Referen e point

(e) (z1, z3), i = 2

6 8 10

0.05

0.1

0.15

0.2

0.25

0.3

PSfrag repla ements

z1

z3

, blo k stru ture



Referen e point

(f) (z1, z3), i = 3

0.1 0.2 0.3

0.05

0.1

0.15

0.2

0.25

0.3

PSfrag repla ements

z2

z3

, blo k stru ture



Referen e point

(g) (z2, z3), i = 1

0.1 0.2 0.3

0.05

0.1

0.15

0.2

0.25

0.3

PSfrag repla ements

z2

z3

, blo k stru ture



Referen e point

(h) (z2, z3), i = 2

0.1 0.2 0.3

0.05

0.1

0.15

0.2

0.25

0.3

PSfrag repla ements

z2

z3

, blo k stru ture



Referen e point

(i) (z2, z3), i = 3

Fig. 8 Evolution of the 95%- redible ellipses with respe t to the iteration number. Blue

ontinuous line: dy = 23 with optimization of the blo k stru ture. Red dotted line: dy = 23without optimization of the blo k stru ture. Green dashed line: dy = 5 without optimization

of the blo k stru ture.


were hosen, introdu ing the partitioning does not make a big dieren e for

the PDF identi ation, whi h explains the similarities between the blue and

the red urves.

This set of gures also emphasizes the importan e of onsidering a high

value of dy , even if it ompli ates the PDF identi ation. For instan e, hoosing

dy = 5 leads to the results in green dashed lines, whi h are learly less relevant

than the results in blue that orrespond to dy = 23. Intermediate results were

obtained for values of dy between 5 and 23, whereas still in reasing dy did not

really hange the results.

In order to emphasize the e ien y of the proposed method to re over

the true underlying sto hasti ity, three additional bat hes of Q = 104 sim-

ulations are laun hed. These simulations are asso iated with the same ubi

system than in Figure 6, but with dierent boundary onditions (by hang-

ing the boundary onditions, we want to verify that the identied values of

Z are not dependent of a xed onguration). While the boundary ondi-

tions on the inferior side of the ube do not hange, the left and right sides

are now free of onstraints, and the displa ements on the superior side are

hosen equal to 0.002e1 − 0.01e2. We then denote by

X(1,q), 1 ≤ q ≤ Q

,

X(2,q), 1 ≤ q ≤ Q

and

X(3,q), 1 ≤ q ≤ Q

the elasti ity elds hara teriz-

ing the material properties of the dierent ubes of the three sets respe tively.

For all 1 ≤ q ≤ Q,

X(1,q)is an independent realization of the true elasti ity eld, X(z⋆);

X(2,q)is an independent realization of X((zq,prior, z⋆4 , z

⋆5)), where zq,prior

is a realization of Z, whose distribution is given by Eq. (29),

X(3,q)is an independent realization of X((zq,post, z⋆4 , z

⋆5)), where z

q,postis

a realization of Z|S(N) after the three formerly presented iterations.

For ea h simulation, we denote by U(X(i,q)), 1 ≤ i ≤ 3, the on atena-

tion of the verti al and horizontal displa ements that are indu ed on the left

and right sides of the ube. To ompare the statisti al information gathered

in these displa ements, we then ompute, for ea h 1 ≤ i ≤ 3, the eigenval-

ues

v(i)j , j ≥ 0

asso iated with the empiri al estimate of their ovarian e

matri es. In addition, we denote by σVM(X(i,q)) the maximum value over the

ubi domain of the Von Mises stress. This Von Mises riterion is ommonly

used to hara terize the resistan e of the system (see [Lai et al., 2010 for more

details). The de rease of these eigenvalues and the PDF of these Von Mises

riteria are nally ompared in Figure 9. Looking at these gures, we see that

the results asso iated with the posterior distribution of Z are very lose to the

ones asso iated with the true elasti ity eld, whi h is not true for the results

asso iated with the prior distribution of Z. This underlines the apa ity of the

proposed method to take into a ount indire t observations for the identi a-


0 50 100 150 200 250 30010

−4

10−3

10−2

10−1

100

101

PSfrag repla ements

j

v(1)j

v(2)j

v(3)j

(a) Eigenvalues

0 5 10 15 20 25 30 35 400

0.02

0.04

0.06

0.08

0.1

0.12

0.14

PSfrag repla ements

σ

V

M

PDF

σVM(X(1,q))

σVM(X(2,q))

σVM(X(3,q))

(b) Von Mises riterion

Fig. 9 Representation of the eigenvalues de reases (a) and of the PDFs of the Von Mises

riteria (b) for the three ompared ongurations. In ea h gure, the blue ontinuous lines are

asso iated with the referen e ase, the red dashed line orrespond to the results asso iated

with the prior distribution of Z, when the bla k two-dashed lines orrespond to the results

asso iated with the posterior distribution of Z.

tion of the parameters hara terizing the distribution of an unknown random

pro ess of interest.


4 Con lusion

The in reasing of the omputational resour es and the generalization of the

monitoring of me hani al systems have en ouraged many s ienti elds to

take into a ount random elds in their modeling. In that prospe t, this work

proposes an adaptive Bayesian framework to e iently identify the statisti al

properties of these random elds when the available information is a redu ed

set of indire t observations. Two examples based on simulated data are nally

presented to show the potential of this approa h.

Extending this approa h to the ases where the number of parameters to

identify and the number of observations are very high would be interesting for

future work.

Appendix

A.1. Proof of the equality of Eq. 9

Let A,B,D be the blo k de omposition matri es of C−1:

C−1

=

[A B

BT D

]. (30)

Using the S hur omplement, if follows that:

C−1

ZZ = D −BTA−1B,

(CY Y − CY ZC−1

ZZC

T

Y Z)−1 = A,

− CY ZC−1

ZZ= A−1B.

(31)

It omes

((y, z)− (Y (ωm),Z(ωm)))T(h2C)−1 ((y, z)− (Y (ωm),Z(ωm)))

=1

h2

((y − Y (ωm))TA(y − Y (ωm)) + 2(y − Y (ωm))TAA−1B (z −Z(ωm))

+ (z −Z(ωm))TD(z −Z(ωm))

)

=1

h2

(y − Y (ωm) +A−1B (z −Z(ωm)))TA(y − Y (ωm) +A−1B (z −Z(ωm)))

+ (z −Z(ωm))T(D −BTA−1B

)(z −Z(ωm))

= (y − µn(z))TC−1

n (y − µn(z)) +1

h2(z −Z(ωm))

TC−1

ZZ(z −Z(ωm))

(32)

This leads to the sear hed result.


Referen es

Akaike, 1974. Akaike, H. (1974). A new look at the statisti al model identi ation. IEEE

Transa tions on Automati Control, 19 (6):716723.

Arnst et al., 2010. Arnst, M., Ghanem, R., and Soize, C. (2010). Identi ation of bayesian

posteriors for oe ients of haos expansions. Journal of Computational Physi s, 229

(9):31343154.

Atwell and King, 2001. Atwell, J. and King, B. (2001). Proper orthogonal de omposition

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