Adaptive method for indirect identification of the statistical properties of random fields in a Bayesian framework

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statistical properties of random fields in a Bayesian

framework

Guillaume 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�

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Adaptive method for indire t identi ation of the

statisti al properties of random elds in a Bayesian

framework

GuillaumePerrin

·

ChristianSoize

Re eived:date/A epted:date

Abstra t Thiswork onsidersthe hallengingproblemofidentifyingthe

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

aBayesianapproa his introdu ed, whose keystep isthe nonparametri

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

likelihoodfun tionisbasedontheevaluation ofanexpensive omputer ode,

thisworkalsoproposesamethodtosele titerativelynewdesignpointsto

re-du etheun ertaintiesontheresultsthatareduetotheapproximationofthe

likelihood.Twoappli ationsarenallypresentedtoillustratethee ien yof

theproposed pro edure:arstonebasedonanalyti data, andase ond one

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

heteroge-neousmi rostru ture.

Keywords Bayesian framework

·

un ertainty quanti ation

·

statisti al inferen e

·

sto hasti pro ess

·

kerneldensityestimation

1Introdu tion

Randomeldanalysishasbe omeamajortoolinmanys ienti elds,su h

asun ertaintyquanti ation, materials ien e, biology,medi ine, signal

pro- essing,quantitativenan e,et . However,inmostofthese 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

fromtheavailableinformation.

G.Perrin

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

E-mail:guillaume.perrin2 ea.fr

C.Soize

UniversitéParis-Est,MSMEUMR8208CNRS,Marne-la-Vallée,Fran e

(3)

Whenthe information is onstituted of dire t measurementsof the eld

to model, several te hniqueshave been proposed to perform su h an

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

[Whittle, 1951,Whittle,1983,BoxandJenkins,1970℄,allowthedes riptionof

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 basedon a two-step

pro e-dure.Therststepistheapproximationoftherandomeldbyitsproje tion

on a redu ed number of deterministi fun tions [GhanemandSpanos,2003,

LeMaîtreand Knio,2010℄,usingforinstan etheproperorthogonal

de ompo-sition[AtwellandKing,2001℄,thepropergeneralizedde omposition[Nouy, 2010℄,

ortheKarhunen-Loèveexpansion[Williams,2011,Perrinet al.,2014,Perrinetal.,2013℄.

These ondstepistheidenti ationofgeneralsto hasti representationsofthe

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

Perrinetal.,2012,NouyandSoize, 2014,SoizeandGhanem,2016,Perrinet 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 therandom eld to identifybelongs to aknown parametri lass.

Thus, identifying the distributionof

X

amountsto identifying the valuesof these parameters,whi haregatheredin theve tor

z

.A Bayesianframework isthen onsidered [MarzoukandNajm,2009,Stuart,2010,Arnstetal.,2010,

Matthiesetal.,2016,Emeryetal.,2016℄:parameter

z

issupposed tobe ran-dom,andwesear hitsposteriordistributiongiventheavailabledata.

MarkovChainMonteCarlo(MCMC)[RubinsteinandKroese,2008,Tianet al.,2016℄

isgenerally onsideredasapowerfultooltoexploretheposteriordistribution

for these parameters. However it an be omputationally prohibitive when

ea hposteriorevaluationrequires evaluationsofa omputationallyexpensive

ode,asitthe asehere.To ir umventthisproblem,astandardapproa histo

repla ethe odebyasurrogatemodel,andtodire tlysamplefromthe

approxi-matedposteriordistributionasso iatedwiththemodiedlikelihoodusing

las-si alMCMCpro edures.Thesurrogatemodel anbebasedonpolynomial

rep-resentations[MarzoukandNajm,2009,MarzoukandXiu,2009,WanandZabaras,2011,

LiandMarzouk,2014,Tsilisetal.,2017℄,Gaussianpro essregression[KennedyandO'Hagan,2001,

Santneret al.,2003,Higdonetal.,2008,BilionisandZabaras,2015,Sinsbe kandNowak,2017,

Damblinetal.,2013℄,orrunsofthe odeatdierentresolutionlevels[Higdonetal.,2003,

ChenandS hwab,2015℄. Alternatively, the surrogate model an be used to

adapt theproposal distribution. Inthat ase, the numberof expensive

pos-terior evaluations perMCMC step an be strongly redu ed, while sampling

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

Fieldingetal.,2011,Conradetal.,2016,Conradetal.,2018℄forfurtherdetails

aboutthisapproa h).

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

(4)

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, wefo us ontheapproximationof the probabilitydensity

fun tion(PDF)of

g(X(z))

.

Torun a MCMCpro edure basedon theasso iated approximated

likeli-hood in areasonable omputational time, this approximationof thePDF of

g(X(z))

inany

z

hastobe onstru tedfromaxednumberofalready

om-puted odeevaluations.Tothisend, werst proposeto dire tlyworkonthe

jointPDFof

(g(X(z)), z)

.Then,wefo usontheGaussiankerneldensity esti-mation(G-KDE)[WandandJones,1995,S ottandSain,2004,Perrinet al.,2018℄

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

forits abilityto model non-Gaussiandistributions with omplexdependen 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- hosengroups of omponents that an

reason-ablybe onsideredasindependent.Se ondly,asequentialstrategyisproposed

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 ev

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

high. We refer to [M Kayetal.,1979,Fangand Lin,2003,Fang etal.,2006,

Dragulji¢etal.,2012,Josephet al.,2015℄ for the onstru tion of the initial

spa e-llingdesignswhentheinputspa esisanhyperre tangle,andto[Stinstraet al.,2003,

Stinstraetal.,2010,Aurayet al.,2012,Dragulji¢et al.,2012,LekivetzandJones,2015,

MakandJoseph,2016,Perrinand Cannamela,2017℄forthegeneral 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

theidenti ation of theme hani alproperties of anunknown heterogeneous

medium.

2Indire tidenti ation ofthe statisti al properties ofrandom

elds

Theobje tiveofthisse tionistodes ribetheadaptivepro edurewepropose

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

availableinformationis asetofindire tobservations.

2.1Denitions andnotations

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P(X, R

d

x

₎

denotesthespa eofallthese ond-orderrandomelds dened

on

(Ω, A, P)

,withvaluesin

R

d

x

,indexedbya ompa tand onne tedspa e

X

;

L

2 _{(X, R}

d

x

₎

isthespa eofallthesquare-integrablefun tionsdened on

X

withvaluesin

R

d

x

;

g

is a nonlinear measurable mapping whose omputational ost an be

high:

g

:

(

L

2 (X, R

d

x

_{) → R}

d

y

h

7→ g(h)

;

(1)

X (R

d

z

_{, R}

d

x

₎

x

₎

an orrespondto the set ofGaussian random

elds,whosemeanand ovarian efun tionsareparameterizedbythesame

d

z

oe ients. Forall

z

in

R

d

z

,

X

(z)

isanelementof

X (R

d

z

X

⋆

, whi h we denote by

X

n

)}

1≤n≤N

, θ

n

∈ Ω,

thepurposeof this work is toproposea Bayesianformalismfor the

identi- ationof

z

⋆

,su hthat theprobabilitydistribution of

X(z

⋆

₎

isthe losestto theoneof

X

⋆

. Remarks

As mentionedin Introdu tion,itis importanttonoti e thatforea h

z

∈

R

d

z

,

g(X(z))

is random. This stronglylimits thepossibilityof repla ing

mapping

z

7→ g(X(z))

byasurrogatemodel,asitis lassi allydonewhen solvinginverseproblemsthat invoke omputationallyexpensivemodels.

Inthefollowing,forthesakeofsimpli ity,weassumethat

X

⋆

∈ X (R

d

z

_{, R}

d

x

₎

.

Ifitwasnotthe ase,it ouldbene essarytointrodu eanerrortermto

modelthedieren ebetween

X

⋆

and

,whi hissupposedtobeknownasapriormodel.Hen e, identify-ing

z

⋆

amountstosear hingtheposteriorPDFof

Z

| S(N )

,whi hwedenote by

f

Z

|S(N )

.UsingtheBayestheorem,it omes:

f

Z

_{|S(N )}

(z) =

L

S(N )

(z)f

Z

(z)

E

L

S(N )

(Z)

, z ∈ R

d

z

_.

(2)

There,

E [·]

is the mathemati al expe tation and

L

S(N )

is the likelihood fun tion.Theelementsof

S(N )

beingstatisti allyindependent,itfollows:

L

S(N )

(z) =

N

Y

n=1

f

Y

_(z)

(Y

⋆

(θ

n

)), z ∈ R

d

z

,

(3)

inwhi h

f

Y

(z)

isthePDFof

Y

(z) := g(X(z))

forgiven

z

in

R

d

z

,andis

un-known.Toapproximate

f

(z)

in anypoint

y

in

R

d

y

anbeapproximatedusinganyparametri or

nonparamet-ri statisti allearningte hnique.However,this meansthatfun tion

g

hasto beevaluated

M

× Q

timesto evaluatefun tion

L

S(N )

in

Q

pointsfor

z

.This qui klybe omesburdensomewhenthe omputational ostforea hevaluation

of

g

isrelativelyhigh(betweenseveralminutestoseveralhoursCPUforthe onsideredappli ations).Onepossibleapproa hto ir umventthisproblemis

to dire tly approximate the jointPDF of the

(d

y

+ d

z

)

-dimensional random ve tor

(Y (Z), Z)

[SoizeandGhanem,2017℄.Indeed,

)

,where

ω

1 , . . . , ω

M

arein

Ω

;

forea hvalueof

z

in

{Z(ω

1 ), . . . , Z(ω

M

)}

,wedraw,atrandomand inde-pendentlytheones fromtheothers, aparti ularrealizationof

X(z)

,and wededu earealizationof

Y

(z)

byevaluating

g

inthisrealizationof

X(z)

.

Forthesakeofsimpli ity,wedenotetheserealizationsby

Y

(ω

m

)

,

1 ≤ m ≤

M

.Basedontheserealizations,thekernelestimatorof

f

Y

,Z

is:

b

f

Y

,Z

(y, z; H) :=

det

(H)

−1/2

M

X

m=1

K

H

−1/2

((y, z) − (Y (ω

m

), Z(ω

m

)))

.

(4)

Here, det

(·)

is the determinant operator,

K

is any positive fun tion whose integralover

R

d

y

+d

z

is one, and

H

is a

((d

y

+ d

z

) × (d

y

+ d

z

))

-dimensional

positive-denitesymmetri matrix, whi h is generallyreferredasthe

"band-widthmatrix".Inthefollowing,wefo usonthe asewhere

K

istheGaussian multidimensionaldensity, andwhere

H

is proportionaltothe empiri al esti-mationofthe ovarian ematrixof

(Y (Z), Z)

,denoted by

C

b

:

(7)

Themain interestofthis hypothesis omes from thefa t that itstrongly

redu esthenumberofparametersthatneedtobeidentiedforthe

onstru -tionof

H

,whilegenerallyleadingtoveryinterestingresultsforthemodeling of multivariatePDFs(see [Perrinetal.,2018℄for moredetails). Other

parsi-moniousparameterizations ouldbeproposed for

H

,su hasdiagonal repre-sentations,butforsu ientlyhighvaluesof

M

,theinuen eofthis hoi eon theidenti ationresultsisexpe tedtobesmall.

Hen e,thePDFof

(Y (Z), Z)

isapproximatedbyamixtureof

M

Gaussian PDFs,forwhi hthemeansaretheavailablerealizationsof

(Y (Z), Z)

andthe ovarian ematri esareallparameterizedbyauniques alar

h

:

b

f

Y

,Z

(y, z; h) =

1 M

M

X

m=1

φ

(y, z); (Y (ω

m

), Z(ω

m

)), h

2 C

b

.

(6)

There,forany

R

d

-dimensionalve tor

µ

andforany

(R

d

_{× R}

d

₎

-dimensional

symmetri positive-denitematrix

C

,

φ(·; µ, C)

isthePDFofany

R

d

-dimensional

Gaussianrandomve torwithmean

µ

and ovarian ematrix

C

:

φ

(x; µ, C) :=

exp

−

1

2 (x − µ)

T

C

−1

(x − µ)

(2π)

d/2

p

det

(C)

,

x

∈ R

d

.

(7)

Inaddition,theblo kde ompositionof

C

b

iswrittenas:

b

C

=

"

b

C

Y Y

C

b

Y Z

b

C

T

_{Y Z}

C

b

ZZ

#

.

(8) Forall

(y, z) ∈ R

d

y

_{× R}

d

z

,thekernelapproximationof

f

Y

(z)

(y)

,whi hwe denote by

f

b

Y

(z)

(y; h)

, antherefore bewritten asfollows(see Appendix for moredetails aboutthisexpression):

b

f

Y

(z)

(y; h) =

b

f

Y

_,Z

(y, z; h)

R

dy

f

b

Y

,Z

(v, z; h)dv

=

M

X

m=1

γ

m

(z; h)

P

M

m

′

=1

γ

m

′

(z; h)

φ

(y; µ

m

(z), C

m

(h)) ,

(9)

γ

m

(z; h) := exp

−

1 2h

2 (z − Z(ω

m

))

T

_b

C

−1

ZZ

(z − Z(ω

m

))

,

(10)

µ

_m

(z) := Y (ω

m

) + b

C

Y Z

C

b

−1

ZZ

(z − Z(ω

m

)),

(11)

C

m

(h) := h

2 b

C

Y Y

− b

C

Y Z

C

b

−1

ZZ

C

b

T

Y Z

.

(12)

ItfollowsthattheposteriorPDFof

Z

isestimated forea h

z

in

R

d

z

(8)

f

Z

|S(N )

(z) ≈

b

L

S(N )

(z; h)f

Z

(z)

E

h

L

b

S(N )

(Z)

i ,

L

b

S(N )

(z; h) :=

N

Y

n=1

b

f

Y

(z)

(Y

⋆

(θ

n

); h).

(13) Remarks

Onekeystepofthesemethodsistheexplorationofthewholespa eofthe

inputvariables.Tomaximizethis overing,it isgenerallyworth hoosing

{Z(ω

1 ), . . . , Z(ω

M

)}

asaspa ellingdesignofexperimentsthatpreserves

goodproje tionpropertiesforea h s alarinput(see[FangandLin,2003,

Fanget al.,2006,PerrinandCannamela,2017℄forthe onstru tionofsu h

designswhenpriordensity

f

Z

isuniformornot).

Another ru ialaspe toftheseBayesianapproa hesisthe hoi eofprior

distribution

f

Z

.Indeed,themoreinformativeitis,thelessmeasurements weneedtogetausefulposteriordistributionfor

Z

.Butifitis over on-dentaroundvaluesthat arepotentiallybiased,theun ertainty arriedby

theposteriordistributionmaynotbelargeenoughtoadequately apture

islikelytobehigher,aswewantthe odeevaluationstoallowapre iseapproximationofthedependen e

stru -turebetween

Y

(Z)

and

Z

inthe onstru tionoftheirjointPDF.Andthe higher

d

y

+ d

z

is,thehighervalueof

M

wemayneed.However,when on-frontedto expensivesimulators,the maximalnumberof odeevaluations

isgenerallylimited(

M

mustbelessthan

1000

forinstan e).Inthat ase, itisparti ularlyimportanttoworkonmethodsthatallowthemostpre ise

identi ationoftheparametersattheminimal ost.Thisistheobje tiveof

thefollowingse tions.InSe tion2.3,werstproposetode ompose

Y

(Z)

inseveralgroupstoimprovetherelevan eofthenonparametri

represen-tation of PDF

f

Y

,Z

for a xed value of

M

. Then, sele tion riteria are proposedinSe tion2.5tosequentially on entratethe odeevaluationsin

themostlikelyregionsfor

Z

,andthereforeredu etheun ertaintiesonits posteriorPDF

f

Z

|S(N )

.

2.3Optimalpartitioning

As it is explained in [Perrinet al.,2018℄, when

d

y

be omeshigh, separating indierentgroupsthe omponentsof

Y

(Z)|Z

that ouldreasonablybe

(9)

number of ode evaluations. Let

b

= (b

1 , . . . , b

d

y

)

be aparti ular group de- omposition of

Y

(Z)|Z

in thesensethat:

if

b

i

= b

j

,

Y

i

(Z)|Z

and

Y

j

(Z)|Z

aresupposedto bedependentand

there-forebelong tothesameblo k,

if

b

i

6= b

j

,

Y

i

(Z)|Z

and

Y

j

(Z)|Z

aresupposedtobeindependentandthey

anbelongtotwodierentblo ks.

Toavoidredundan ies inthis blo k by blo krepresentation,ve tor

b

an be hosenin theset:

B(d

y

) :=

b

∈ {1, . . . , d

y

}

d

y

| b

1 = 1, 1 ≤ b

j

≤ 1 + max

1≤i≤j−1

b

i

,

2 ≤ j ≤ d

y

.

(14)

Hen e,forany

b

in

B(d

y

)

,we andene Max

(b)

asthemaximalvalueof

b

,

Y

(ℓ)

_{(z, b)}

astherandomve torthatgathersallthe omponentsof

Y

(Z)|Z =

z

withablo kindex equalto

ℓ

,

y

(ℓ)

_{(y, b)}

as theve torthat gathersallthe omponentsof

y

withablo k index equalto

ℓ

. For all

b

in

B(d

y

)

,

z

in

R

d

z

and

h

:= (h

1 , . . . , h

Max

(b)

)

in

R

Max

(b)

, if

b

f

leadingto anotherapproximationof

f

Z

|S(N )

(z)

forea h

z

in

R

d

z

:

f

Z

|S(N )

(z) ≈ e

f

Z

|S(N )

(z) :=

e

L

S(N )

(z; h, b)f

Z

(z)

E

h

e

L

S(N )

(Z)

i

,

(16)

e

L

S(N )

(z; h, b) :=

N

Y

n=1

e

f

Y

(z)

(Y

⋆

(θ

n

); h, b).

(17)

2.4Estimationofthekernelparameters

Toevaluate

L

e

S(N )

, the valuesof

h

and

b

haveto beidentied. This anbe donebysolvingthefollowingoptimization problem:

(h

AIC

, b

AIC

) ≈ arg

min

h

∈]0,+∞[

Max

(b)

, b∈B(d

y

)

AIC LOO

(h, b),

(18)

(10)

AIC LOO

(h, b) := 2

Max

(b)−2 log





M

Y

m=1

Max

(b)

Y

ℓ=1

b

f

_Y

(−m)

(ℓ)

_(Z(ω

m

),b)

(Y

(ℓ)

_{(Y (ω}

m

), b); h

ℓ

)



 ,

(19) where

f

b

(−m)

Y

(ℓ)

_(Z(ω

m

),b)

isthekernelestimatorofthePDFof

Y

(ℓ)

_(Z(ω

m

), b)

that

isbasedonalltheevaluationsof

g

butthe

m

th

one.Indeed,givenEq.(9),this

amountstominimizing a"Leave-One-Out"versionoftheAkaikeinformation

riterion(AIC)[Akaike,1974℄asso iatedwiththePDFof

Y

(Z)|Z

(very lose results would beobtained by onsidering another information riterionsu h

astheBayesianinformation riterion(BIC)). Wereferto[Perrinetal.,2018℄

formoredetails aboutthesolvingofthis optimizationproblem.

2.5Adaptivestrategy

By onstru tion, the pre isionof the estimation of

z

⋆

depends on the

num-berof experimental measurements,

N

, and the number of ode evaluations,

M

. Classi ally, thevalue of

N

is xed, whereas it should bepossibleto im-prove the a ura y of

f

e

Y

(z)

, whi h is dened by Eq. (15), by adding new odeevaluations inthelearningset.For instan e,

M

new

newpoints ouldbe

addedtothelearningsetbyevaluatingthe odein

M

new

independent

realiza-tionsof

Z|S(N )

(weremindthat no odeevaluationsare requiredto hoose these new points). However,as the kernel density estimator is based onthe

post-pro essingof independentand identi allydistributed realizations ofthe

randomve torto model, non onsistentresults ould beobtainedby mixing

realizationsof

Z|S(N )

withrealizationsof

Z

.Ifsu hasele tion riterionwas hosen,thiswouldmeanthatthe

M

odeevaluationsattheinitialstepshould notbeusedfortherening.

Asanalternative,weproposeto evaluatethefun tion

z

7→ e

f

(z) := e

L

S(N )

(z; h

AIC

, b

AIC

)f

Z

(z)

in ea h value of

{Z(ω

1 ), . . . , Z(ω

M

)}

. For ea h

1 ≤ m ≤ M

, let

π

m

be the followingweights:

0 ≤ π

m

:=

e

f

(Z(ω

m

))

P

M

m

′

₌₁

f

e

(Z(ω

m

′

))

≤ 1.

(20)

Withoutlossof generality, theseweightsare assumed to besorted in

de- reasingorder,

π

1 ≥ π

2 ≥ . . . ≥ π

M

.Hen e,for

0 < α < 1

,ifwedenoteby

Q

α

thesmallestintegersu hthat:

Q

X

α

−1

m=1

(11)

thedomain

Z

α

:= {z ∈ R

d

z

_{| e}

_f

_{(z) ≥ e}

_f

_(Z(ω

Q

α

)}

anbeseenasa onservative

α

- redible set for

Z|S(N )

, in the sense that the probability for

Z|S(N )

to bein

Z

α

islikely tobehigherthan

α

. Therefore, addingnewrealizations of

Z|Z ∈ Z

α

seemsagoodmeantoenri hthesetofpointsonwhi hthekernel

densityestimator isbased. Indeed,the mostlikelyvaluesof

z

at theformer steparekeptintheadaptivepro edure,whileagoodexplorationoftheinput

⋆

.

isas lose tothetrue(butunknown)posterior

f

Z

|S(N )

aspossible.Choosingavalue of

α

thatisstri tlyinferiortooneonlyaimsatlimitingthenumberofnew odeevaluationsthat willbein theregionwheretrueposterior

f

Z

|S(N )

is almostzero.However,thisvaluehasnottobe hosentoosmall,asitwould

arti iallyredu etheun ertaintyasso iatedwiththeestimationof

z

⋆

by

uttingtoomu hthetailsofthetrueposterior.Hen e,in theappli ations

thatwillbepresentedin Se tion3,

α

is hosenequalto

0.99

.

A ording to Eq. (21), we deliberatelyadd oneto the valueof

Q

α

to be onservativefor the estimation of the

α

- redible set. This is parti ularly important for aseswhen after therst iteration,

π

1 ≈ 1

.Indeed, evenif onevalueof

z

appearsto be mu h morerelevant thanthe others, wedo notwantto fo ustoomu haroundasinglemode.

3Appli ations

Thepurpose ofthisse tion isto illustratethe methodproposedin Se tion 2

ontwoappli ations.

3.1Analyti alappli ation

In this rst appli ation,

X

(z)

refers to the Gaussian random elds whose meanisequalto

t

7→ sin(2πz

3 t

+ z

4 )

,and whose ovarian efun tionis equal

to

(t, t

′

_{) 7→ z}

2

1 exp

−

(t−t

_2z

2 ′

)

2

.

(z)

bythefollowingnonlinearmapping:

(12)

0

0.2

0.4

0.6

0.8

1

0

0.5

1

1.5

PSfragrepla ements

t

U(t, ω

1 )

U(t, ω

2 )

U(t, ω

3 )

(a)Realizationsof

U(X(Z))

0

0.2

0.4

0.6

0.8

1

0

0.5

1

1.5

PSfragrepla ements

t

U

⋆

_{(t, θ}

1 )

U

⋆

_{(t, θ}

2 )

U

⋆

_{(t, θ}

3 )

(b)Realizationsof

U(X(Z))|Z = z

⋆

Fig.1 Comparisonofindependentrealizations

U(X(Z))

and

U(X(Z))|Z = z

⋆

.

U

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

(22)

Thevalueof

z

⋆

is hosenequalto

(0.3, 0.2, 2, 1)

,anditisapriori modeled byauniformlydistributedover

[0.1, 1] × [0.05, 1] × [1, 3] × [0, 2]

randomve tor, denoted by

Z

. Toidentify

z

⋆

, theavailable information is made of

N

= 10

U

(X(Z))|Z = z

⋆

,denotedby

U

⋆

_(θ

1 ), . . . , U

⋆

(θ

N

)

. Tosolvetheinferen e problem,

M

= 500

Z

have beendrawn, whi hwewrite

{Z(ω

1 ), . . . , Z(ω

⋆

are omparedinFigure 1.

Inprin iple,theBayesianformulation anbeappliedtoanymulti-variate

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

itis possible)the statisti al ontentofthe ode outputin alow-dimensional

ve tor[Perrin,ress℄.Inour ontext,itisevenmoreimportant,asakeystepof

theproposedmethodistheidenti ationofthejointdistributionbetweenthe

parameterstobeidentiedandtheasso iated odeoutput,whose omplexity

stronglyin reaseswiththedimensionofthe odeoutput.Inthatprospe t,we

introdu e

ψ

p

,

p

≥ 1

asthesolutionsofthefollowingeigenvalueproblem:

Z

1

0 M

X

m=1

U

(t, ω

m

)U (t

′

, ω

m

)ψ

p

(t

′

)dt

′

= λ

p

ψ

p

(t),

(23)

(13)

λ

1 ≥ λ

2 ≥ · · · → 0,

Z

1

0 ψ

p

(t

′

)ψ

q

(t

′

)dt

′

= δ

pq

,

(24)

where

δ

pq

istheKrone kersymbolthatisequalto1ifp=qand0otherwise.To solvetheinferen eproblem,wenallyintrodu e

Y

(z)

astheve torgathering theproje tion oe ientsof

U(X(Z))

ontheformereigenfun tionsasso iated withthe

d

y

highesteigenvalues:

Y

(Z) :=

Z

1

0 U

(t; X(Z))ψ

1 (t)dt, . . . ,

Z

1

0 U

(t; X(Z))ψ

d

y

(t)dt

.

(25)

Thevalueof

d

y

anthenbe hosentoguaranteearelevantrepresentation oftheobservations.Tothisend, weintrodu e

ε

2

asthefollowingquantity:

ε

2 (d

y

) :=

P

N

n=1

R

1

0 U

⋆

_{(t, θ}

n

) − b

U

⋆

(t, θ

n

; d

y

)

2 dt

P

N

n=1

R

1

0 (U

⋆

(t, θ

n

))

2 dt

,

(26)

b

U

⋆

(t, θ

n

; d

y

) :=

d

y

X

p=1

ψ

p

(t)

Z

1

0 U

⋆

(t

′

_{, θ}

n

)ψ

p

(t

′

)dt

′

.

(27)

that islessthan

1%

.

Basedonthese

M

realizationsof

(Y (Z), Z)

,andonthese

N

realizationsof

Y

(z

⋆

_{) := Y (Z)|Z = z}

⋆

,theadaptiveBayesianformalismpresentedinSe tion

2isnowapplied. Forthisappli ation,theparameter

α

,whi hwasintrodu ed in Se tion 2.5, is hosen equal to

0.99

. At ea h iteration, new samples are thereforeaddedintheregionwhere

f

Z

|S(N )

isnottoosmallusingareje tion approa h untilwegetatotalof

M

points(in ludingthepoints omputedat theformeriterations)in the

α

- redibleset

Z

α

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

equalto

2300

,whi hmeans that around450newpoints havebeenadded at ea h iteration. The results are summarized in Table 3.1 and Figures 3 and

4. As arst omment, we verify that the identi ation of

z

⋆

after only one

iterationisnotverypre ise, inthesense thatthepredi tionun ertaintiesare

veryhigh. This is notsurprising, as weare trying to approximate the PDF

of a16-dimensionalrandomve tor(

d

y

= 12

,

d

z

= 4

)onits whole denition

domain from only 500 realizations. Moving from

M

= 500

to

M

= 2300

,

that istosayspending thetotalbudgetat therstiteration,doesnotreally

help. Indeed, the results we get in terms of mean and varian e of

Z|S(N )

are approximativelythe same.This is explained bythe fa t that even ifthe

(14)

0

10

20

30

40

50

60

70

80 −12

−11

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

PSfragrepla ements

lo

g

1

0 (ε

2 )

(

%

)

d

y

(a)Error onvergen e

0

0.5

1 −0.5

0

0.5

1

PSfragrepla ements

t

U

⋆

_{(t, θ}

1 )

b

U

⋆

_{(t, θ}

1 ; d

y

= 15)

b

U

⋆

_{(t, θ}

1 ; d

y

= 5)

b

U

⋆

_{(t, θ}

1 ; d

y

= 10)

(b)Redu edbasisrepresentation

Fig.2 Evolutionoftheproje tionerrorwithrespe tto

d

domainstaysverysparse.Onthe ontrary,addingiterativelyaround

450

new odeevaluationsinthemostlikelyregion,whosevolumeismu hsmallerthan

theinitialvolume,allows

E [Z|S(N )]

totendto

z

⋆

, andstronglyredu esthe

redibleintervals.This onvergen e is qui kerforthe meanparametersthan

forthe ovarian eparameters,whi hwasalsoexpe ted,asthemeanfun tion

isgenerallyeasiertoidentify thanthe ovarian e.Fo using onFigure 4, itis

also interestingtonoti e that thereferen e valuedoesnotneed tobein the

99

%

- redibleellipseasso iatedwith

Z|S(N )

attherstiterationtobeinthe 99

%

- redibleellipsesatthenextiterations.

(15)

1

2

3

4

5

0

1

2

3

PSfragrepla ements V alues of

Z

1 ,

Z

2 ,

Z

3 ,

Z

Z

4 |S(N )

z

⋆

1 z

⋆

2 z

⋆

3 z

⋆

4

Fig.3 Evolutionofthe

99%

- redibleintervalsandofthe meanvaluesofthe omponents of

Z|S(N )

withrespe ttotheiterationnumber.

0

0.2

0.4

0.6

0.8

0

0.2

0.4

0.6

0.8

1

PSfragrepla ements

z

2 z

1 i

= 1

i

= 2

i

= 3

i

= 4

i

= 5

Referen epoint

Fig. 4 Evolution of the

99%

- redible ellipses of the two rst elements of

Z|S(N )

with respe ttotheiterationnumber.

3.2Appli ationtotheidenti ationoftheme hani alpropertiesofan

unknownanisotropi material

These ondappli ationdealswiththeidenti ationoftheme hani al

proper-tiesofanheterogeneousmi ro-stru ture,whi hismodeledbyarandomelasti

medium. Tothis end, severalexperimental testsare performed onaseriesof

spe imensmadeof thesamematerial.Tobe oherentwiththe notations

(16)

Champ nu

> 2.87E−01

< 3.15E−01

0.29

0.30

0.31

0.32

(a)EvolutionofthePoissonratio

Champ young

SCAL

> 2.04E+02

< 2.17E+02

2.05E+02

2.06E+02

2.07E+02

2.08E+02

2.09E+02

2.10E+02

2.11E+02

2.12E+02

2.13E+02

2.14E+02

2.15E+02

2.16E+02

2.17E+02

(b)EvolutionoftheYoungmodulus

Fig.5 OnepotentialspatialvariationoftheYoungmodulusandthePoissonratio

asso i-atedwiththesto hasti model

X(z

⋆

₎

.

(17)

me hani alpropertiesof thematerial that onstitutesthespe imens.Several

sto hasti modelshavebeenproposedin theframeworkoftheheterogeneous

anisotropi linearelasti ity[Soize,2006,Soize, 2008,Clouteauetal.,2013,GuilleminotandSoize,2013℄.

Itshouldbenotedthattheelasti ityeldisnotareal-valuedrandomeld,but

atensor-valued randomeld, andthat thedierent omponents of this

ran-domeld annotbeidentiedseparatelyduetoalgebrai onstraints.Forthis

appli ation,thesto hasti modelfortheelasti ityeldisbasedonthemodel

proposedin[Soize,2006℄and[Guilleminot andSoize,2013℄in2Dplanstresses

forthe sakeofsimpli ity. Hen e,thedistribution of

X

isnon-Gaussian,and it is parameterizedby a5-dimensionaldeterministi ve tor

z

= (z

1 , . . . , z

5 )

, where:

z

1

isapositivedispersion oe ientthat ontrolsthelevelofu tuations,

z

2 , z

3

aretwospatial orrelationlengths,

z

4

isthemeanvalueoftheYoungModulus(

×10

9

Pa);

z

5

isthemeanvalueofthePoissonratio.

Wethen assume that

N

= 100

ubi spe imens are available,whose re-spe tiveme hani alpropertiesare hara terizedbyoneparti ularrealization

of

X(z

⋆

₎

, with

z

⋆

_{= (2000, 0.1, 0.15, 210, 0.3)}

. As an illustration, Figure 5

shows, foroneparti ularspe imen, theevolutionof theYoungmodulusand

thePoissonratioin ea hpointof

[0, 1]

2

.Thesamepressureeld

f

S

= −f

S

e

2

isthen imposedonthetopofea hspe imen,andweonlyhavea ess tothe

indu eddispla ement eld onthe boundariesof these spe imens(see Figure

6foranillustrationoftheexperimentalproto ol).Let

U

⋆

_(θ

1 ), . . . , U

⋆

(θ

N

)

be

thesemeasureddispla ements.

hara -terized by its young modulus

E

and its Poisson ratio

ν

, it is well known [Laiet al.,2010℄thattheindu eddispla ementinea hpoint

s

∈ [0, 1]

2

would beequalto

u

homo

(s) = (as

1 , bs

2 )

,with

a

b

=

λ

+ 2µ

λ

+ 2µ

λ

−1

0 −f

S

, µ

=

E

2(1 + ν)

, λ

=

2µν

(1 − 2ν)

.

(28)

Hen e,asweare onsidering a lass of stationary random pro esses,the

valuesof

z

⋆

4

and

z

⋆

5

anbe identiedastheargumentsthat minimizetheL2 distan ebetweenthe

N

measureddispla ementsandtheasso iated

(18)

homoge-nolonger arriedoutindimension5,butindimension3.Thisstronglyredu es

thenumberof odeevaluationsthatwillbeneededfora orre tidenti ation

of

(z

⋆

1 , z

2 ⋆

, z

3 ⋆

)

.

Thus, in the following, only

z

⋆

1 , z

⋆

2

and

z

⋆

3

are modeled byrandom

quan-tities. They are gathered in the ve tor

Z

, whose omponents are assumed independentanddistributeda ordingto thefollowingdistributions:

log(Z

1 ) ∼ U(4.6, 11.5), Z

2 ∼ U(0.01, 0.3), Z

3 ∼ U(0.01, 0.3),

(29)

where forall

a < b

,

U(a, b)

istheuniform distributionover

[a, b]

.Foragiven valueof

Z

, it is possible to simulate independent realizations of

X

(Z)

, and toapproximate(usingtheFiniteElementMethod)thedispla ementsindu ed

bytheexperimentalfor eeld,whi hwewrite

U

(X(Z))

.Thus, forthis se -ondappli ation,werst hoseatrandom

M

= 1000

valuesof

Z

a ordingto itspriordistribution.Forea h ofthese values,aparti ularrealization ofthe

elasti ity tensorwas thengenerated over

[0, 1]

2

,and theme hani alproblem

that orrespondstotheexperimentalproto olwassolved(usingthesoftware

Cast3M) to getthe displa ementsat the boundary ofthe ube. Inthe same

manner than in Se tion 3.1, we nally introdu e

Y

(Z)

as the proje tionof

U

(X(Z))

onthe

d

y

rsteigenfun tionsasso iatedwiththeempiri al

estima-tionof the ovarian eof

U(X(Z))

basedon the

M

ode evaluations. Inthe samemanner,wegatherin

S(N )

theproje tion oe ientsofea hmeasured

displa ement

U

⋆

_(θ

n

)

on this redu ed basis. To hoose the value of

d

y

, the normalizederrordened byEq.(26)ison eagain onsidered.Forthis

appli- ation,

d

y

is hosenequalto23inorderto orre tlyrepresentmostofthelo al os illationsofthedispla ements.A ordingtoFigure7,this orrespondstoa

proje tionerrorthatislessthan

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 moreoverintrodu ed to better on entratethe distributionof

Z|S(N )

onthetruevalueof

z

⋆

.Tobemorepre ise,

900

new odeevaluationswereaddedbetweenthetworstiterations,and

620

between thetwolastiterations,leadingtoatotalbudgetof

2520

odeevaluations.The relevan e of this approa h is shown in Figure 8, where the blue ontinuous

lines orrespondto the

95%

- redibleellipsesasso iated withthe distribution

of

Z|S(N )

.Afterthreeiterations,thevaluesof

z

⋆

1 , z

2 ⋆

and

z

⋆

3

areindeed iden-tiedwithahighpre ision.Toemphasizetheinterestofthepartitioning

pre-sentedinSe tion3.2,theseresultsare omparedtothe asewherethereisno

optimizationoftheblo kstru ture(theellipsesinreddottedlines).Although

thesetwoapproa hesarebasedonthesameinformation,there isnodenying

thatsear hinggroupsofindependent omponentsof

Y

(Z)|Z

isreallyhelpful. This isespe ially truefortherstiteration,where 23groupsofindependent

(19)

in-0

20

40

60

80

100 −5

−4

−3

−2

−1

0

PSfragrepla ements

lo

g

1

0 (ε

2 )

d

y

(a)Error onvergen e

0

0.2

0.4

0.6

0.8

1 −1.5

−1

−0.5

0

0.5

1

1.5 x 10

−4

PSfragrepla ements

U

⋆

1 s

2

Measureddata Proje tionwith

d

y

= 10

Proje tionwith

d

y

= 20

Proje tionwith

d

y

= 30

(b)Redu edbasisrepresentationof

U

⋆

1 (1, s

2 ; θ

1 ) − u

homo

(1, s

2 )

0

0.2

0.4

0.6

0.8

1 −5

0

5 x 10

−5

PSfragrepla ements

U

⋆

2 s

1

Measureddata Proje tionwith

d

y

= 10

Proje tionwith

d

y

= 20

Proje tionwith

d

y

= 30

( )Redu edbasisrepresentationof

U

⋆

2 (s

1 ,

1; θ

1 ) − u

homo

(0, 1)

(20)

6

8

10

0.05

0.1

0.15

0.2

0.25

0.3

PSfragrepla ements

z

1 z

2

,blo kstru ture ,noblo kstru ture ,noblo kstru ture Referen epoint (a)

(z

1 , z

2 )

,

i

= 1

6

8

10

0.05

0.1

0.15

0.2

0.25

0.3

PSfragrepla ements

z

1 z

2

,blo kstru ture ,noblo kstru ture ,noblo kstru ture Referen epoint (b)

(z

1 , z

2 )

,

i

= 2

6

8

10

0.05

0.1

0.15

0.2

0.25

0.3

PSfragrepla ements

z

1 z

2

,blo kstru ture ,noblo kstru ture ,noblo kstru ture Referen epoint ( )

(z

1 , z

2 )

,

i

= 3

6

8

10

0.05

0.1

0.15

0.2

0.25

0.3

PSfragrepla ements

z

1 z

3

,blo kstru ture ,noblo kstru ture ,noblo kstru ture Referen epoint (d)

(z

1 , z

3 )

,

i

= 1

6

8

10

0.05

0.1

0.15

0.2

0.25

0.3

PSfragrepla ements

z

1 z

3

,blo kstru ture ,noblo kstru ture ,noblo kstru ture Referen epoint (e)

(z

1 , z

3 )

,

i

= 2

6

8

10

0.05

0.1

0.15

0.2

0.25

0.3

PSfragrepla ements

z

1 z

3

,blo kstru ture ,noblo kstru ture ,noblo kstru ture Referen epoint (f)

(z

1 , z

3 )

,

i

= 3

0.1

0.2

0.3

0.05

0.1

0.15

0.2

0.25

0.3

PSfragrepla ements

z

2 z

3

,blo kstru ture ,noblo kstru ture ,noblo kstru ture Referen epoint (g)

(z

2 , z

3 )

,

i

= 1

0.1

0.2

0.3

0.05

0.1

0.15

0.2

0.25

0.3

PSfragrepla ements

z

2 z

3

,blo kstru ture ,noblo kstru ture ,noblo kstru ture Referen epoint (h)

(z

2 , z

3 )

,

i

= 2

0.1

0.2

0.3

0.05

0.1

0.15

0.2

0.25

0.3

PSfragrepla ements

z

2 z

3

,blo kstru ture ,noblo kstru ture ,noblo kstru ture Referen epoint (i)

(z

2 , z

3 )

,

i

= 3

Fig. 8 Evolutionof the

95%

- redible ellipseswithrespe tto the iterationnumber.Blue ontinuousline:

d

y

= 23

withoptimizationoftheblo kstru ture.Reddottedline:

d

y

= 23

withoutoptimizationoftheblo kstru ture.Greendashedline:

d

y

= 5

withoutoptimization

(21)

were hosen,introdu ing thepartitioning doesnot makea bigdieren e for

thePDF identi ation, whi h explainsthe similaritiesbetween theblue and

thered urves.

.Intermediateresultswere obtainedforvaluesof

d

y

between5and23,whereasstillin reasing

d

y

didnot really hange theresults.

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

= 10

4

sim-ulationsare laun hed. These simulationsare asso iated withthe same ubi

system thanin Figure 6, but with dierent boundary onditions (by

hang-ing the boundary onditions, we want to verify that the identied valuesof

Z

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

ondi-tionson the inferior sideof the ubedo not hange, theleft and rightsides

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

(3,q)

,

1 ≤ q ≤ Q

o

theelasti ityelds hara teriz-ingthematerialpropertiesofthedierent ubesofthethreesetsrespe tively.

Forall

1 ≤ q ≤ Q

,

X

(1,q)

isanindependentrealizationofthetrueelasti ityeld,

X(z

⋆

₎

;

X

(2,q)

isan independent realization of

X((z

q,

prior

, z

z

q,

post is

arealizationof

Z|S(N )

afterthethreeformerlypresentediterations.

Forea h simulation, we denote by

U

(X

(i,q)

)

,

1 ≤ i ≤ 3

, the

on atena-tionoftheverti alandhorizontal displa ementsthat areindu edontheleft

and right sides ofthe ube. To omparethe statisti al information gathered

in these displa ements, we then ompute,for ea h

1 ≤ i ≤ 3

, the

eigenval-ues

n

v

j

(i)

, j

≥ 0

o

asso iated with the empiri al estimate of their ovarian e

matri es.Inaddition,wedenoteby

σ

VM

(X

(i,q)

)

themaximumvalueoverthe ubi domain of theVonMises stress.This VonMises riterionis ommonly

usedto hara terizetheresistan eofthesystem(see[Laietal.,2010℄formore

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

riteriaarenally omparedinFigure9.Lookingatthesegures,weseethat

theresultsasso iatedwiththeposteriordistributionof

Z

arevery losetothe onesasso iatedwiththetrueelasti ityeld,whi h isnottruefortheresults

(22)

0

50

100

150

200

250

300

10 −4

10 −3

10 −2

10 −1

10

0

10

1

PSfragrepla ements

j

v

(1)

j

v

(2)

j

v

(3)

j

(a)Eigenvalues

0

5

10

15

20

25

30

35

40

0

0.02

0.04

0.06

0.08

0.1

₎

(b)VonMises riterion

Fig.9 Representation oftheeigenvaluesde reases(a)andofthePDFsofthe VonMises

riteria(b)forthethree ompared ongurations.Inea hgure,theblue ontinuouslinesare

asso iatedwiththereferen e ase,thereddashedline orrespondtotheresultsasso iated

withthepriordistributionof

Z

,whenthebla ktwo-dashedlines orrespondtotheresults asso iatedwiththeposteriordistributionof

Z

.

(23)

4Con lusion

The in reasing of the omputational resour esand the generalizationof the

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

takeintoa ountrandomeldsin theirmodeling.Inthat prospe t,thiswork

proposesanadaptiveBayesianframeworktoe ientlyidentifythestatisti al

propertiesofthese randomeldswhentheavailable informationisaredu ed

setofindire tobservations.Twoexamplesbasedonsimulateddataarenally

presentedtoshowthepotentialofthisapproa h.

Extending thisapproa hto the aseswhere thenumberofparametersto

identifyandthenumberofobservationsareveryhighwouldbeinterestingfor

futurework.

Appendix

A.1.Proofoftheequalityof Eq.9

Let

A, B, D

betheblo kde ompositionmatri esof

C

b

−1

:

b

C

−1

=

A B

B

T

D

.

(30)

UsingtheS hur omplement,iffollowsthat:











b

C

−1

ZZ

= D − B

T

A

−1

B,

( b

C

Y Y

− b

C

Y Z

C

b

−1

ZZ

C

b

T

Y Z

)

−1

= A,

− b

C

Y Z

C

b

−1

ZZ

= A

−1

B.

(31) It omes

((y, z) − (Y (ω

m

), Z(ω

m

)))

T

(h

2 C)

b

−1

((y, z) − (Y (ω

m

), Z(ω

m

)))

=

1 h

2 (y − Y (ω

m

))

T

A(y − Y (ω

m

)) + 2(y − Y (ω

m

))

T

AA

−1

B

(z − Z(ω

m

))

+ (z − Z(ω

m

))

T

D(z − Z(ω

m

))

!

=

1 h

2 



(y − Y (ω

m

) + A

−1

B

(z − Z(ω

m

)))

T

A(y − Y (ω

m

) + A

−1

B

(z − Z(ω

m

)))

+ (z − Z(ω

m

))

T

D

− B

T

A

−1

B

(z − Z(ω

m

))





= (y − µ

n

(z))

T

C

−1

n

(y − µ

n

(z)) +

1 h

2 (z − Z(ω

m

))

T

C

−1

ZZ

(z − Z(ω

m

))

(32)

(24)

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