Champs aléatoires et problèmes statistiques inverses associés pour la quantification des incertitudes : application à la modélisation de la géométrie des voies ferrées pour l'évaluation de la réponse dynamique des trains à grande vitesse et l'analyse

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Champs aléatoires et problèmes statistiques inverses

associés pour la quantification des incertitudes :

application à la modélisation de la géométrie des voies

ferrées pour l’évaluation de la réponse dynamique des

trains à grande vitesse et l’analyse

Guillaume Perrin

To cite this version:

Guillaume Perrin. Champs aléatoires et problèmes statistiques inverses associés pour la quantification

des incertitudes : application à la modélisation de la géométrie des voies ferrées pour l’évaluation de

la réponse dynamique des trains à grande vitesse et l’analyse. Other. Université Paris-Est, 2013.

English. �NNT : 2013PEST1137�. �pastel-01001045�

Do toral Thesis Spe iality: Me hani s

presentedby Guillaume PERRIN

Random elds and asso iated statisti al inverse problems for un ertainty quanti ation - Appli ation to railway tra k geometries for high-speed trains dynami al responses and risk

assessment

Didier CLOUTEAU ECP - MSSMAT Examiner

Denis DUHAMEL ENPC - Navier Supervisor

Christine FUNFSCHILLING SNCF - I&R Examiner

Jean GIORLA CEA - DCSA Examiner

Olivier LE MAÎTRE CNRS - LIMSI Reporter

Anthony NOUY ECN - GeM Reporter

Thisdo toralthesis arisedwithintheframeworkof a ontra t betweenthe laboratoriesNavier andModélisationSimulationMulti-É helleatUniversitéParisEstandtheresear hdepartment ofSNCF,andwasfundedbySNCFandtheFren hministryofe ology,sustainmentdevelopment and energy.

I feel very grateful to everybodywho ontributedto this work. In parti ular, I would like to thank my supervisorsat SNCF and at Université ParisEst. With theirstronginvolvement in this work, their high interest for the subje t and their wise advi e, they reated a very onstru tive andstimulatingenvironmentforthisthesis to keepbringingforward. I wouldlike to thank them for all their ideas and proposals, for their required level and pre ision, whi h have beenendless sour es of motivation and innovation duringthiswork. I also would like to thank them forthe ordialrelations we had and forthetaste forresear hthey gave to me.

I would like to express my gratitude to all the members of the jury. It was a pleasureto ex hange ideaswithsu hdistinguishedresear hers.

Finally,Iwouldliketoaddressmythankstoallmy olleaguesatSNCFandattheUniversité ParisEstfortheirsupport,butalso forall thegood timewehad.

A knowledgments 2

Introdu tion and obje tives 8

Industrial obje tives . . . 8

S ienti obje tives. . . 8

State of theart . . . 9

Main s ienti andindustrial ontributions . . . 11

Outline ofthe thesis . . . 12

General theoreti al frameand orrespondingnotations . . . 12

1 Short review of the methods for modeling random elds 15 1.1 Introdu tion. . . 15

1.2 Classi al methodsto generaterandomelds . . . 15

1.3 The optimalityofthe Karhunen-Loève expansion . . . 16

1.3.1 Denition oftheKarhunen-Loèveexpansion . . . 16

1.3.2 Optimalityof theKLexpansion . . . 17

1.3.3 Pra ti al solvingof theFredholmequation . . . 17

1.3.4 ApproximatedKL expansion . . . 18

1.4 Identi ationand generationof randomve tors . . . 18

1.5 PCE identi ationof randomve tors . . . 21

1.5.1 Theoreti al frame. . . 21

1.5.2 Identi ation ofthepolynomial haos expansion oe ients . . . 22

1.5.3 Pra ti al solvingof thelog-likelihood maximization. . . 23

1.5.4 Identi ation ofthePCE trun ationparameters . . . 25

1.6 Con lusions . . . 27

2 Optimal redu ed basis for random elds dened by a set of realizations 29 2.1 Introdu tion. . . 29

2.2 Theoreti alframe . . . 29

2.2.1 Quanti ationof therelevan eof a proje tionbasis . . . 29

2.2.2 Optimalityof theKarhunen-Loève expansion . . . 31

2.2.3 Identi ation di ultieswhen theinformationis partial . . . 31

2.3 Identi ationof optimalbasisfrom aniteset ofindependentrealizations . . . . 34

2.3.1 Reformulation of theproje tionerrorminimization . . . 34

2.3.2 Restri tion ofthesear h spa e . . . 35

2.3.3 A posteriorievaluationof therepresentativenesserror . . . 37

2.4 Appli ations. . . 39

2.4.1 Generation of independent realizationsof therandomeld. . . 39

2.4.4 Relevan e oftheLOO error . . . 44

2.5 Con lusions . . . 45

3 PCE identi ation in high dimension from a set of realizations 46 3.1 Introdu tion. . . 46

3.2 Optimizedtrialsof independentrealizations ofrandom matri es. . . 47

3.2.1 Reformulation of the orrelation onstraints . . . 47

3.2.2 Notations anddenitions . . . 48

3.2.3 Theoreti al frame. . . 50

3.2.4 Iterative algorithms . . . 51

3.2.5 Adaptationsto the ase

ν < M

. . . 53

3.3 Numeri al stabilizationofthepolynomialbasisinhigh dimension . . . 55

3.3.1 De ompositionofthe matrixofindependentrealizations . . . 56

3.3.2 Inuen e of thetrun ationparameters . . . 58

3.3.3 Adaptation ofthe optimizationproblem . . . 60

3.3.4 Remarks onthenew optimization problem . . . 62

3.4 Appli ation . . . 63

3.4.1 Appli ationinlowdimension . . . 63

3.4.2 Appli ationinhigh dimension. . . 74

3.4.3 Appli ationinvery highdimension withlimited information . . . 77

3.5 Con lusions . . . 80

4 Karhunen-Loèveexpansion revisited for ve tor-valued random elds 82 4.1 Introdu tion. . . 82

4.2 S aledexpansion . . . 83

4.2.1 Lo al-global errors andoptimalbasis . . . 83

4.2.2 S aledexpansion . . . 85

4.2.3 Propertiesof thes aled expansion . . . 86

4.2.4 Minimization of aweightedsum of lo alerrors . . . 89

4.2.5 Minimization of themaximalvalue of thelo alerrors . . . 90

4.3 Appli ation . . . 92

4.3.1 Generation of ave tor-valued randomeld . . . 92

4.3.2 Inuen e of thes aling ve tor onthe lo alerrors . . . 93

4.3.3 Identi ation oftheoptimalbasis. . . 93

4.4 Con lusions . . . 97

5 Experimental identi ation of the railway tra k sto hasti modeling 100 5.1 Introdu tion. . . 100

5.2 Experimental measurementsand signalpro essing . . . 102

5.2.1 Colle tion oftheexperimentalinputsforthe modeling . . . 102

5.2.2 Lo al-global approa h . . . 102

5.3 Optimal redu edbasis . . . 106

5.3.1 Dire t KL expansionand proje tionbiases . . . 106

5.3.2 Optimization oftheproje tionbasis . . . 107

5.3.3 Choi e of thedimensionof thespatialproje tionparameter . . . 109

5.4 PCE identi ationinveryhigh dimension . . . 109

5.4.1 Sorting withrespe tto thehorizontal urvature. . . 109

5.4.2 PCE identi ation . . . 111

5.7 Con lusions . . . 121

6 Sto hasti dynami s of high-speed trains and risk assessment 122 6.1 Introdu tion. . . 122

6.2 Des riptionof therailwaydynami problem . . . 122

6.2.1 Deterministi railwayproblem . . . 122

6.2.2 Domain of validityforthe deterministi problem . . . 125

6.3 Denitionof thesto hasti problemand validationof themodeling . . . 128

6.3.1 Sto hasti problem . . . 128

6.3.2 Validationof thesto hasti problem . . . 128

6.4 Propagation of thevariability . . . 132

6.4.1 Inuen e of thetra k design. . . 132

6.4.2 Inuen e of anin reaseof thespeedon thequantitiesofinterest . . . 134

6.4.3 Comparison ofthree highspeedtrains . . . 136

6.5 Sensitivityanalysis . . . 136

6.5.1 Nonlinearities and importan eof the onjun tion oftra kirregularities. . 137

6.5.2 KL-basedsensitivityanalysis . . . 138

6.6 Con lusions . . . 141

Con lusions and prospe ts 144 Summary oftheindustrial ontext . . . 144

S ienti and industrial ontributions . . . 145

Prospe ts . . . 146

Publi ationsand external ommuni ations . . . 150

Appendix 153 A Proof of Lemma2 . . . 153

B Generation ofthe matrix-valuedauto orrelation matrix . . . 155

Industrial obje tives

High speed trains are urrently meant to run faster and to arry heavier loads, while being less energy onsuming and stillensuring the safety and omfort erti ation riteria. In order to optimize the on eption of su h high te hnology trains, we need a pre ise knowledge of the realm of possibilitiesof tra k onditions that the trainis likelyto be onfrontedto during its life y le.

In parallel, sin e 2012, European high speed railway networks are meant to have gone to market. Several high speed trains, su h as ICE, TGV, ETR 500..., for whi h me hani al properties and stru tures aredierent,are likelyto runon thesame tra ks,whereasthey may have beenoriginally designedforspe i anddierent railwaynetworks. European highspeed railwaynetworksarethereforeboundto besubje tedtoan in reasingvariabilityofme hani al loads. Tooptimizethetra kmaintenan eandtoadjustthetollsa ordingtotheaggressiveness of a parti ular train toward the tra k, a better understanding of the intera tion between the traindynami behavior and thetra k geometryis ne essary.

Simulation is a very usefultool to fa e these hallenges. However, it hasto be very repre-sentative ofthephysi albehaviorof thesystem. The modelsof thetrain,of therailwaytra k, and of the wheel/rail onta ts have thus to be fully validated and the simulations have to be raisedon realisti and representative sets of ex itations.

Hen e, based on experimental measurements, a omplete parametrization of the tra k ge-ometryand ofitsvariabilitywouldbeofgreat on ernto analyzethe omplexlinkbetweenthe traindynami sand thephysi aland statisti al properties ofthetra kgeometry.

S ienti obje tives

From a s ienti pointof view, a railwaysimulation an be seenas thedynami response of a omplexme hani alsystemex itedbyamultivariaterandomeld,forwhi hstatisti al proper-tiesare onlyknown througha set of independent realizations. Due to thespe i intera tions betweenthetrainand thetra k, thisrandomeldis neitherstationary norGaussian.

In order to propagate the tra k geometry variability to the train response, methods to identify in inverse, from a nite set of experimental data, the statisti al properties of non-stationary and non-Gaussianrandomeldswillbe analyzed inthismanus ript.

Thetrainbehaviorbeingverynonlinearandverysensitivetothetra kgeometry,therandom eldhastobedes ribedverypre iselyfromfrequen yandstatisti alpointsofview. Asaresult, the statisti al dimension of this random eld is very high. Hen e, a parti ular attention will bepaid inthisthesis to statisti al redu tionmethodsand to statisti alidenti ationmethods that an be numeri allyappliedto thehigh dimensional ase.

The general s heme for probabilisti analysis is usuallydividedin three steps (see [1 , 2, 3 ℄for furtherdetails). First, the me hani al modeland theasso iatedinputparameters and output riteria(safety riteria forinstan e)have to be denedpre isely. Then,thedierentsour es of un ertaintyhavetobeidentiedandmodeled arefully. Atlast,theinputun ertaintyhastobe propagated throughthedeterministi model, inorder to hara terize thestatisti al properties of theoutputquantitiesof interest.

These three stepsarerapidlydes ribedhereunder forthestudiedrailwaysystem.

Me hani al model. In this work, the rea tions of trains ex ited by the tra k geometry throughthespe i wheel/rail onta ts arestudied. Threekindsofinputsarethereforeneeded insu h simulations:

•

the vehi le model. Multibody simulations are usually employed to model the train dy-nami s (see [4 ℄). Carbodies, bogies and wheelsets are modeled by rigid bodies linked with onne tions representedbyrheologi models(damper, springs,...). Thisleadsusto several hundredsof degreesof freedom.

•

the tra k model. A double s ale parametrization is usually introdu ed to des ribe the tra kgeometry(see[5℄): ea hrailpositionis hara terized byamean-lineposition,whi h onlydepends onthe verti aland horizontal urvatures, on thetra ksuper-elevation and the tra k gauge of the tra k, and by a deviation towards thismean-line position, whi h anbedes ribedbyfour urvilinearirregularityelds. Whilethemeanpositionisde ided on efor all at the buildingof a new line,the tra k irregularities an evolve withrespe t to the tra k substru ture, to the weather onditions and to the train dynami s. Seven urvilineareldsareneededto ompletely hara terizethepositionsofthetworigidrails.

•

thewheel/rail onta tmodel. Thewheel/rail onta tfor esare omputedforanyposition of thetrain from thewheel and therail proles thanksto the Hertz and Kalkertheories ([6 ,7℄). The onta t propertiesaremoreovergenerally re ordedina onta ttable. Giventhese threeinputs,thetrainresponse anbe omputedasthesolutionof asystemof oupledequations thatare strongly nonlinear. This systemis usuallysolved with an expli it s heme. On e these equations have been solved, the spatial a elerations of ea h mass body, as well as the internal and external loads are available. These railway outputs an then be post-pro essedto dene safety, omfortand maintenan e riteria.

In thiswork,the ommer ial ode Vampire isusedtosolvetheseequations. Themovement equations of the railway dynami s are thus not available. Moreover, the duration of a whole railwaysimulationoveralengthof 5kmis approximately120 se ondsona standard omputer. Un ertainty quanti ation. Several sour esof un ertainty an be ategorized:

•

Model un ertainty. Inea h model,simplifyinghypotheses areintrodu ed. In thestudied system,therigidbodymodelingofthetrainand theHertz formulationforthewheel/rail onta t aretwo examplesof su h model simpli ations.

•

Parameter un ertainty. The hosen model to des ribe the onsidered systemis generally based on parameters, forwhi h exa t values are unknown and annot exa tly be exper-imentally measured. Forinstan e, the total mass of a trainis, inpra ti e, impossibleto pre iselyevaluate.

•

Parametervariability,whi h omesfromthephysi alvariabilityoftheinputparametersof themodel. Exampleisthetrainsuspensions,forwhi hthepro essofmanufa turingleads ustome hani al hara teristi sthatarenotexa tlyasdesignedsu hthattheperforman e an varyfrom one suspensionto anotherone.

•

Algorithmi un ertainty, whi h omes from numeri al approximations. In the railway eld, thisun ertainty omes mostlyfrom thetime dis retizationin the expli itsolver of the movement equations. A onvergen e analysis has thus to be performed to hoose a relevant timestep.

•

Measurements un ertainty. Experimental dataare imagesof the reality,forwhi h biases have to be minimizedasmu h aspossible.

Dueto theseun ertainties,dis repan ieswillalways be observed when omparingthe mod-eledandmeasureddeterministi responsesofatrain. Onthe ontrary,sto hasti models,whi h would be able to take into a ount these un ertainties, should lead to a better representation ofthebehaviorofthesystem. Thisexplainsthevery highinterest forthese methodsthat have spreadforthelastde adesto mostof thes ienti elds.

Some spe i elds of the probability theory have therefore fo used on parti ular sour es of un ertainty. First, the methodsbased on InformationTheory and on theMaximal Entropy prin iple (see [8 ℄ and [9℄) have been ontinuously improved to always better hara terize the parameter un ertainty and variabilityfrom the only available and usable information. In the same manner, the use of methods based on the Bayesian method (see [10 , 11 , 12 ℄) has kept in reasingtoupdatetheinputsto hasti modelinginthelightofnewandrelevantdata. Then, when the movement equations of the systemare available, non-parametri probabilitymodels (see[13 , 14℄)havebeenintrodu edtotakeintoa ount notonlytheinputun ertaintybutalso themodeland algorithmi un ertainties.

In this work, it is supposed that a nominal model of a train is available, for whi h me- hani al parameters are xed and have been a urately identied. In the same manner, the onta t properties are omputed on e for all from a new rail prole and a new wheel prole. Given a parti ular des ription of the tra k geometry and these onta t and vehi le models, it is assumed that boththe railway model and the numeri al solving are su iently relevant to a urately ompute theresponse of thesystem: theapproximationsintrodu edin the ompu-tational s heme are supposed to be ontrolledand the movement equations areassumed to be pre ise enoughto represent thephysi alphenomena.

Hen e,onlytheun ertaintyinthetra kgeometryisaddressedinthisthesis. Inthisprospe t, thistra k geometrywill be seen as a multivariaterandom eld. To identifythiseld, a set of experimental measurements of thetra k geometry is used. It is assumed that the experimen-tal un ertainties for these measurements are negligible, su h that no distin tion will be made betweenthetra kmeasurementsand therealtra kgeometry inthe following. These measure-mentsdenethemaximalavailableinformationaboutthetra k-geometryrandomeld. Un ertainty propagation and risk assessment. On e the parameter un ertainty and variabilityhavebeen hara terized,thevariabilityhastobepropagatedthroughtheme hani al

on the omputational ost ofthesimulation. In thiswork,we willfo usonthe a elerationsof thetrainmassbodiesandtheloadsbetweenthetrainandthetra k. Wearemoreoverinterested inprobabilitiesforthese outputsto ex eednormalized thresholds.

Re allthattherailwayme hani alisbasedonavery highnumberofvariableinput param-eters, that thetrain responseis very sensitive, very non linear,and very fuzzywith respe tto these inputparameters,that themovement equationsare notavailable, and that theduration of onesimulationisrather heap. Thebestmethodto omputesu hprobabilitiesofex eeding thresholdsforthese railwayoutputsistherefore theMonteCarlo(MC) method ([15 ℄). Indeed, thestatisti al onvergen eofsu hamethoddependsneitheronthedimensionoftheinput,nor on the omplexity and the nonlinearity of the me hani al model, and is parti ularly adapted to systems that are ontrolled by bla k-box odes, that is to say, odes for whi h movement equationsare notavailable,as itisthe asehere.

In ordertogeta urateresults, mu hattentionhastobepaidto themodelingoftheinput variability, as any error on the input will be propagated to the output. In addition, the MC method asksforthegenerationof sets ofindependentrealizations oftheinputparameters. As this work fo us on the tra k geometry variability, methods to generate independent realisti and representative tra k onditionswillbeneeded inthiswork.

At last, based on this MC method, ea h railway simulation gives a ess to a parti ular realizationofthetimerea tionsofthetra k. Theriskassessmenthasthereforetobeperformed usingstatisti al methodsbased on sto hasti pro esses(see [16 ℄ forfurtherdetails).

Main s ienti and industrial ontributions

The developmentsof thiswork werea hieved to answer thefourfollowingquestions.

•

Virtual erti ation. Howtodevelopatra kgenerator,whi hwouldbeabletogenerate tra k onditions, whi h are on the one hand realisti from a statisti al, frequen y and dynami al point of view, and from the other hand representative of a measured set of experimentaldata? Thenumeri al erti ationindeedrequiresalargesetofrepresentative tra k onditionsto apturerare events[3 ℄.

•

Optimization of the system. How to propagate thetra kgeometry variabilityto the traindynami al quantitiesof interest, whi haremostly lateraland verti ala elerations and loads? The knowledge of the link between the tra k variability and the response of thetrain ould indeedhelp usto propose optimizedmaintenan e poli ies.

•

Railway eld going to market. How to develop a method to evaluate and ompare theaggressivenessofseveral trains thatwouldbelikelyto runon thesame network?

Fours ienti mains ienti ontributionsare summarizedhereunder.

1. The statisti al dimension of the tra k-geometry randomeld is very high, su h that ad-van edredu tionte hniqueswillbeneededtooptimally ondensethestatisti alproperties oftherandomeldto beidentied. Inparti ular,theimportan eof theKarhunen-Loève (KL)expansionwillbeanalyzed indetailin thiswork.

to its statisti al dimension. The statisti al moments of this random eld, su h as the empiri al estimators of the mean fun tion or the ovarian e operator, on whi h the KL expansion is based, are not onverged. A method to adapt the KL formulation to this kindofproblemswillthusbe proposedinthisthesis.

3. The tra k-geometry random eld is multivariate, and its dierent omponents are very statisti ally dependent. A ve torial approa h has therefore to be onsidered in order to a urately take into a ount thedependen iesbetween these dierent omponents of the tra k-geometry random eld. Moreover, the amplitudes of these omponents are dierent and their importan es on the dynami al quantities are a priori unknown. An other adaptation of the lassi al KL expansion has thus to be introdu ed in order to identifyaredu edbasisthatallowsthedes riptionofea h omponentoftherandomeld of interestwith thesame pre ision.

4. Duetothespe i intera tionbetweenthetrainandthetra k,thetra k-geometryrandom eld is neither stationary nor Gaussian, su h that a parti ular attention has to be paid totheidenti ationofthemultidimensionaldistributionofthe oe ientsoftherandom eld on the redu ed proje tion basis. Due to the omplexity of the random eld to be modeled, these oe ients denea very high dimensionrandom ve tor. To this end, an adaptationto theveryhighdimensionoftheidenti ationininverse methodsbasedon a polynomial haos expansionwillbe presentedinthiswork.

Outline of the thesis

From these obje tives,the do ument isorganized insix haptersthat arenowpresented. Chapter 1 ontainsareviewofwell-knownmethodsforrandomeldidenti ationand gen-eration. Inparti ular,theKarhunen-Loève(KL)expansionandthepolynomial haosexpansion (PCE)identi ationininverse willbepresentedindetail.

The next haptersaredevotedtotheauthor ontributionsintheeldofun ertainty propa-gation. Chapter 2dealswiththeadaptationof theKLmethodto ases forwhi h themaximal availableinformationabouttherandomeldto identifyislimitedtoaniteset ofindependent realizations.

Chapter3addressestheadaptationofthepolynomial haosexpansionidenti ationmethods to thevery highdimensional ase.

Chapter4presentsanoriginals aledKLexpansionfortheanalysisofve tor-valuedrandom elds.

Chapter 5 onsidersthe appli ationof thetheoreti al developments of Chapters2,3 and 4 to identify,in inverse, from experimental data, the statisti alproperties of thetra k-geometry randomeld.

At last, Chapter 6 shows in what extent su h a sto hasti modeling of the tra k geometry opensnewopportunitiesfortherailwayeldin erti ation,maintenan e,andsafetyprospe ts.

General theoreti al frame and orresponding notations

Thisse tion aimsat summarizing themainnotations thatwillbeused inthismanus ript.

• R

denotestheset of real numbers.

• Ω ⊂ R

refersto a subsetof

R

.

• (Θ, T , P)

is aprobabilityspa e.

• E [·]

isthemathemati al expe tation.

• H = L

2

P

Θ, R

M

isthespa e ofallthese ond-orderrandomve torsdenedon

(Θ,

T , P)

withvaluesin

R

M

,equippedwiththe innerprodu t

h., .i

:

hA, Bi =

Z

Θ

A

T

(θ)B(θ)dP (θ) = E



A

T

B



,

∀ A, B ∈ L

2

_P

Θ, R

M

.

(1)

• P

(Q)

_{([0, S])}

,where

S < +

∞

,isthespa eofallthese ond-order

R

Q

-valuedrandomelds, indexedbythe ompa t interval

[0, S]

.

•

For

Q

≥ 1

,

X

= (X

1

, . . . , X

Q

) =

{(X

1

(s), . . . , X

Q

(s)) , s

∈ [0, S]}

isin

P

(Q)

_{([0, S])}

.

•

Let

H

= L

2

_{([0, S], R}

Q

₎

be the spa e of square integrable fun tions on

[0, S]

, with values in

R

Q

,equipped withtheinnerprodu t

(

·, ·)

,su hthat, forall

u

and

v

in

H

,

(u, v) =

Z

[0,S]

u(s)

T

v(s)ds.

(2)

• k·k

_P

(Q)

_([0,S])

denotesthe

L

2

norm in

P

(Q)

_{([0, S])}

, su h that:

kXk

2

_P

(Q)

_([0,S])

= E

Z

Ω

X(s)

T

X(s)ds



,

X

_{∈ P}

(Q)

([0, S]).

(3)

• δ

mp

isthe krone kersymbolthat isequalto 1 if

m = p

and 0otherwise.

• Tr [·]

isthetra e operatorforsquare matri es.

• a

,

b

orrespondto onstants in

R

.

• a

,

b

refer to ve tors withvaluesin

R

Q

,

Q

≥ 1

.

• ×

is theve torialprodu tbetween ve tors.

• a

T

isthetranspose of

a

.

• ⊗

is thetensorialprodu tsu h that

a

⊗ b = ab

T

.

•

A,B orrespond to randomvariableswith valuesin

R

.

• A

,

B

denoterandomve tors withvaluesin

R

Q

,

Q

≥ 1

.

• [A]

,

[B]

refer to real matri es.

• k·k

F

istheFrobeniusnormof matri es.

• P

A

and

p

A

denote respe tively the multidimensional probability distribution and the multidimensionalProbabilityDensityFun tion (PDF)of randomve tor

A

.

•

If random ve tor

A

is of se ond order, we denote by

µ

A

and

[R

AA

]

the mean and the ovarian e matrixof

A

respe tively.

• (s, s

′

₎

_{7→ [R}

_XX

_{(s, s}

′

_)]

orresponds to the matrix-valued ovarian e fun tion of

X

, su h thatforall

s

,

s

′

in

Ω

,

[R

XX

(s, s

′

_{)] = E [(X(s)}

_{− E [X(s)]) ⊗ (X(s}

′

₎

_{− E [X(s}

′

_)])]

.

•

When

Q = 1

,

P

(1)

_(Ω)

,

X

and

[R

XX

]

are written

P(Ω)

,

X

and

R

XX

respe tivelyforthe sakeof simpli ity.

• F

(M )

denotesa subsetof

H

thatgathers

M

fun tionswithvaluesin

R

Q

that aredened on

Ω

.

• c

X

F

(M )

refersto theproje tionof

X

on thesubspa espannedby

F

(M )

Short review of the methods for

modeling random elds

1.1 Introdu tion

As presented in Introdu tion, the goal of this work is to quantify the inuen e of the tra k geometry variabilityon the train dynami al responses. A good approa h to take into a ount this inputvariability is to onsiderthe tra k geometry as a multivariate random eld. It has moreover been shown that the most appropriate method to propagate the tra k variability through the me hani al model is the Monte Carlo (MC) method. For su h a method to be implemented, one has therefore to be able to generate independent realizations of this tra k-geometry random eld. Due to the spe i intera tions between the train and the tra k, this random eld is neither Gaussian nor stationary. In this prospe t, several existing methodsto identifyandgeneratenon-Gaussianrandomeldsareaddressedinthis hapter. Morepre isely, this hapter des ribes in detail the method on whi h the sto hasti modeling of the tra k geometry will be based in the next hapters, whi h is based on the oupling of a Karhunen-Loève expansionand apolynomial haosexpansion.

1.2 Classi al methods to generate random elds

For the last de ades, the random elds analysis has been used in an in reasing number of s ienti elds, su h as un ertainties quanti ation, material s ien es, seismology, geophysi s, quantitative nan e,signal pro essing, ontrol engineering et . It is indeed a very interesting toolforsto hasti modeling,fore asting, lassi ation,signaldete tion and estimation. Let

X =

_{{X(s), s ∈ Ω ⊂ R} ,}

(1.1)

bea randomeld forwhi hwe want to generatesample paths. For thesake of simpli ity,and withoutanylossofgenerality,only entered randomelds

X

are onsideredinthiswork:

E [X(s)] = 0,

_{∀ s ∈ Ω,}

(1.2)

where

E [

·]

is themathemati alexpe tation.

The Gaussian ase is a well-posed problem, as the Gaussian random elds are ompletely hara terized only by their mean fun tion and their auto orrelation fun tion. It exists there-fore manyee tive methods to simulate Gaussian random elds. In parti ular, when

Ω = R

, AutoRegressive-Moving-Average (ARMA) models, that were rst introdu ed by Whittle for

stationary random elds as a parameterized integral of a Gaussian white noise random eld. Basedonlimitedknowledgeofrandomeld

X

,thesemodels anthereforebeusedtoemphasize parti ular propertiesof

X

andto extrapolate itsvalue.

On the ontrary, the random eld simulation problem is an ill-posed problem. To har-a terize a non-Gaussian random eld, we need to know the entire family of joint probability distributions

{(X(s

1

), . . . , X(s

n

)) , n

≥ 1, (s

1

, . . . , s

n

)

∈ Ω

n

_}

. As this information is most of thetime nota essible, onlypartialdes riptionof non-Gaussianrandomeld an be given.

Two lassesofmethodsaregenerallyusedto hara terizesu hnon-Gaussianrandomelds. On the rst hand, translation methods allow the identi ation and the generation of a non-GaussianrandomeldfromamemorylessnonlineartransformationofaknownGaussianrandom eld(see forinstan e [20 ℄).

On the other hand, in the general ase, spe tral methods ([21, 22 ℄) based on a two-step approa hhavegivenverypromisingresultstoidentifythedistributionofa priori non-Gaussian andnon-stationaryrandomelds. Therststepofthesemethodsisgenerallytheapproximation oftherandomeld,

X

,byitsproje tion

X

b

B

(M )

ona

M

-dimensionsetofdeterministi fun tions,

B

(M )

₌

_{b

m

(s), s

∈ Ω}

_1≤m≤M

,thataresupposedtobesquareintegrableon

Ω

andorthonormal su h that:

b

X

B

(M )

=

M

X

m=1

C

m

b

m

,

Z

Ω

b

m

(s)b

p

(s)ds = δ

mp

,

C

m

=

Z

Ω

X(s)b

m

(s)ds,

(1.3)

where

δ

mp

is the krone ker symbol. The ve tor

C

= (C

1

, . . . , C

M

)

is thus a

M

-dimension random ve tor, for whi h omponents are a priori dependent. The se ond step is then the identi ationofthemultidimensionaldistributionof

C

.

When the knowledge of the random eld is limited to a set of independent realizations, as it is the ase for the modeling of the tra k geometry, su h spe tral methods present many advantages. First, no hypothesis on therandom eldis required to implement these methods. Then,byproposingadis retizeddes riptionoftherandomeld,theytakeadvantageofallthe developmentsthat havebeendoneinthe hara terization ofthemultidimensionaldistribution of non-Gaussianrandomve tors.

1.3 The optimality of the Karhunen-Loève expansion to gener-ate approximated realizations of random elds

1.3.1 Denition of the Karhunen-Loève expansion

Mathemati ally,theKarhunen-Loève(KL)expansion orrespondsto theorthogonal proje tion theorem in separable Hilbert spa es. In this ase, the Hilbertian basis,

{k

m

, m

≥ 1}

, is onstru ted as the eigenfun tions of the ovarian e operator of

X

, dened by the ovarian e fun tion,

R

XX

, whi h is assumed, for instan e, to be square integrable on

Ω

× Ω

. Therefore, forall

(s, s

′

₎

in

Ω

× Ω

and

m

≥ 1

and

p

≥ 1

, we get:

R

XX

(s, s

′

)

def

= E



X(s)X(s

′

)



=

X

m≥1

λ

m

k

m

(s)k

m

(s

′

),

(1.4)

Z

Ω

R

XX

(s, s

′

)k

m

(s

′

)ds

′

= λ

m

k

m

(s),

(1.5)

(k

m

, k

p

) = δ

mp

,

λ

1

≥ λ

2

≥ . . . → 0,

X

m≥1

λ

2

_m

< +

∞.

(1.6)

1.3.2 Optimality of the KL expansion

In order to represent theeld

X

witha small numberof ve tors

M

, it isimportant to hoose a relevantbasis regarding

X

. Indeed, themore relevant theproje tion basis

B

(M )

is,the lower thedimension

M

hasto be,to guarantee thattheamplitudeof theresidue,

N

2

(X

_{− b}

X

B

(M )

),

(1.7)

islowerthana given threshold, andsotheeasier andthe morepre ise theidenti ationofthe distributionof

C

willbe.

N

2

isa norm thathasto beadapted to thestudied problem. If

N

2

(

·) = E

Z

Ω

(

·)

2



,

(1.8)

duetotheorthogonalproje tiontheoreminHilbertspa es,foranyinteger

M

,the

M

-dimension family

K

(M )

₌

_{k

m

, 1

≤ k ≤ M}

,whi hgathersthe

M

rstelementsoftheKLbasisasso iated with

X

, minimizes theamplitude

N

2

_(X

_{− b}

_X

F

(M )

)

among all the

M

-dimensionfamilies

F

(M )

, where

X

b

F

(M )

isthe proje tionof

X

on

F

(M )

. In other words,forany

M

≥ 1

,it an be shown that:

N

2

(X

_{− b}

X

K

(M )

)

_{≤ N}

2

(X

_{− b}

X

F

(M )

),

(1.9) where

X

b

K

(M )

istheproje tionof

X

on

K

(M )

.

Due to this optimality property, the Karhunen-Loève (KL) basis has played, for the last de ades, amajorrole and hasbeenapplied inmanyworks(seeforinstan e [23 , 24,25 , 26 ,27 , 28 ,29 , 30 ,31 ,32 , 33 ,34,35 , 36 ,37 , 38 ,39 ,40 , 41,42 ,43 ℄).

1.3.3 Pra ti al solving of the Fredholm equation

Equation(1.5)is ommonlyreferredtoasFredholmequation,andissues on erningthesolving of this integral eigenvalue problem an be foundin [21 , 44 , 45 ℄. The idea of this se tion is to des ribethe dierentstepsto solve theFredholmproblemthanks to a niteelement approa h when

Ω = [0, S]

. Tothisend,thefun tions

k

m

,

1

≤ m ≤ M

,aresear hedastheirniteelement estimator

k

FE

m

,su h that,forall

s

in

Ω

:

k

m

(s)

≈ k

FE

m

(s) =

N

S

X

j=1

d

m

_j

h

j

(s),

(1.10)

d

m

= d

m

₁

, . . . , d

m

_N

_S

,

h(s) = (h

1

(s), . . . , h

N

S

(s)) ,

(1.11) where

d

m

is the unknown ve tor to be identied, and

{s 7→ h

j

(s), 1

≤ j ≤ N

S

}

are shape fun tions su hthat:







s

1

= 0, s

N

S

= S, s

j+q

− s

j

= qh,

h

j

(s

k

) = δ

jk

, 1

≤ j, k ≤ N

S

,

P

N

S

j=1

h

j

(s) = 1, s

∈ Ω = [0, S],

(1.12)

with

h = S/ (N

S

− 1)

theniteelement dis retizationlength. Theniteelementdis retization of Eq. (1.5 ) yields:

([K]

_{− λ}

m

[M ]) d

m

= 0,

(1.13)

inwhi h thepositive-denitesymmetri

(N

S

× N

S

)

real matri es

[K]

and

[M ]

are denedby

[K] =

Z

Ω

Z

Ω

h(s)

T

[R

XX

(s, s

′

)]h(s

′

)ds

′

ds,

(1.14)

[M ] =

Z

Ω

h(s)

T

h(s)ds.

(1.15)

This approa h is parti ularly well adapted to the modeling of random elds, for whi h experimental values are re orded every

eh

meters. Spatial dis retization step

h

is thus hosen equalto

eh

to limitthe errorintrodu ed bythe niteelement approa h. Moreover, it hasto be noti edthattheregularityoftheshapefun tionshastobeadaptedtotheregularityofrandom eld

X

. Inparti ular,iftherstand se ondorderspatialderivativesoftherandomeldpaths area priori non zero, at least ubi shapefun tions willbe needed.

1.3.4 Approximated KL expansion

Aspresentedin Se tion1.3.1,the KL expansionof a entered random eld

X

is based on the knowledgeofitsauto ovarian efun tion,

R

XX

. Whenthemaximalavailableinformationabout

X

isasetof

ν

independentrealizations,

{X(θ

1

), . . . , X(θ

ν

)

}

,thisfun tionisnotexa tlyknown, but an beapproximatedbyitsempiri alestimation,

R

b

XX

(ν)

,su h that:

R

XX

(s, s

′

)

≈ b

R

XX

(ν, s, s

′

) =

1

ν

ν

X

n=1

X(θ

n

, s)X(θ

n

, s

′

), (s, s

′

)

∈ Ω × Ω.

(1.16)

By solving the Fredholmproblem asso iated with

R

b

XX

(ν)

instead of

R

XX

, it is therefore possible to identify a rather good approximation of the KL basis of

X

, whi h is denoted by

n

bk

m

(ν), 1

≤ m

o

,espe iallywhen

ν

is high,as:

lim

ν→+∞

R

b

XX

(ν) = R

XX

,

lim

ν→+∞

bk

m

(ν) = k

m

.

(1.17)

1.4 Dire t and indire t methods for the identi ation of the distribution of random ve tors and their generation

On erandomeld

X

hasbeenproje tedona hosendeterministi

M

-dimensionfamily,

B

(M )

₌

{b

m

(s), s

∈ Ω}

_1≤m≤M

, su hthat

X

≈ b

X

B

(M )

=

M

X

m=1

C

m

b

m

,

(1.18)

identifyingits statisti aldistribution amountsto identifyingthemultidimensionaldistribution ofrandomve tor

C

= (C

1

, . . . , C

M

)

,denotedby

P

C

. Themeanvalueandthe ovarian ematrix of

C

aremoreoverdenoted by

µ

µ

_C

= E [C] ,

[R

CC

] = E [(C

− µ

C

)

⊗ (C − µ

C

)] .

(1.19) Inthiswork,itisassumedthat

P

C

(dx) = p

C

(x)dx

,inwhi htheprobabilitydensityfun tion (PDF)

p

C

is a fun tionin theset

F(D, R

∗

₎

of all the positive-valued fun tions denedon any part

D

of

R

M

and forwhi h integralover

D

is1.

Twokindsofmethods anbeusedtobuildsu haPDF:thedire tandtheindire tmethods. Amongthedire tmethods,thePriorAlgebrai Sto hasti Modeling(PASM)methodspostulate analgebrai representation

C

≈ t

alg

_{(Ξ, w)}

,with

t

alg

apriortransformation,

Ξ

arandomve tor and

w

a ve tor of parameters to be identied. For instan e, we an suppose that

C

an be writtenunder theform:

C

_{≈ t}

_alg

(Ξ, w) = w

1

+ [w

2

]Ξ,

w

=

{w

1

, [w

2

]

} ,

(1.20) with

Ξ

a

M

-dimension random ve tor for whi h omponents are independent, normally dis-tributedwithzero meanand unit varian e. It an dire tlybeseen that:

E [t

alg

(Ξ, w)] = w

1

,

E [(t

alg

(Ξ, w)

− w

1

)

⊗ (t

alg

(Ξ, w)

− w

1

)] = [w

2

][w

2

]

T

.

(1.21)

Hen e, supposingthat

C

≈ t

alg

(Ξ, w)

amountsto supposingthat

C

is a Gaussianrandom ve tor, su h thatthemost a uratevaluesfor

w

1

and

[w

2

]

orrespondto the meanvalue of

C

andto theCholeskyde ompositionmatrixofmatrix

[R

CC

]

. If

C

isa tuallynotGaussian, this transformation is not relevant, and another one has to be introdu ed to better represent the behaviorof

C

,su h asforinstan e:

C

_{≈ t}

(2)

_alg

(Ξ, w) = w

1

+ [w

2

]Ξ + (Ξ

⊗ Ξ) w

3

,

(1.22) where

w

=

{w

1

, [w

2

], w

3

}

has on e again to be identiedto represent as well as possiblethe behaviorof

C

.

In the same ategory, the methods based on the Information Theory and the Maximum Entropy Prin iple (MEP) have beendeveloped (see [8 ℄ and [9℄) to ompute

p

C

from theonly available statisti al informationof the randomve tor

C

. Thisinformation an be seen asthe admissibleset

C

ad

for

p

C

:

C

ad

=



p

C

∈ F(D, R

∗

)

|

Z

D

p

C

(x)dx = 1,

∀ 1 ≤ n ≤ N,

Z

D

g

_n

(x)p

C

(x)dx = f

n

,

(1.23) where

{f

n

, 1

≤ n ≤ N}

gathers

N

ve tors whi h are respe tively asso iated with the ve tor-valued fun tions

{g

n

, 1

≤ n ≤ N}

. Hen e,the MPE allows building

p

C

asthe solutionof the optimizationproblem:

p

C

= arg max

p

C

∈C

ad



−

Z

D

p

C

(x) log (p

C

(x)) dx



.

(1.24)

Asanexample,ifthemaximumavailableinformationabout

C

isthefa tthatitsrealizations areinthe hyper ube

[

−1, 1]

M

,theadmissible set

C

ad

C

ad

=

(

p

C

∈ F([−1, 1]

M

, R

∗

),

|

Z

[−1,1]

M

p

C

(x)dx = 1

)

,

(1.25)

andit an beshownthatthePDF

p

C

thatmaximizes theoptimizationproblemdenedbyEq. (1.24) is theuniformPDF over

[

−1, 1]

M

:

p

C

(x) =

1

2

M

.

(1.26)

On the other hand, the indire t methods allow the onstru tion of the PDF

p

C

of the onsidered random ve tor

C

thanks to a transformation

T

of a known PDF

p

ξ

of a random ve tor

ξ

= ξ

1

, ..., ξ

N

g

of givendimension

N

g

≤ M

:

C

= t (ξ) ,

(1.27)

p

C

= T (p

ξ

) .

(1.28)

The onstru tionofthetransformation

t

isthusthekeypointoftheseindire tmethods. In this ontext,theisoprobabilisti transformationssu hastheNataftransformation(see[46 ℄)or theRosenblatttransformation(see [47 ℄) haveallowed thedevelopment of interestingresultsin these ondpartof thetwentieth enturybutarestilllimitedto very smalldimension ases and not to the high dimension ase onsidered inthis work. Nowadays, the mostpopularindire t methodsarethepolynomial haosexpansion(PCE)methods, whi hhavebeenrstintrodu ed byWiener [48 ℄ for sto hasti pro esses,and pioneered byGhanem and Spanos [49,22 ℄ forthe use of it in omputational s ien es. In the last de ade, this very promisingmethod has thus beenappliedinmanyworks(see,forinstan e [50 , 51 , 52 ,53 , 54 , 55,56 ,57 , 58 , 59 ,21 ,60 , 32 , 61 , 62 , 63 ,64 , 65 , 66 , 67 , 68 , 69 , 70,71, 72 ℄). The PCE is based on a dire t proje tionof the random ve tor

C

on a hosen Hilbertian basis

B

orth

=

{ψ

j

(ξ), 0

≤ j}

of all the se ond-order randomve tors withvaluesin

R

M

:

C

=

+∞

X

j=0

y

(j)

ψ

j

(ξ),

(1.29)

E [ψ

j

(ξ)ψ

k

(ξ)] = δ

jk

.

(1.30)

In pra ti alterms,the PCEof

C

hasto be trun ated to its

N + 1

mostinuentialterms:

C

_≈

N

X

j=0

y

(j)

ψ

j

(ξ).

(1.31)

In parti ular,inthefollowing, itwillbeassumed that

ψ

0

(ξ) = 1

,su h that:

y

(0)

= E [C] = µ

_C

.

(1.32)

Amethodto hoosethese

N

parti ulartermsandtoquantitytheamplitudeofthetrun ation residue,

P

_+∞

j=N +1

y

(j)

ψ

j

(ξ)

,hastherefore to be dened. Buildingthetransformation

t

requires at lastthe onstru tion of

N

deterministi oe ients,



y

(j)

, 1

≤ j ≤ N

, from the available informationabout

C

.

It has to be noti ed that in su h an approa h, any distribution for

ξ

an be hosen. For instan e, if the omponents of

ξ

are independent and uniformly distributed between -1 and

1, the orresponding Hilbertian basis,

{ψ

j

(ξ), 1

≤ j}

, is the set of the normalized Legendre polynomials.

When trying to identify in inverse the multidimensional distribution of an a priori non-Gaussianrandomve tor,thePCEmethodappearstobeverye ient,evenwhenthestatisti al dimensionof

C

ishigh. Indeed, thismethod an beappliedto anyrandomve tor,isnotbased on a priori formulations, and allows a very easy generation of independent realizations of

C

, on e the proje tion oe ients are identied. Indeed ea h independent realization of germ

ξ

leadsto an independentrealization of

C

.

1.5 PCE identi ation of random ve tors

In thisse tion,a des riptionof thePCE identi ationwith respe tto an arbitrary measure is given. The obje tive isto summarize thedierent key stepsof thePCE identi ationmethod and thewaythey an bepra ti ally implemented.

After havingdenedthetheoreti alframe ofthePCEidenti ation,the ost-fun tionsthat lead us to the omputation of thePCE oe ients



y

(1)

, . . . , y

(N )

arepresented, fora given trun ationparameter

N

. Two asesaredistinguished: thedire t ase, forwhi hthePCEgerm

ξ

isknown,and theindire t ase, forwhi hthe PCE germ isunknown. At last, to justify the hoi eofthistrun ationparameter,amethodtoperformthe onvergen eanalysisisintrodu ed. 1.5.1 Theoreti al frame

Let

C

= (C

1

, . . . , C

M

)

be an element of the spa e

L

2

P

Θ, R

M

of all the se ond-order

M

-dimensionrandomve torsdenedontheprobabilityspa e

(Θ,

T , P)

withvaluesin

R

M

,equipped withtheinnerprodu t

h·, ·i

. Itisassumedthat

ν

independentrealizations,

{C(θ

1

), . . . , C(θ

ν

)

}

, of

C

areknown and gatheredinthe

(M

× ν)

real matrix

[C

exp

_(ν)]

:

[C

exp

(ν)] = [C(θ

1

)

· · · C(θ

ν

)] .

(1.33)

Equation (1.31) an be rewrittenas:

C

_{− µ}

_C

_{≈ C}

chaos

(N ) = [y]Ψ(ξ),

(1.34)

[y] =

h

y

(1)

_{· · · y}

(N )

i

,

Ψ

_{(ξ) = (ψ}

₁

_{(ξ), . . . , ψ}

_N

_{(ξ)) .}

(1.35) The orthonormalityproperty of the proje tion basis

{ψ

j

(ξ), 1

≤ j ≤ N}

yields the ondi-tion:

E [Ψ(ξ, p)

_{⊗ Ψ(ξ, p)] = [I}

N

],

(1.36)

where

[I

N

]

isthe

(N

× N)

identitymatrix. Let

[R

chaos

CC

(N )]

bethe ovarian ematrixof entered randomve tor

C

chaos

_{(N )}

:

h

R

_CC

chaos

(N )

i

= E

h

C

chaos

(N )

_{⊗ C}

chaos

(N )

i

= [y]E [Ψ(ξ, p)

_{⊗ Ψ(ξ, p)] [y]}

T

= [y][y]

T

.

(1.37)

Tosimplifythenotations,itissupposedinthefollowingthat

C

isa enteredrandomve tor, su h that:

C

inthenext se tions.

1.5.2 Identi ation of the polynomial haos expansion oe ients Inthisse tion,aparti ular hoi eforthe

N

g

-dimensionPCEgerm,

ξ

= ξ

1

, . . . , ξ

N

g

,anda par-ti ularvalueofthetrun ationparameter

N

are onsidered. Let

[Ψ(ν

chaos

_)]

bethe

N

× ν

chaos

real matrixof independentrealizations of thetrun atedPCE basis

Ψ

(ξ)

:

[Ψ(ν

chaos

)] = [Ψ(ξ(Θ

1

))

· · · Ψ(ξ(Θ

ν

chaos

))] ,

(1.39) wheretheset

{ξ (Θ

1

) ,

· · · , ξ (Θ

ν

chaos

)

}

gathers

ν

chaos

independentrealizationsofrandomve tor

ξ

. As a dire t onsequen e of the orthonormalityof thePCE ve tor

Ψ

(ξ)

, matrix

[Ψ(ν

chaos

_)]

veries theasymptoti property:

lim

ν

chaos

_→+∞

1

ν

chaos

[Ψ(ν

chaos

_)][Ψ(ν

chaos

_)]

T

_{= E [Ψ(ξ)}

_{⊗ Ψ(ξ)] = [I}

N

].

(1.40) Dire t identi ation

Iftherealizationsof

C

aresolutionsofame hani alsystem,andif

ξ

orrespondstothevariable inputsofthissystem,then

ν = ν

chaos

andbothrealizationsof

C

,

{C(Θ

1

), . . . , C(Θ

ν

chaos

)

}

,and

Ψ

_(ξ)

,

{Ψ(ξ(Θ

1

)), . . . , Ψ(ξ(Θ

ν

chaos

))

}

,areknownat thesame time. Theyverify:

[C

ν

chaos

] = [C(Θ

1

)

· · · C(Θ

ν

chaos

)]

≈ [C

chaos

(N )] = [y][Ψ(ν

chaos

)].

(1.41) In this ase, two lassi almethodsaregenerallyusedto identifysu h oe ient matrix

[y]

:

•

Methods based on the empiri al estimation of the mean fun tion. From Eq. (1.31), asfamily

{ψ

j

(ξ), 1

≤ j}

is orthonormal,it an be seenthat forall

1

≤ j ≤ N

:

[y] = E [C

⊗ Ψ(ξ)]

≈ [y

opt

1

(ν

chaos

)] =

1

ν

chaos

ν

chaos

X

p=1

C(Θ

p

)

⊗ Ψ(ξ(Θ

p

)) =

1

ν

chaos

[C

chaos

_{(N )][Ψ(ν}

chaos

_)]

T

_.

(1.42)

•

Regression-based methods. Let

C([y], ν

chaos

₎

be the ost fun tionthat quantiesthe mean-squaredistan e between

C

and itsPCE approximation,

C

chaos

_{(N )}

,denedby:

C([y], ν

chaos

) =

_[C

chaos

(N )]

_{− [y][Ψ(ν}

chaos

)]

2

def

= Tr



[C

chaos

(N )]

− [y][Ψ(ν

chaos

)]

 

[C

chaos

(N )]

− [y][Ψ(ν

chaos

)]



T



,

(1.43) with

Tr [

·]

thetra e operator. PCE matrix

[y]

an therefore be sear hedastheargument that minimizes

C([y], ν

chaos

₎

. The ost fun tion

C([y], ν

chaos

₎

being onvex, it admits a minimum,

[y

opt

[y]

≈ [y

2

opt

(ν

chaos

)] = arg min

[y]

n

C([y], ν

chaos

)

o

,

(1.44)

[y

opt

₂

(ν

chaos

)] = [C

chaos

(N )][Ψ(ν

chaos

)]

T



[Ψ(ν

chaos

)][Ψ(ν

chaos

)]

T



−1

.

(1.45)

FromEqs (1.40),(1.42)and (1.45),it an be dire tly veriedthat thetwo formermethods give asymptoti allythesame results:

lim

ν

chaos

_→+∞

[y

opt

1

(ν

chaos

)] =

lim

ν

chaos

_→+∞

[y

opt

2

(ν

chaos

)].

(1.46) Indire t identi ation

If

C

is a random ve tor that gathers the proje tion oe ients of a random eld

X

on a parti ular basis, as it is the ase in this thesis, the realizations of

C

are dedu ed from the available realizations of

X

, su h that there is a priori no dire t link between the two sets of realizationsof

ξ

and

C

. Alternative methodshave thusto beused to identify

[y]

.

To this end, let

M

M,N

be the spa e of all the

(M

× N)

real matri es. For a given value of

[y

∗

_]

in

M

M,N

, the random ve tor

U

([y

∗

_{]) = [y}

∗

_{]Ψ (ξ)}

is a entered

M

-dimension random ve tor, for whi h theauto orrelationis equalto

[y

∗

_][y

∗

_]

T

. Let

p

U([y

∗

_])

be its multidimensional PDF.

Whentheonlyavailableinformationabout

C

islimitedtoasetof

ν

independentrealizations, themostgeneral andrelevant method to identifyininverse theoptimal oe ientsmatrix

[y]

, isto sear h itasthe argument thatmaximizes thelog-likelihood

L

U([y

∗

_])

([C

exp

(ν)])

of

U

([y

∗

])

at theexperimentalpointsgatheredin

[C

exp

_(ν)]

:

[y] = arg

max

[y

∗

_]∈M

M,N

L

U([y

∗

_])

([C

exp

(ν)]) ,

(1.47)

L

U([y

∗

_])

([C

exp

(ν)]) =

ν

X

n=1

log p

_U([y

∗

_])

(C(θ

_n

)) .

(1.48)

1.5.3 Pra ti al solving of the log-likelihood maximization

SolvingtheoptimizationproblemdenedbyEq. (1.47)hasrequiredthedevelopmentofspe i algorithms,whi hare des ribedinthisse tion.

The need for statisti al algorithms to maximize the log-likelihood Thelog-likelihood

L

U([y

∗

_])

([C

exp

(ν)])

being non onvex, deterministi algorithmssu has gradi-ent algorithms annot be applied to solve Eq. (1.47), and random sear h algorithms have to be used. Hen e, the pre ision of the PCE has to be orrelated to a numeri al ost

Z

, whi h orrespondsto a number ofindependenttrials of

[y

∗

]

in

M

M,N

. The higherthe value of

Z

is, thebetter the PCE identi ationshouldbe. Therefore, thisvaluehas to be hosen as highas possiblewhilerespe tingthe omputational resour e limitation. Let

Y =



[y

∗

]

(z)

, 1

≤ z ≤ Z

beasetof

Z

elements,whi hhavebeen hosenrandomlyin

M

M,N

. Foragivennumeri al ost

Z

,the mosta uratePCE oe ientsmatrix

[y]

isapproximatedby:

[y]

_{≈ [y}

_Y

] = arg max

[y

∗

]∈Y

L

U([y

Fromthe

ν

independentrealizations

{C(θ

1

), . . . , C(θ

ν

)

}

,the ovarian ematrix

[R

CC

]

of

C

an beestimated by:

[R

CC

]

≈ [ b

R

CC

(ν)] =

1

ν

ν

X

n=1

C(θ

n

)

⊗ C(θ

n

) =

1

ν

[C

exp

_(ν)][C

exp

_(ν)]

T

_.

(1.50) A goodwaytoimprovethee ien yofthenumeri al identi ationof

[y]

isthento restri t theresear hset to

O

C

⊂ M

M,N

,with:

O

C

=

n

[y

∗

] =

h

y

∗,(1)

,

· · · , y

∗,(N)

i

∈ M

M,N

| [y

∗

][y

∗

]

T

= [ b

R

CC

(ν)]

o

,

(1.51)

whi h,taking into a ount Eq. (1.37), guarantees by onstru tion that:

[R

_CC

chaos

(N )] = [ b

R

CC

(ν)].

(1.52) Hen e, the PCE oe ients matrix

[y]

an be approximated as the argument in

O

C

that maximizesthelog-likelihood

L

U([y

∗

_])

([C

exp

(ν)])

. Bydening

W

thesetthatgathers

Z

randomly raisedelementsof

O

C

,

[y]

an thenbeassessedasthesolutionofthenewoptimizationproblem:

[y]

_{≈ [y}

_W

] = arg max

[y

∗

]∈W

L

U([y

∗

_])

([C

exp

(ν)]) .

(1.53)

Approximation of the log-likelihood fun tion Froma parti ularmatrixof realizations

[Ψ(ν

chaos

_)]

(whi hisdenedinEq. (1.39)),if

[y

∗

_]

isan element of

O

C

,

ν

chaos

independent realizations



U

([y

∗

], Θ

p

) = [y

∗

]Ψ (ξ(Θ

p

)) , 1

≤ p ≤ ν

chaos

of randomve tor

U

([y

∗

_])

an be omputedand gathered inthematrix

[U ]

:

[U ] = [U ([y

∗

], Θ

1

)

· · · U ([y

∗

], Θ

ν

chaos

)] = [y

∗

][Ψ(ν

chaos

)].

(1.54) Hen e,usingGaussianKernels,thePDF

p

U([y

∗

_])

of

U

([y

∗

])

an bedire tlyestimatedbyits non parametri estimator

p

b

U

:

∀ x ∈ R

M

, p

_U([y

∗

_])

(x)

≈

b

p

U

(x) =

1

(2π)

M/2

ν

chaos

Q

M

m=1

h

m

ν

chaos

X

p=1

exp

₋

1

2

M

X

m=1



x

m

− U

m

([y

∗

], Θ

p

)

h

m



2

!

_,

(1.55)

where

h

= (h

1

,

· · · , h

M

)

isthe multidimensionaloptimalSilvermanbandwidthve tor (see [2 ℄) of theKernelsmoothingestimationof

p

U([y

∗

_])

:

∀ 1 ≤ m ≤ M, h

m

=

bσ

U

m



4

(2 + M )ν

chaos



1/(M +4)

,

(1.56) where

σ

b

U

m

istheempiri alestimationofthestandarddeviationofea h omponent

U

m

of

U

. It hastobenoti edthat

p

b

U

onlydependsonthebandwidthve tor

h

,andthetwomatri es

[y

∗

_]

and

[Ψ(ν

chaos

)]

. Hen e, a ording to theEqs. (1.48), (1.54) and (1.55), for a given valueof

ν

chaos

, themaximizationofthelog-likelihoodfun tion

L

U([y

∗

_])

anberepla edbythemaximizationof the ost-fun tion

C([C

exp

_{(ν)], [y}

∗

_{], [Ψ(ν}

chaos

_)])

su hthat:

[y]

≈ [y

O

C

] = arg max

[y

∗

]∈O

C

C([C

exp

(ν)], [y

∗

], [Ψ(ν

chaos

)]) =

_C

C

+

C

V

([C

exp

(ν)], [y

∗

], [Ψ(ν

chaos

)]),

(1.58)

C

C

=

−ν ln

(2π)

M/2

ν

chaos

M

Y

m=1

h

m

!

,

(1.59)

C

V

([C

exp

(ν)], [y

∗

], [Ψ(ν

chaos

)]) =

ν

X

n=1

ln





ν

chaos

X

p=1

exp

−

1

2

M

X

m=1



C

m

(θ

n

)

− U

m

([y

∗

], Θ

p

)

h

m



2

!

 .

(1.60) Hen e, theoptimizationproblemdenedbyEq. (1.53) an nallybe estimatedby:

[y]

≈ [y

_O

Z

C

] = arg max

[y

∗

]∈W

C



[C

exp

(ν)], [y

∗

], [Ψ(ν

chaos

)]



.

(1.61)

A ura y of the PCE identi ation

Foragiven omputation ost

Z

andagiven value forthetrun ationparameter

N

,let

[y

Z

O

C

]

be anoptimalsolutionofEq. (1.61).

[y

Z

O

C

]

isanumeri alestimationofthePCE oe ientsmatrix

[y]

. Foranew

N

× ν

chaos,∗

real matrix

[Ψ

∗

_(ν

chaos,∗

_)]

of independentrealizations (

ν

chaos,∗

an behigherthan

ν

chaos

),therobustnessof

[y

Z

O

C

]

regardingthe hoi eof

[Ψ(ν

chaos

_)]

anthenbe es-timatedby omparing

C



[C

exp

(ν)], [y

Z

_O

_C

], [Ψ(ν

chaos

)]



and

C



[C

exp

(ν)], [y

_O

Z

_C

], [Ψ

∗

(ν

chaos,∗

)]



. If

ν

newindependentrealizationsof

C

wereavailableandgatheredinthematrix

[C

exp,new

_(ν)]

,the over-learning of the method ould be measured by omparing

C



[C

exp

(ν)], [y

Z

_O

_C

], [Ψ(ν

chaos

)]



and

C



[C

exp,new

(ν)], [y

Z

_O

_C

], [Ψ(ν

chaos

)]



. Atlast,forthesamevaluefor

Z

,if

[y

Z,new

O

C

]

isanew op-timalsolutionofEq. (1.61),theglobala ura yoftheidenti ationstemsfromthe omparison between

C



[C

exp,new

(ν)], [y

_O

Z

_C

], [Ψ

∗

(ν

chaos,∗

)]



and

C



[C

exp,new

(ν)], [y

_O

Z,new

_C

], [Ψ

∗

(ν

chaos,∗

)]



.

1.5.4 Identi ation of the PCE trun ation parameters

Asshown inSe tion1.4 , twotrun ationparameters,

N

g

and

N

,appearinthetrun atedPCE,

C

chaos

(N )

,of

C

. Amethodto hoosethesize

N

g

andthese

N

elementsfrombasis

B

orth

aswell asa methodto quantifytherelevan eof su h a

N

-dimensionbasishave thusto bedened. Restri tion of the admissible proje tion basis

In this work, only polynomial basis are addressed, su h that for

1

≤ j

, a parti ular element

ψ

j

(ξ)

in

B

orth

an be writtenunder theform:

ψ

j

(ξ) =

+∞

X

q=1

c

(j)

_q

ξ

α

(q)

1

1

× · · · × ξ

α

(q)

_Ng

N

g

,



α

(q)

₁

, . . . , α

(q)

_N

_g



_{∈ N}

N

g

_.

(1.62) In addition,the lassi alassumption thatthemostinuentialelementsof

B

orth

orrespond totheelementsoflowesttotalpolynomialorderisintrodu edinthiswork. Let

p

bethemaximal polynomialorderof theproje tionbasis, su h thatfor

1

≤ j ≤ N

,we hoose: