Identification of polynomial chaos representations in high dimension from a set of realizations

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Submitted on 4 Jan 2013

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high dimension from a set of realizations

Guillaume Perrin, Christian Soize, Denis Duhamel, Christine Fünfschilling

To cite this version:

Guillaume Perrin, Christian Soize, Denis Duhamel, Christine Fünfschilling. Identification of poly- nomial chaos representations in high dimension from a set of realizations. SIAM Journal on Sci- entific Computing, Society for Industrial and Applied Mathematics, 2012, 34 (6), pp.A2917-A2945.

�10.1137/11084950X�. �hal-00770006�

G. PERRIN

∗† ‡

,C. SOIZE

∗

, D. DUHAMEL

†

, AND C. FUNFSCHILLING

‡

Abstrat.

Thispaper deals with the identiation in high dimensionof polynomial haos expansionof

randomvetorsfromasetofrealizations. Duetonumerialandmemoryonstraints,theusualpoly-

nomialhaosidentiation methodsarebasedon aseriesof trunations thatinduesanumerial

bias. Thisbiasbeomesverydetrimentaltothe onvergeneanalysisofpolynomialhaosidenti-

ationinhighdimension. Thispaperthereforeproposesanewformulationoftheusualpolynomial

haosidentiationalgorithmstoavoidthisnumerialbias.Afterareviewofthepolynomialhaos

identiationmethod,theinueneofthenumerialbiasontheidentiationaurayisquantied.

Thenewformulationisthendesribedindetails,andillustratedontwoexamples.

Keywords.polynomialhaosexpansion,highdimension,omputation.

AMSsubjet lassiations.60H35,60H15,60H25,60H40,65C50

1. Introdution. Inspiteofalwaysmoreauratenumerialsolvers,determin-

istimodelsarenotabletorepresentmostoftheexperimental data,whiharevari-

ableandoftenunertainbynature. Hene,theappliationeldsofnondeterministi

modeling,whihantakeintoaountthemodelparametersvariabilityaswellasthe

modelerrorunertainties,haskeptinreasing. Unertaintiesarethereforeintrodued

inomputationalmehanialmodelswithmoreandmoredegreesoffreedom. Inthis

ontext,theharaterizationoftheprobabilitydistribution

P

η

(dx)

^of

N

η^-dimension

randomvetor

η

^fromsetsofexperimentalmeasurementsisboundtoplayakeyrole, inpartiular,inhighdimension,thatistosayforalargevalueof

N

η^{. Inthiswork,it}

isassumed that

P

η

(dx) = p

η

(x)dx

^inwhihtheprobabilitydensityfuntion(PDF)

p

η ^{is afuntion in theset}

F ( D , R

⁺

)

^{of all the}positive-valued funtions dened on anypart

D

^of

R

^Nη andforwhihintegralover

D

^is1.

Two kinds of methods an be used to build suh a PDF: the diret and the

indiretmethods. Amongthediretmethods,thePriorAlgebraiStohastiModeling

(PASM) methods postulate an algebrai representation

η ≈ t

^alg

(Ξ, w)

^{, with}

t

^{alg a}

prior transformation,

Ξ

^{a given random vetor and}

w

^{a vetor of parameters to}

identify. Inthe same ategory, the methods based on the Information Theory and

theMaximumEntropyPriniple (MEP) havebeendevelopped(see[13℄ and[27℄)to

ompute

p

η^{fromtheonlyavailable}informationofrandomvetor

η

^{. This}information anbeseenastheadmissibleset

C

^adfor

p

η^:

C

^ad

=

p

η

∈ F ( D , R

⁺

) | Z

D

p

η

(x)dx = 1,

∀ 1 ≤ m ≤ M, Z

D

g

_m

(x)p

η

(x)dx = f

_m

,

^(1.1)

∗

UniversitéParis-Est, Modélisationet SimulationMulti-Éhelle (MSMEUMR8208 CNRS),5

Bd.Desartes,77454Marne-la-Vallée,Frane(hristian.soizeuniv-paris-est.fr).

†

UniversitéParis-Est,Navier(ENPC-IFSTTAR-CNRSUMR8205),EoleNationaledesPontset

Chaussées,6et8AvenueBlaisePasal,CitéDesartes,ChampssurMarne,77455Marne-la-Vallée,

Cedex2,Frane(denis.duhamelenp.fr)

‡

SNCF,Innovation and Researh Department, Immeuble Lumière, 40avenue desTerroirs de

Frane,75611,Paris,Cedex12,Frane(guillaume.perrinsnf.fr,hristine.funfshillingsnf.fr).

where

{ f

_m

, 1 ≤ m ≤ M }

^gathers

M

^{givenvetorswhih are}respetivelyassoiated with a given vetor-valued funtions

{ g

_m

, 1 ≤ m ≤ M }

^{. Hene, the MPE allows}

building

p

η ^{asthesolutionofthe}optimizationproblem:

p

η

= arg max

pη∈C^ad

− Z

D

p

η

(x) log (p

η

(x)) dx

.

^(1.2)

Ontheotherhand,theindiretmethodsallowtheonstrutionofthePDF

p

η^of

theonsidered random vetor

η

^{from a}transformation

t

^{of aknownrandom vetor}

ξ = ξ

1

, ..., ξ

Ng

ofgivendimension

N

g

≤ N

η^:

η = t (ξ) ,

^(1.3)

dening atransformation

T

between

p

η ^andthePDF

p

ξ ^of

ξ

^:

p

η

= T (p

ξ

) .

^(1.4)

The onstrution of the transformation

t

^{is thus the key point of these indi-}

retmethods. Inthis ontext,theisoprobabilistitransformationssuh astheNataf

transformation(see[20℄)ortheRosenblatttransformation(see[23℄)haveallowedthe

developmentofinterestingresultsintheseondpartofthetwentiethenturybutare

stilllimitedtoverysmalldimensionasesandnottothehighdimensionaseonsid-

eredinthiswork. Nowadays,themostpopularindiretmethodsarethepolynomial

haosexpansion(PCE)methods,whihhavebeenrstintroduedbyWiener[33℄for

stohastiproesses,andpioneered byGhanemandSpanos[10℄ [11℄fortheuseof it

in omputational sienes. In thelast deade, this verypromising method has thus

been applied in many works (see, for instane [1℄, [2℄, [3℄, [4℄, [5℄, [7℄, [8℄, [9℄, [12℄,

[14℄,[15℄, [16℄,[19℄,[18℄,[17℄,[21℄,[22℄,[24℄, [26℄,[28℄,[31℄,[32℄, [25℄,[34℄). ThePCE

is basedon adiret projetionof therandom vetor

η

^{on ahosen hilbertian basis}

B

^orth

=

ψ

α

(ξ), α ∈ N

^Ng ofalltheseond-orderrandomvetorswithvaluesin

R

^Nη:

η = X

α∈NNg

y

^(α)

ψ

α

(ξ),

^(1.5)

ξ 7→ ψ

α

(ξ) = X

α1

(ξ

1

) ⊗ ... ⊗ X

α_Ng

(ξ

Ng

),

^(1.6)

where

x 7→ X

αℓ

(x)

^{isthenormalizedpolynomialbasisofdegree}

α

ℓ^{assoiatedwiththe}

PDF

p

ξℓ ^{oftherandomvariable}

ξ

ℓ^,and

α

^isthemulti-indexofthemultidimensional polynomialbasiselement

ψ

α

(ξ)

^{. Buildingthe}transformation

t

^{requiresthereforethe}

onstrutionoftheprojetionvetors

y

^(α)

, α ∈ N

^Ng .

Thepresentwork is devoted tothe identiationin high dimensionof thePCE

oeients

y

^(α)

, α ∈ N

^Ng , when the only available information on the random vetor

η

^isasetof

ν

^expindependentrealizations

η

⁽¹⁾

, · · · , η

^{(νexp) .}

Inpratie,thePCEof

η

^{hasrsttobetrunated:}

η ≈ η

^chaos

(N) = X

α∈A_p

y

^(α)

ψ

α

(ξ),

^(1.7)

A

_p

=

 

 α = α

1

, ..., α

Ng

| | α | =

Ng

X

ℓ=1

α

ℓ

≤ p

 

 = n

α

⁽¹⁾

, · · · , α

^(N)

o

,

^(1.8)

where

η

^chaos

(N)

^{is the projetion of}

η

^{on the}

N

^{-dimension subspae spanned by}

{ ψ

α

(ξ), α ∈ A

_p

} ⊂ B

^{orth. It an be notied that}

N

^{inreases veryquikly with}

respet tothe dimension

N

g ^of

ξ

^{and themaximumdegree}

p

^{ofthetrunated basis}

{ ψ

α

(ξ), α ∈ A

^p

}

^,as:

N = (N

g

+ p)!/ (N

g

! p!) .

^(1.9)

Methodstoperformtheonvergeneanalysisinhighdimensionwithrespettoa

givenerrorthresholdonthePCEresidue

η − η

^chaos

(N )

^{arethereforeofgreatonern}

tojustifythetrunationparameters

N

g ^and

p

^.

In this prospet, the artile [29℄ provides advaned algorithms to ompute the

PCEoeientsfromthe

ν

^exp independentrealizations

η

⁽¹⁾

, · · · , η

^{(νexp) byfous-}

ing on the maximization of the likelihood. In partiular, one of the key point of

these algorithms is the alulation of

N × ν

^chaos

real matrix

[Ψ]

^of independent realizationsofthetrunatedPCEbasis

{ ψ

α

(ξ) , α ∈ A

_p

}

^:

[Ψ] = [Ψ (ξ (θ

1

) , p) · · · Ψ (ξ (θ

ν^chaos

) , p)] ,

^(1.10)

Ψ (ξ, p) = ψ

_α(1)

ξ

1

, · · · , ξ

Ng

, · · · , ψ

_α(N)

ξ

1

, · · · , ξ

Ng

,

^(1.11)

where the set

{ ξ (θ

1

) , · · · , ξ (θ

ν^chaos

) }

^gathers

ν

^chaos independent realizations of the randomvetor

ξ

^.

Reurrene formula or algebrai expliit representations are generally used to

omputesuhmatrix

[Ψ]

^{,whih aresupposedtoverifythe}asymptotialproperty:

ν^chaos

lim

→+∞

1

ν

^chaos

[Ψ][Ψ]

^T

= [I

N

],

^(1.12)

asadiretonsequeneoftheorthonormalityofthePCEbasis

{ ψ

α

, α ∈ A

_p

}

^,where

[I

N

]

^isthe

N

^{-dimensionidentitymatrix.}

However,fornumeriallyadmissiblevaluesof

ν

^{chaos(between1000and10000),it}

hasbeenshownin[30℄thatthedierene

1

ν^chaos

[Ψ][Ψ]

^T

− [I

N

]

^{anbeverysigniant}

whenhigh valuesofthemaximumdegree

p

^{anbeenountered with}simultaneously signiantvaluesof

N

g^{. Thisdiereneinduesa}detrimentalbiasinthePCEidenti-

In[30℄,itisthereforeproposedamethod usingsingularmatrixdeompositiontonu-

meriallyadaptlassialgenerationsof

[Ψ]

^{,and makethisdierenebezeroforany}

valuesof

p

^and

N

g^. Nevertheless,thisonditionningon

[Ψ]

^{modiestheinitialstru-}

tureof

[Ψ]

^{, and makestheidentied PCE oeients}

y

^(α)

, α ∈ A

_p impossibleto bereusedonanothermatrix

[Ψ

^∗

]

^of

ν

^{chaos,∗ new}realizationsof

Ψ(ξ, p)

^.

As an extension of the works desribed in [29℄ and [30℄, this artile proposes

anoriginal deomposition of the PCE oeients

y

^(α)

, α ∈ A

_p , that redues the numerial bias introdued during the identiation by the nite dimension of

[Ψ]

andfor largevaluesof degree

p

^{. Thisnew}formulationispartiulary adaptedto the highdimension,andallowstheidentiedoeientstobereusedforothermatrixof

realizations

[Ψ

^∗

]

^.

InSetion2,thePCEidentiationfromasetofexperimentaldatawithanarbi-

trarymeasureisdesribed. Inpartiular,theroleplayedbythematrixofindependent

realizations

[Ψ]

^isemphasized. Setion3fousesontheonvergenepropertiesofthis matrix

[Ψ]

^{with respet to three statistial measures, and desribes an innovative}

method togeneratethismatrixwithoutusing omputationalreurreneformulanor

algebraiexpliitrepresentation. InSetion4,thenewformulationofthePCEiden-

tiationproblemisgiven. Finally,arepresentedinSetion5twoappliationsofthe

formermethodwithaGaussianmeasure.

2. PCE identiation of random vetors from a set of independent

realizations. Inthissetion,adesriptionofthePCEidentiationwithrespetto

anarbitrarymeasureis given. The objetiveis tosummarizethedierentkeysteps

ofthePCEidentiationmethodandthewaytheyarepratiallyimplemented.

After having dened the theoretial frame of the PCE identiation, the ost-

funtion that leads to the omputation of the PCE oeients

y

^(α)

, α ∈ A

_p is presented, for given trunation parameters

N

g ^and

p

^{. At last, to justify the hoie}

of these trunation parameters, a method to perform the onvergene analysis is

introdued.

2.1. Theoretialframe. Let

(Θ, T , P )

^beaprobabilityspae. Let

L

²_P

Θ, R

^Nη

bethespaeofalltheseond-order

N

η^{-dimensionrandomvetorsdenedon}

(Θ, T , P )

withvaluesin

R

^Nη, equippedwiththeinnerprodut

h ., . i

^:

h U , V i =

Z

Θ

U

^T

(θ)V (θ)dP (θ) = E U

^T

V

, ∀ U, V ∈ L

²_P

Θ, R

^Nη

,

^(2.1)

where

E (.)

^isthemathematialexpetation.

Let

η = η

1

, · · · , η

Nη

be an element of

L

²_P

Θ, R

^Nη

. It is assumed that

ν

^exp

independent realizations

η

⁽¹⁾

, · · · , η

^{(νexp) of}

η

^{are known and gathered in the}

(N

η

× ν

^exp

)

^{real matrix}

[η

^exp

]

^:

[η

^exp

] = h

η

⁽¹⁾

· · · η

^(νexp)

i

.

^(2.2)

Equation(1.7)anberewrittenas:

η

^chaos

(N ) = [y]Ψ(ξ, p),

^(2.3)

[y] = h

y

^(α(1))

· · · y

^(α(N))

i

.

^(2.4)

Theorthonormalitypropertyoftheprojetionbasis

{ ψ

α

(ξ), α ∈ A

_p

}

^yieldsthe

ondition:

E Ψ(ξ, p)Ψ(ξ, p)

^T

= [I

N

].

^(2.5)

Sine

ψ

_α⁽¹⁾

(ξ) = 1

^{,itanbeseenthat:}

E η

^chaos

(N )

= y (

^α(1)

) .

^(2.6)

Let

[R

η

]

^and

[R

^chaos_η

(N)]

^betheautoorrelationmatrixoftherandomvetors

η

and

η

^chaos

(N )

^:

[R

η

] = E ηη

^T

,

^(2.7)

R

^chaosη

(N)

= E

η

^chaos

(N ) η

^chaos

(N )

^T

= [y]E Ψ(ξ, p)Ψ(ξ, p)

^T

[y]

^T

= [y][y]

^T

.

(2.8)

2.2. Identiation of the polynomial haos expansion oeients. In

thissetion,partiularvaluesofthetrunationparameters

N

g ^and

p

^{areonsidered.}

Let

M

^{NηN bethespaeofallthe}

(N

η

× N )

^{realmatries. Foragivenvalueof}

[y

^∗

]

ⁱⁿ

M

^NηN^{, therandomvetor}

U ([y

^∗

]) = [y

^∗

]Ψ (ξ, p)

^isa

N

η^{-dimensionrandomvetor,}

forwhihtheautoorrelationisequalto

[y

^∗

][y

^∗

]

^{T. Let}

p

U([y^∗])^beitsmultidimensional PDF.

Whentheonlyavailableinformationon

η

^isasetof

ν

^expindependentrealizations, theoptimaloeientsmatrix

[y]

^{ofitstrunated PCE,}

η

^chaos

(N ) = [y]Ψ(ξ, p)

^,an

beseenastheargumentwhihmaximizesthelog-likelihood

L

^U([y∗])

([η

^exp

])

^of

U ([y

^∗

])

^:

[y] = arg max

[y^∗]∈MNη N

L

^U([y∗])

([η

^exp

]) ,

^(2.9)

L

^U([y∗])

([η

^exp

]) =

ν

X

^exp

i=1

ln p

U([y^∗])

η

⁽ⁱ⁾

.

^(2.10)

2.3. Pratial solving ofthe log-likelihoodmaximization.

2.3.1. Theneedforstatistialalgorithmstomaximizethelog-likelihood.

Thelog-likelihood

L

U([y^∗])

([η

^exp

])

^{beingnon-onvex,}deterministialgorithmssuhas gradient algorithms annot be applied to solveEq. (2.9), and random searh algo-

rithms have to beused. Hene, thepreision ofthe PCE has to beorrelated to a

numerial ost

M

^{, whih orresponds to a number of} independent trials of

[y

^∗

]

ⁱⁿ

M

^NηN^{. Let}

Y =

[y

^∗

]

^(r)

, 1 ≤ r ≤ M

^{be a set of}

M

^{elements, whih have been}

hosenrandomly in

M

^{NηN. Foragiven numerialost}

M

^{,the mostauratePCE}

oeientsmatrix

[y]

^isapproximatedby:

[y] ≈ [y

_Y

] = arg max

[y^∗]∈Y

L

^U([y∗])

([η

^exp

]) .

^(2.11)

2.3.2. Restritionofthemaximizationdomain. Fromthe

ν

^expindependent realizations

η

⁽¹⁾

, · · · , η

^{(νexp) ,themeanvalue}

E(η)

^andtheautoorrelationmatrix

[R

η

]

^of

η

^{anbeestimatedby:}

E (η) ≈ η(ν b

^exp

) = 1 ν

^exp

ν^exp

X

i=1

η

⁽ⁱ⁾

,

^(2.12)

[R

η

] ≈ [ R b

η

(ν

^exp

)] = 1 ν

^exp

ν^exp

X

i=1

η

⁽ⁱ⁾

η

⁽ⁱ⁾

T

= 1

ν

^exp

[η

^exp

][η

^exp

]

^T

.

^(2.13)

Agoodwaytoimprovetheeienyofthenumerialidentiationof

[y]

^isthen

torestrittheresearhsetto

O

^η

⊂ M

^NηN,with:

O

^η

= n [y] = h

y

^(α(1))

, · · · , y

^(α(N))

i

∈ M

^NηN

| y

^(α(1))

= b η(ν

^exp

), [y][y]

^T

= [ R b

η

(ν

^exp

)] o

,

(2.14)

whih,takingintoaountEqs. (2.6)and(2.8),guaranteesbyonstrutionthat:

[R

^chaos_η

(N )] = [ R b

η

(ν

^exp

)], E η

^chaos

(N )

= η(ν b

^exp

).

^(2.15)

Hene,thePCE oeientsmatrix

[y]

^anbeapproximatedastheargumentin

O

^{η that maximizes the} log-likelihood

L

^U([y∗])

([η

^exp

])

^{. By dening}

W

^{the set that}

gathers

M

^{randomlyraisedelementsof}

O

^η,

[y]

^{anthen be assessedasthesolution}

ofthenewoptimization problem:

[y] ≈ [y

_W

] = arg max

[y^∗]∈W

L

U([y^∗])

([η

^exp

]) .

^(2.16)

2.3.3. Approximation of the log-likelihood funtion. From a partiular

matrix of realizations

[Ψ]

^{(whih is dened in Eq. (1.10)), if}

[y

^∗

]

^{is an element of}

O

^η,

ν

^chaos independent realizations

U ([y

^∗

], θ

n

) = [y

^∗

]Ψ (ξ(θ

n

), p) , 1 ≤ n ≤ ν

^chaos

oftherandomvetor

U ([y

^∗

])

^{anbeomputedandgatheredinthematrix}

[U ]

^:

[U ] = [U ([y

^∗

], θ

1

) · · · U ([y

^∗

], θ

ν^chaos

)] = [y

^∗

][Ψ].

^(2.17)

Hene,usingGaussian Kernels,thePDF

p

_U([y^∗])^of

U ([y

^∗

])

^{anbediretlyesti-}

matedbyitsnonparametriestimator

p b

U^:

∀ x ∈ R

^Nη

, p

U([y^∗])

(x) ≈ b

p

U

(x) = 1

(2π)

^Nη/2

ν

^chaos

Q

Nη

k=1

h

k ν

X

^chaos

n=1

exp



 − 1 2

Nη

X

k=1

x

k

− U

k

([y

^∗

], θ

n

) h

k

2



 ,

(2.18)

where

h = h

1

, · · · , h

Nη

isthemultidimensionnaloptimalSilvermanbandwithvetor

(see[6℄)oftheKernelsmoothingestimationof

p

U([y^∗])^:

∀ 1 ≤ k ≤ N

η

, h

k

= b σ

Uk

4 (2 + N

η

)ν

^exp

1/(Nη+4)

,

^(2.19)

where

b σ

Uk ^{is theempirial estimation of thestandard deviation of eah omponent}

U

k ^of

U

^{. Ithastobenotied that}

p b

U ^{onlydependsonthebandwidthvetor}

h

^,and

thetwomatries

[y

^∗

]

^and

[Ψ]

^{. Hene,aordingtotheEqs. (2.10),(2.17)and(2.18),}

for a given value of

ν

^{chaos, the} maximizationof the log-likelihood funtion

L

^U([y∗])

anbereplaedbythemaximizationoftheost-funtion

C ([η

^exp

], [y

^∗

], [Ψ])

^suhthat:

[y] ≈ [y

_Oη

] = arg max

[y^∗]∈O^η

C ([η

^exp

], [y

^∗

], [Ψ]),

^(2.20)

where:

C ([η

^exp

], [y

^∗

], [Ψ]) = C

^C

+ C

^V

([η

^exp

], [y

^∗

], [Ψ]),

^(2.21)

C

^C

= − ν

^exp

ln



(2π)

^Nη/2

ν

^chaos

Nη

Y

k=1

h

k



 ,

^(2.22)

C

^V

([η

^exp

], [y

^∗

], [Ψ]) =

ν

X

^exp

i=1

ln





ν

X

^chaos

n=1

exp



 − 1 2

Nη

X

k=1

η

_k⁽ⁱ⁾

− U

k

([y

^∗

], θ

n

) h

k

!

2







 .

^(2.23)

Hene,theoptimization problem dened byEq. (2.16)annallybeestimated

by:

[y] ≈ [y

_O^M_η

] = arg max

[y^∗]∈W

C ([ν

^exp

], [y

^∗

], [Ψ]) .

^(2.24)

The optimization problem dened by Eq. (2.24) is now supposed to be solved

withtheadvanedalgorithmsdesribedin[29℄tooptimizethetrialsoftheelementsof

W

^{foragivenomputationost}

M

^{. Thehigherthevalueof}

M

^{is,thebetterthePCE}

identiation should be. Therefore, this value has to behosenas high aspossible

2.3.4. Auray of the PCE identiation. For agiven omputation ost

M

^{, let}

[y

^M_Oη

]

^{be an optimal solution of Eq. (2.24).}

[y

^M_Oη

]

^{is a numerialestimation}

of the PCE oeients matrix

[y]

^{. For a new}

N × ν

^chaos,∗

real matrix

[Ψ

^∗

]

^of

independentrealizations(

ν

^{chaos,∗ anbe higherthan}

ν

^{chaos),therobustnessof}

[y

^M_Oη

]

regardingthehoieof

[Ψ]

^{anthenbeestimated byomparing}

C

[η

^exp

], [y

_O^M_η

], [Ψ]

and

C

[η

^exp

], [y

^M_Oη

], [Ψ

^∗

]

. In addition, if

ν

^{exp new} independent realizations of

η

wereavailableandgatheredin thematrix

[η

^exp,new

]

^,theover-learningofthemethod ould be measured by omparing

C

[η

^exp

], [y

^M_Oη

], [Ψ]

and

C

[η

^exp,new

], [y

_O^M_η

], [Ψ]

.

Atlast,forthesameomputationost

M

^,if

[y

^M,new_Oη

]

^{isanewoptimalsolutionofEq.}

(2.24), the globalauray ofthe identiationstems from theomparison between

C

[η

^exp,new

], [y

^M_Oη

], [Ψ

^∗

]

and

C

[η

^exp,new

], [y

^M,new_Oη

], [Ψ

^∗

]

.

2.4. Identiation of the PCE trunation parameters. As shown in In-

trodution, two trunation parameters,

N

g ^and

p

^{, appear in the trunated PCE,}

η

^chaos

(N ) = [y]Ψ(ξ, p)

^{, of}

η

^{. Thevaluesofthese parametershaveto bedetermined}

from aonvergene analysis. Theobjetiveofthis setionisthus togivethe funda-

mentalelementstoperformsuhaonvergeneanalysis.

2.4.1. Denitionof a log errorfuntion. Foreahomponent

η

_k^chaos

(N)

^of

the trunated PCE,

η

^chaos

(N ) = [y]Ψ(ξ, p)

^{, of}

η

^{, the}

L

^{1-log error funtion}

err

k ^is

introduedasdesribedin [29℄:

∀ 1 ≤ k ≤ N

η

, err

k

(N

g

, p) = Z

BIk

| log

10

(p

ηk

(x

k

)) − log

10

p

_η^chaos

k

(x

k

)

| dx

k

,

^(2.25)

where:

• BI

k ^{isthesupportof}

η

^exp_k ^;

• p

ηk ^and

p

_η^chaos

k

arethePDF of

η

k ^and

η

^chaos_k respetively.

Themultidimensional errorfuntion

err(N

g

, p)

^{isthen dedued from theunidi-}

mensional

L

^{1-logerrorfuntion as:}

err(N

g

, p) =

Nη

X

k=1

err

k

(N

g

, p).

^(2.26)

The parameters

N

g ^and

p

^{havethus to be determined to minimizethe multidi-}

mensional

L

^{1-logerrorfuntion}

err(N

g

, p)

^.

For given values of trunation parameters

N

g ^and

p

^{, it is reminded that PCE}

oeients matrix

[y]

^{is searhed in order to maximize the} multidimensional log- likelihoodfuntion,whihallowsustoonsideraprioristronglyorrelatedproblems.

One this matrix

[y]

^{is identied, it is possible to generate as many} independent realizationsoftrunatedPCE

η

^chaos

(N )

^{asneededtoestimateaspreiselyaspossible}

thenon parametriestimator

p b

U ^{of its} multidimensionalPDF. Thenumber

ν

^{exp of}

available experimentalrealizationsof

η

^{ishoweverlimited. Thisnumberisgenerally}

toosmall for thenon parametri estimatorof multidimensional PDF

p

η ^of

η

^{to be}

relevant,whereasit ismostof thetime largeenoughtodene theestimatorsof the

marginalsof

p

η^{. Therefore,thelog-errorfuntions denedbyEqs. (2.25)and(2.26)}