Non-polynomial expansion for stochastic problems with non-classical pdfs

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non-classical pdfs

Remi Abgrall, Pietro Marco Congedo, Gianluca Geraci, Gianluca Iaccarino

To cite this version:

Remi Abgrall, Pietro Marco Congedo, Gianluca Geraci, Gianluca Iaccarino. Non-polynomial ex- pansion for stochastic problems with non-classical pdfs. [Research Report] RR-8191, INRIA. 2012.

�hal-00766760�

0 2 4 9 -6 3 9 9 IS R N IN R IA /R R -- 8 1 9 1 -- F R + E N G

RESEARCH REPORT N° 8191

December 18, 2012

Non-polynomial

expansion for stochastic problems with

non-classical pdfs

Remi Abgrall, Pietro Marco Congedo, Gianluca Geraci, Gianluca

Iaccarino

RESEARCH CENTRE BORDEAUX – SUD-OUEST 351, Cours de la Libération

RemiAbgrall,Pietro Maro Congedo, GianluaGerai,

GianluaIaarino

Projet-TeamBahus

ResearhReport n°8191Deember18,201223pages

Abstrat: Inthisstudy, somepreliminary resultsaboutthepossibilitytoextendthelassial

polynomialChaos(PC) theorytostohastiproblemswithnon-lassialprobabilitydistributions

ofthevariables,i.e. outsidetheframeworkofthelassialWiener-Askeysheme[1℄,arepresented.

Theproposedstrategyallowstoobtainananalytialrepresentationofthesolutioninordertobuild

a metamodel or to ompure onditional statistis. Various numerial results obtained on some

analytialproblemsarethenprovidedtodemonstratetheorretnessofthepresentedapproah.

Key-words: PolynomialChaos,Colloation,Unertainty Quantiation,ANOVA,Conditional

statistis

non-lassiques

Résumé : Dans ette étude, on présente des résultats préliminaires sur la

possibilité d'utiliser la théorie lassique du Chaos Polynomial pour des prob-

lèmesstohastiquesavedesfontionsdensitédeprobabiliténon-lassiques. La

stratégie proposéepermet de aluler une représentation analytique de laso-

lution pour onstruire un metamodèle ou aluler les statistiques. Plusieurs

résultatsnumériquessontprésentés pourillustrerlavaliditédel'approhepro-

posée.

Mots-lés : ChaosPolynomial,Colloation,Quantiationdesinertitudes,

ANOVA

Contents

1 Introdution 5

2 Mathematialand problem setting 5

2.1 ANOVAdeompositionandsensitivityindexes . . . 7

2.1.1 Sobolsensitivityindies . . . 7

3 ClassialPolynomial Chaosapproah 8

3.1 SensitivityomputationfromPCexpansion . . . 9

4 Anon polynomialapproahfor highorder statistis 10

4.1 PolynomialVsNon-Polynomialapproah. . . 12

4.2 ComputationoftheonditionalstatistisbythenPC. . . 14

5 Numerialresults 16

6 Conlusionsand perspetive 20

7 Aknowledgements 22

List of Figures

1 ExtratermswiththenPCapproah . . . 14

2 ColloaltionvsnPCforthemonodimensionalproblem . . . 17

3 ColloaltionvsnPCfor

f pol

^withpdf

p 1

⁽

d = 2

^{). . . . . . . . . . 18}

4 ColloaltionvsnPCfor

f pol

^withpdf

p 2

⁽

d = 2

^{). . . . . . . . . . 18}

5 ColloaltionvsnPCfor

f pol

^withpdf

p 3

⁽

d = 2

^{). . . . . . . . . . 19}

6 ColloaltionvsnPCfor

f pol

^withpdf

p 1

⁽

d = 3

^{). . . . . . . . . . 19}

7 ColloaltionvsnPCfor

f pol

^withpdf

p 2

⁽

d = 3

^{). . . . . . . . . . 20}

8 ColloaltionvsnPCfor

f np

^withpdf

p 1

⁽

d = 2

^{) . . . . . . . . . . 20}

9 ColloaltionvsnPCfor

f np

^withpdf

p 2

⁽

d = 2

^{) . . . . . . . . . . 21}

10 ColloaltionvsnPCfor

f np

^withpdf

p 3

⁽

d = 2

^{) . . . . . . . . . . 21}

1 Introdution

Inreentyearsthe growinginterestin UnertaintyQuantiation (UQ)in the

numerial eld has leadto manydierent numerialmethods or strategiesto

quantify the statistisin preseneof stohasti inputs. Oneof the mostpop-

ular method is the Polynomial Chaos (PC) both in its version intrusive or

non-intrusive [2℄. This lass of methods has bee shown to be very eient,

omparedto MonteCarloorolloationstrategies,ifthemodeltorepresentis

smoothenough. Morereentlywiththeaim toextendthelassialPCtheory

to realappliation asesthesoalled multi-elementPC(me-PC)has beenin-

troduedbyKarniadakisetal. (seeforinstane[3℄)allowingtherepresentation

ofprobabilitydistributionsthatfalloutsidetheso-alledWiener-Askeysheme.

Thisopenthewaytotherepresentationofinputsharaterizedbyprobability

distributions that anbeextratedfrom experiments. Howevertheimplemen-

tation of a me-PC ode ould be not so straightforward even if aPC ode is

already at disposal and, until this moment, the eetiveness of this approah

to obtainstatistial momentshigherthanthe varianehasnotbeen doneyet.

In partiular a simple extension of the simple link between the ANOVA rep-

resentation andthe PCexpansion(seefor instane [4℄) isstill missing. Inthe

presentworkwewouldliketoreinterprettheproblemtoextendthePCtonon

lassial pdf in a more diret way reovering all the ommon features of the

lassial PC, as for instane the link between theexpansion and theANOVA

deomposition to omputeonditional statistis. Weproposeastrategybased

onamappingoftheoriginalproblemonanequivalentuniformstohastispae

on whih applying the PC analysis. This strategy leads to a non orthogonal

andnonpolynomial(inthegeneralase)representationthatreetsinaloseof

eienyifomparedtothePCintheaseoflassialpdfs. Howeveraswillbe

learlatertheloseoforthogonalitydoesnotaettheomputationoftheoef-

ientoftherepresentationbutonlythestatistial momentsomputation. We

expet anywayto improvethese preliminary resultswith some simple further

steps, as aoupling with aSparse Grid algorithm anda parallel implementa-

tion,inashortterm. Theremainingpartoftheworkisorganizedasfollows. In

thesetion Ÿ2 themathematialsetting is presentedand thehigher statistial

moments are dened. An analytial denition of the ANOVA expansion and

howitsreetin theomputationofonditional statistisisalsoprovided. An

introdutiononthelassialPCisfurnishedwiththeaimtomaketheworkas

possibleself-ontainedinseetionŸ3. ThehearthoftheworkisthesetionŸ4in

whihourstrategy(nPC)isproposed.SomeresultsonthelinkbetweenthePC

and nPCare thenpresentedand thelink betweenthenPC expansionandthe

ANOVA deompositionisdesribedfortherstordertermsoftheonditional

varianes. Finallysomenumerialresultsarepresentedforanalytialproblems

indimensionuptothreefordierentkindofustomdened pdfsinsetionŸ5.

Conludingremarksandfutureperspetiveworksarereportedaslosurein Ÿ6.

2 Mathematial and problem setting

Considerthefollowingproblem foranoutputofinterest

u(x, t, ξ(ω))

^:

L(x, t, ξ(ω); u(x, t, ξ(ω))) = S(x, t, ξ(ω)),

⁽¹⁾

wheretheoperator

L

^{anbeeitheranalgebraioradierentialoperator(inthis}

aseweneedappropriateinitialandboundaryonditions). Theoperator

L

^and

thesoureterm

S

^{aredenedonthedomain}

D × T × Ξ

^{, where}

x ∈ D ⊂ R ^{n d}

^,

with

n d ∈ {1, 2, 3}

^,and

t ∈ T

^{are the spatialand temporal} dimensions. Ran- domnessisintroduedin(1)anditsinitialandboundaryonditionsintermof

d

seondorder randomparameters

ξ(ω) = {ξ 1 (ω 1 ), . . . , ξ d (ω d )} ∈ Ξ

^withparam-

eterspae

Ξ ⊂ R ^d

^{. Thesymbol}

ω = {ω 1 , . . . , ω d } ∈ Ω ⊂ R

^denotesrealizations inaompleteprobabilityspae

(Ω, F , P )

^{. Here}

Ω

^{isthesetofoutomes,}

F ⊂ 2 ^Ω

isthe

σ

^{-algebraofeventsand}

P : F → [0, 1]

^isaprobabilitymeasure. Random parameters

ξ(ω)

^{an haveany arbitrary}probability density funtion

p(ξ(ω))

^,

in thisway

p(ξ(ω)) > 0

^forall

ξ(ω) ∈ Ξ

^and

p(ξ(ω)) = 0

^forall

ξ(ω) ∈ / Ξ

^{; we}

annowdroptheargument

ω

^{forbrevity. The}probabilitydensityfuntion

p(ξ)

is dened as ajointprobability density funtion from the independentproba-

bilityfuntion of eahvariable:

p(ξ) = Q d

i=1 p i (ξ i )

^{. Thisassumption allowsto}

anindependentpolynomialrepresentationforeverydiretionintheprobabilis-

tispaewiththepossibilitytoreoverthemultidimensionalrepresentationby

tensorization. Inthepresentworkthetestasesarealgebrai,steadyequations

withnophysialspaedependene(weandropthespatialargument

x

^),sowe

anwrite

L(ξ; f (ξ)) = 0

⁽²⁾

thentheaimistondthestatistialmomentsofthesolution

f (ξ)

^.

The(entered)statistial momentsofdegree

n

^tharedenedas

µ ⁿ (f ) =

Z

Ξ

(f (ξ) − E(f )) ⁿ p(ξ)dξ,

⁽³⁾

where

E(f )

^{representstheexpetedvalueofthesolution}

f (ξ) E(f ) =

Z

Ξ

f (ξ)p(ξ)dξ.

⁽⁴⁾

Inthis work moments upto thedegree four areonsidered and in the fol-

lowing we refer to the variane

Var

^{, skewness}

s

^{and kurtosis}

k

^{to indiate,}

respetively,theseond,thirdandfourthorder statistialmoments.

Higherorder statistis(fromordertwo)anbeomputedknowingonlyex-

petanies of the model funtion

f

^{and its values raised to the desired order}

Var = E(f ² ) − E(f ) ²

s = E(f ³ ) − 3E(f )Var − E(f ) ³

k = E(f ⁴ ) − 4E(f ³ )E(f ) + 6E(f ² )E(f ) ² − 3E(f ) ⁴ .

(5)

This representation of the higher moments will be employed to ompare the

strategywithaolloationapproah. Thekeyidea,leadingtothislastapproah,

is to ompute the expetany for the funtions

f

^,

f ²

^,

f ³

^and

f ⁴

^{, obtained}

diretlyfrom thevaluesof themodel

f

^{andthenombinethese expetanyto}

obtainhigh order statistis. Adiret, brutal-fore,olloationapproah ould

betoomputediretlyalltheintegralsontainedintheequations(5)evaluating

eahtermbythepreedentstatistialmoment. Theproblemreduetoompute

only four integral. This an be done eiently if the number of quadrature

points is high enough. Anyway this approah annot provide ametamodel of

the funtion and an expansionof the solution on whih onditional statistis

anbeomputed(seeforinstane[5℄). Thetheoretialframeworkonwhihthe

onditionalstatistisanbeobtainedbasedonanANOVArepresentationisthe

subjetofthefollowingsetion.

2.1 ANOVA deomposition and sensitivity indexes

Let us onsider to havea given equation, or asystems of equations, to solve

and to have an output of interest

f = f (ξ)

^{. The output of the system is}

dependentby

d

unertaintiesparameters

ξ i

^{assumedsothat}

ξ = {ξ 1 , . . . , ξ d } ∈ Ξ ⊂ R ^d

^{. In this work we assume} independent distributed random variables

ξ i ∈ Ξ i

^and,onsequently,thespae

Ξ

^{anbeobtainedby}tensorizationoftheir monodimensional spaes,i.e.

Ξ i ⊂ R

^,

Ξ = Ξ 1 × · · · × Ξ d

^.

From the independene of the random variables follows diretly

p(ξ) = Q

i p(ξ i )

^{. Assuming}

f (ξ) ∈ L ² (ξ, p(ξ))

^{thenaSobolunique funtionaldeom-}

positionexists:

f (ξ) = X

u ⊆{1,...,d}

f u (ξ u )

⁽⁶⁾

where

u

^{is aset of integerswith ardinality}

v = |u|

^and

ξ u = {ξ u 1 , . . . , ξ u v }

^.

Eahfuntion

f u

^{isomputedbytherelation[4℄:}

f u (ξ u ) = Z

Ξ u ¯

f (ξ)p(ξ u _¯ )dξ u _¯ − X

w ⊂ u

f w (ξ w )

⁽⁷⁾

where

Ξ u ¯

^{is thespae}

Ξ

^{withoutthedimensionsontainedin}

u

^and

ξ u _¯

^isthe

vetor

ξ

^{withoutthevariablesin}

u

^.

Bydenition

f 0 = Z

Ξ

f (ξ)p(ξ)dξ

⁽⁸⁾

is the mean of the funtion

f (ξ)

^{. This funtional} deomposition is alled ANOVA if eah of the

2 ^d

^{elements of the} deomposition, exept

f 0

^{, veries}

forevery

ξ i

^:

Z

Ξ i

f ^u (ξ ^u )p(ξ i )dξ i = 0, ∀i ∈ u

⁽⁹⁾

Diretlyfromeq. 9followstheorthogonality:

Z

Ξ

f u (ξ u )f w (ξ w )p(ξ)dξ = 0, u 6= w

⁽¹⁰⁾

2.1.1 Sobolsensitivity indies

EmployingtheANOVAdeompositionitispossibleto deomposethevariane

of

f = f (ξ)

^:

Var(f ) = X

u ⊆{1,...,d}

u 6=0

D u (f u )

⁽¹¹⁾

where

D u (f u ) = Z

Ξ u

f u ² (ξ u )p(ξ u )dξ u

⁽¹²⁾

and

Ξ u = Ξ u 1 × · · · × Ξ u v

^.

TheSobolsensitivityindies(SI),aredenedas:

S ^u = D ^u

Var

⁽¹³⁾

measuringthesensitivityofthevarianeduetothe

v

^-order(

v = |u|

^)interation

betweenthevariablesin

ξ u

^{. Itisevidentthatthesummationofthe}

2 ^d −1

^Sobol

indiesisequaltoone.

3 Classial Polynomial Chaos approah

In this setion we refer briey to the main results relative to the high order

statis omputation by the lassial PC approah with further extensions for

higherstatistialmomentsprovidedbythesameauthors;forexhaustivedetails

referto[5℄.

Thesolutionisexpandedas

f (ξ) =

P

X

k=0

β k Ψ k (ξ),

⁽¹⁴⁾

where

Ψ k

^{is thepolynomialbasisorthogonal to the}probabilitydensity distri- bution

p(ξ)

^.

Eahofthe

P + 1 = (n 0 + d)!/(n 0 !d!)

^{oeientsoftheexpansion(oftotal}

degree

n 0

^{)anbeomputedasprojetionofthemodelfuntion}

f (ξ)

^withthe

polynomial basis exploiting the orthogonality onditions, i.e. the polynomial

basisis hosen in aordto the Wiener-Askeysheme to be orthogonalto the

probabilitydistribution

p(ξ) β k =

R

Ξ f (ξ)Ψ k (ξ)p(ξ)dξ R

Ξ Ψ k (ξ)Ψ(ξ)p(ξ)dξ .

⁽¹⁵⁾

Theproblemistoomputetheintegralatnumeratorwhileforthedenominator

thevalueouldbeknownanalytially.

Assumingto know the oeients

β k

^{of the expansionthe four statistial}

momentsare[5℄

E(f ) = β 0

Var =

P

X

i=1

β ² _i hΨ ² _i (ξ)i

s =

P

X

k=1

β ³ _k hΨ ³ _k (ξ)i + 3

P

X

i=1

β _i ²

P

X

j=1 j6=i

β j hΨ ² _i (ξ), Ψ j (ξ)i

+ 6

P

X

i=1 P

X

j=i+1 P

X

k=j+1

β i β j β k hΨ i (ξ), Ψ j (ξ)Ψ k (ξ)i

k =

P

X

i=1

β ⁴ _i hΨ ⁴ _i i + 4

P

X

i=1

β _i ³

P

X

j=1 j6=i

β j hΨ ³ _i , Ψ j i + 3

P

X

i=1

β ² _i

P

X

j=1 j6=i

β _j ² hΨ ² _i , Ψ ² _j i

+ 12

P

X

i=1

β _i ²

P

X

j=1 j6=i

β j P

X

k=j+1 k6=i

β k hΨ ² _i , Ψ j Ψ k i

+ 24

P

X

i=1 P

X

j=i+1 P

X

k=j+1 P

X

h=k+1

β i β j β k β h hΨ i Ψ j , Ψ k Ψ h i

(16)

Theexpressionreportedin(16)areobtainedthankstotheorthogonalityof

thepolynomialbasis,i.e.

Z

Ξ

Ψ i (ξ)Ψ j (ξ)p(ξ)dξ = δ ij hΨ ² _i i.

⁽¹⁷⁾

3.1 Sensitivity omputation from PC expansion

TheomputationoftheSobolindiesis possibleusingeverysamplestohasti

method (Monte Carlo, quasi-Monte Carlo)but anbedone in averyeient

waywhenapolynomialexpansionofthesolutionisadopted. Theideaistoom-

pute theexpansionof thesolution(trunated) andompute theSobolindies

fromtheexpansioninsteadofomputingthemontherealfuntion. Remember

thepolynomialexpansion:

f (ξ) = ˜ f (ξ) + O T =

P

X

k=0

β k Ψ k (ξ) + O T ,

⁽¹⁸⁾

withanumberoftermsrelatedtothemaximumdegreeofthepolynomialreon-

strution

n o

^{andthedimensionofthesystem}

d

^:

P + 1 = ⁽ⁿ _n ^o _o ^+d)! _!d!

^{. Eahelement}

f u

^{ofthefuntional}deompositionof

f (ξ)

^isapproximatedbytherelativeterm

f ˜ u

^:

f u (ξ u ) ≈ f ˜ u (ξ u ) = X

k∈K u

β k Ψ k (ξ u ),

⁽¹⁹⁾

wheretheset ofindies

K u

^isgivenby

K u =







k ∈ {1, . . . , P }|Ψ k (ξ u ) =

| u |

Y

i=1

φ _α ^k _i (ξ u i ), α ^k _i > 0







(20)

and

φ _α ^k _i (ξ u i )

^{are the} monodimensional polynomials for every diretion

ξ i

^of

degree

k

^{hosenwithrespettotheso-alled}Wiener-Askeysheme[1℄.

Thanks to the orthogonality it is possible to obtain diretly the variane

Var(f ˜ ) = Var( ˜ f ) ≈ Var(f )

^{and the onditional variane}

D ˜ u (f u ) = D u ( ˜ f u ) ≈ D ^u (f ^u )

^{from thefollowingrelations:}

Var(f ˜ ) =

P

X

k=1

β _k ² hΨ k , Ψ k i

⁽²¹⁾

D ˜ ^u (f ^u ) = X

k∈K u

β _k ² hΨ k , Ψ k i.

Sobolsensitivityindiesfollowsdiretly fromeq. 21:

S u ≈ S ˜ u = D ˜ u (f u ) Var(f ˜ ) =

P

k∈K u β ² _k hΨ k , Ψ k i P P

k=1 β _k ² hΨ k , Ψ k i .

⁽²²⁾

If ustom dened probability distribution funtion are employed the or-

thonality properties of thebasisannot beexploited anymore to omputethe

oeientsofthepolynomialexpansion. Inthefollowingsetionwedesribea

strategytoreoverasolutionexpansionwithrespetanonpolynomialbasis,in

the generalase,in whih theortoghonalityof asupport basisis employedto

omputetheoeientsoftheexpansion.

4 Handling whatever form of pdf for the ompu-

tation of higher order statistis

Thestate-of-the-artto omputestatistismomentsfornon lassialpdf isthe

so-alledmulti-elementmethodme-PC[3℄. This familyof tehniques allowsto

eiently ompute, thanks to a partition of the stohasti spae, the (piee-

wise) polynomial approximationsof the model funtion

f (ξ)

^{. Despite to the}

possibilitytoomputeexpetanyandvarianeeiently toomputehigh or-

der statistis this method enounter the same problems of the lassial PC

approah: high number ofintegralsto omputeasin the diret approah (see

equations (16))orsampling onvergene issues asametamodel approah. At

thesametimeaverystrongomputationaleortmustbedevotedto adaptan

existingPCodetoamulti-elementapableone. Anywaytheeetivepossibil-

ityto linkthe me-PCmethod tothe ANOVAdeomposition hasnotbeenyet

providedtoomputeonditionalstatistis.

Totaklethisproblemweproposedamapping proedurethatallowstoob-

tain apolynomialrepresentation ofafuntion depending onaustomdened

distributed randomvetor. Thisproedure allowsastraightforwardimplemen-

tationstartingfromanexistingPCode.

Theproedure is thesameof thelassial PCapproah, i.e. a polynomial

approximation ofthe model is omputed andthen all thestatistial moments

are omputed diretlyfrom this polynomialexpansion. Obviously onditional

statistisanbeomputedtoo.

First of all the

P + 1

^{oeients must be omputed. If the} probability distributionfalloutsidetheso-alledWiener-Askeyshemeaproperorthogonal

basisannotbeemployedasin thegeneralized-PCmethod.

Letus assumetoreasttheprobabilitydensityfuntion

p(ξ)

^as

p(ξ) = p(ξ)¯ p

¯

p ,

⁽²³⁾

where

p ¯

^{isanequivalentuniform}distributiononthespae

Ξ

¯ p = 1

R

Ξ dξ .

⁽²⁴⁾

All theintegralofthekind

R f (ξ)p(ξ)dξ

^anbere-interpretedasstatistis ofafuntion

f ¯ (ξ)

^denedby

f ¯ (ξ) = f (ξ)p(ξ)

¯

p ,

⁽²⁵⁾

withanuniformdistribution

p ¯

^on

Ξ

^{oftheparametervetor.}

At this stage a polynomial representation of the funtion

f ¯ (ξ)

^{an be ob-}

tainedemployingaLegendrebasisorthogonalonastohastispaeembedded

withanuniformdistribution.

The

P + 1

^{oeientsoftheseriesanbeomputedas}

β k =

R

Ξ f ¯ (ξ)Ψ k (ξ)¯ pdξ R

Ξ Ψ k (ξ)Ψ(ξ)¯ pdξ ,

⁽²⁶⁾

wheretheorthogonalityofthepolynomialbasisallowstoavoidtoomputethe

mixed terms,i.e. termsofthekind

R

Ξ Ψ i Ψ j pdξ ¯

^.

The

P + 1

^{terms of the polynomial} approximation of the funtion

f ¯ (ξ)

^,

obtainedontheLegendrebasis,ouldbeemployedeasilytoobtainapolynomial

expansionof

f (ξ)

^onasetof

P +1

^(nonorthogonal)termonthestohastispae withdistribution

p

^{. Theproedureisthefollowing}

f ¯ (ξ) = f (ξ) p(ξ)

¯

p =

P

X

i=0

β k Ψ k (ξ) → f (ξ) =

P

X

i=0

β k Ψ k

¯ p p(ξ) =

P

X

i=0

β k Ψ ¯ k ,

(27)

this is made possible if, as is the ase, no holes in the domain are present,

p(ξ) > 0

^{. Weremarkthatthebasis}

Ψ ¯ i P

i=0

^{isnotpolynomialifthe}probability distribution

p(ξ)

^{is notpolynomialand, in the general ase}

R

Ξ Ψ ¯ i Ψ ¯ j p(ξ)dξ 6=

δ ij

R

Ξ Ψ ¯ ² _i p(ξ)dξ

^.

Howeverthisexpansionallowstoomputestatistis(eventuallyonditional)

and also the metamodel for the funtion

f (ξ)

^{. Even in the ase of time de-}

pendent pdf the points on whih evaluate the model are always the zeros of

theLegendre polynomialassoiatedto thespae. Despiteto thepossibilityto

ompute the expansion even for a known distribution as for instane a Beta

distribution,thistehniqueisexpetedtoperformworstinthisaseinwhiha

properoptimalbasis,intermofonvergene,anbehosen. Adisussiononthe

relationofthelassialgPCmethodwiththisnonpolynomialhaosexpansion

isreportedin thenextsetion.

4.1 On the relation between polynomial and non polyno-

mial haos approah

In the previous setion a novel tehnique has beenpresented to ompute the

expansionofthesolutionforawhateverformofpdf.

Letassumetohaveapolynomialfuntion

f (ξ) ∈ P ^r

^withadistributionofthe vetorparameter

ξ ∼ Beta(α, β)

^{. Wereallherethata}

Beta(α, β)

distribution isdesribedas

p(ξ) = (ξ + 1) ^α−1 (1 − ξ) ^β−1

B(α, β)2 ^α+β−1 ,

⁽²⁸⁾

wherethefuntion

B(α, β)

^is

B(α, β) = Γ(α)Γ(β)

Γ(α + β)

⁽²⁹⁾

andthegammafuntion is

Γ(n) = (n − 1)!

Inthegeneralized-PCasetheJaobibaseanbeemployedandthenumber

ofpoints

n

^{neededtoomputetheintegraloforder}

2r

^{(produtofthefuntion}

and the Jaobipolynomialof degree

r

^{th) is}

2r = 2n − 1

^{, then}

n = r + 1

^(for

eahdimension). Ifthenonpolynomialhaosisemployedtheintegrandshould

beof theorder

2r + α + β

^{. Ofoursethe regularityof theintegrandbeomes}

the sameof the PC asein aseof uniform distribution (

α = β = 0

^{). In the}

generalaseanhighernumberofpointsisneededtosolveorretlytheintegrals

withrespetthegeneralase. Howeverin ageneralasethemodelfuntion is

notpolynomialandthentheadvantageofaGaussianquadraturebasedonthe

polynomial orthogonal basis is less evident. Of ourse this tehniques make

sense in the ase of non standard (eventually time dependent) pdfs in whih

ase thePC tehniquesannot beapplied without amulti-elementtehnique.

Weremarkherethat intheaseofdisontinuouspdfthisproedure ouldstill

be applied even if the onvergene ould be slower or even prevented in the

worstases.

Theother dierene with respet a standardPC tehnique is in the om-

putation of the statistial moments. The basis

Ψ ¯ i P

i=0

^{is not orthogonal on}

thespae

Ξ

^{withrespetthe}distribution

p(ξ)

^{andthenthemixed termsofthe}

expansion must be onsidered. When theexpansion for

f (ξ)

^{is obtained (see}

equation(27))thestatistisanbeomputedas(see[5℄fortheorthogonalase)

E(f ) = β 0

Var = β ₀ ² (h Ψ ¯ ² ₀ − 1i) +

P

X

k=1

β _k ² h Ψ ¯ ² _k i + 2

P

X

i=0 P

X

j=i+1

β i β j h Ψ ¯ i , Ψ ¯ j i

s =

P

X

k=0

β _k ³ h Ψ ¯ ³ _k (ξ)i + 3

P

X

i=0

β _i ²

P

X

j=0 j6=i

β j h Ψ ¯ ² _i (ξ), Ψ ¯ j (ξ)i

+ 6

P

X

i=0 P

X

j=i+1 P

X

k=j+1

β i β j β k h Ψ ¯ i (ξ), Ψ ¯ j (ξ) ¯ Ψ k (ξ)i

k =

P

X

k=0

β _k ⁴ h Ψ ¯ ⁴ _k (ξ)i + 4

P

X

i=0

β _i ³

P

X

j=0 j6=i

β j h Ψ ¯ ³ _i , Ψ ¯ j i + 3

P

X

i=0

β _i ²

P

X

j=0 j6=i

β ² _j h Ψ ¯ ² _i , Ψ ¯ ² _j i

+ 12

P

X

i=0 P

X

j=0 j6=i

P

X

k=j+1 k6=i

β _i ² β j β k h Ψ ¯ i (ξ) ² , Ψ ¯ j (ξ) ¯ Ψ k (ξ)i

+ 24

P

X

i=0 P

X

j=i+1 P

X

k=j+1 P

X

h=k+1

β i β j β k β h h Ψ ¯ i (ξ), Ψ ¯ j (ξ) ¯ Ψ k (ξ) ¯ Ψ h (ξ)i.

(30)

Obviously the number of integral to ompute depends diretly form the

numberofoeientsthatisdiretlyrelatedtothedimension

d

^oftheproblem

and the total degreeof approximation

n 0

^{. The numberof termsfor the non}

polynomialapproahanbeevaluatedas

T ¯ Var = P + 1 + P

2

T ¯ s = P + 1 + P (P + 1) +

P + 1 3

T ¯ k = P + 1 + P (P + 1) + P (P + 1) + (P + 1) P

2

+

P + 1 4

,

(31)

wherethesymbols

T ¯ Var

^,

T ¯ s

^and

T ¯ k

^indiaterespetivelythenumberofintegral toomputeforvariane,skewnessandkurtosisforthenonpolynomialapproah.

The number of the mixed term to ompute, whih is equal to the dier-

ene between thenumber ofintegralto omputein thelassialPC ase (see

equations(16)),anbeshowntobyequalto

∆ Var = P

2

+ 1

∆ s = 2P + 1 +

P + 1 3

− P

3

∆ k = P ² + 3P + 1 + (P + 1) P

2

+

P + 1 4

− P

2

− P P − 1

2

− P

4

,

(32)

where

P

^{dependsfromboththestohastidimension}

d

^{andthetotaldegreeof}

theapproximation(ofthefuntion

f ¯ (ξ)

^).

Theevolutionofthe numberof theextratermsto omputeisreported for

theasewith

n 0 = 2

^{anddimensionbetween2and 10in gure1.}

d

∆ T

2 4 6 8 10

10 ¹ 10 ² 10 ³ 10 ⁴

∆T Var

∆T s

∆T k

Figure 1: Evolution ofthe extratermsto omputein thenon polynomialap-

proahwith respetthe standardgPC method. Theasereported isobtained

with

n 0 = 2

^{andadimension}

d

^{between2and10.}

4.2 Computation of the onditional statistis by the nPC

The approah presented in this work allowsto reprodue thestruture of the

solution

f (ξ)

^{intermofomponentseveniftheseomponentsarenotorthogonal}

eah other. For this reason the funtional ANOVA deomposition annot be

derivedin astraightforwardwayasshowed in setionŸ3.1 identifyingdiretly

thesubsetofindexes

K u

^{asshowedintheequation(20). Howevertheorthogonal}

basisofLegendreunderlinedabovetheseriesexpansionofthesolutionallowsto

obtaininarelativesimplewayeahrstordertermoftheANOVAexpansion,

i.e. termsassoiated to thesubset

u

^{with ardinalityequal to one. Realling}

thedenitionofthese

d

^{termsoftheANOVAexpansion,theequation(7)here}

reportedonlyforonveniene

f u (ξ u ) = Z

Ξ u ¯

f (ξ)p(ξ u _¯ )dξ u _¯ − X

w ⊂ u

f w (ξ w ),

intheaseoftherstorderterms,i.e. termswith

card(u) = 1

^,anberedued

in

f u (ξ u ) = Z

Ξ u ¯

f (ξ)p(ξ u _¯ )dξ u _¯ − f 0 .

⁽³³⁾

Inatingtheexpansion(27)into(33)oneobtains

f ^u (ξ ^u ) =

P

X

k=0

β k

Z

Ξ u ¯

Ψ ¯ k (ξ)p(ξ u _¯ )dξ u _¯ − f 0

=

P

X

k=0

β k

Z

Ξ u ¯

Ψ k (ξ) p(ξ u _¯ )

p(ξ) pdξ ¯ u _¯ − f 0 .

(34)

Theratiobetweenthetwoprobabilitydensity

p(ξ u _¯ )

^and

p(ξ)

^anbeomputed

as

p(ξ u _¯ )

p(ξ) = p(ξ u _¯ )

p(ξ u )p(ξ u _¯ ) = 1

p(ξ u ) ,

⁽³⁵⁾

where

p(ξ u )

^istheprobabilitydensityrelativeto thesubset

u

^{. Theequivalent}

uniform distribution

p ¯

^{, in the same way, an be deomposed in the produt}

betweentheequivalentuniformdistribution relativetotheset

u

^andtheother

variablesas

¯

p = ¯ p u p ¯ u ¯ .

⁽³⁶⁾

Theselast twoequationallowtoreasttherstorderANOVAtermas

f ^u (ξ ^u ) =

P

X

k=0

β k

¯ p u

p(ξ u _¯ ) Z

Ξ u ¯

Ψ k (ξ)¯ p u ¯ dξ u ¯ − f 0 .

⁽³⁷⁾

Thevalueoftheintegralin thelastequationdepends ontheset assoiated

totheindex

k

^{. Wean write}

Z

Ξ u ¯

Ψ k (ξ)¯ p u ¯ dξ u _¯ =

Ψ k (ξ) if k ∈ K u ⁰

0 if k / ∈ K u ⁰ ,

⁽³⁸⁾

where theset

K u ⁰

^{isobtainedasunionof theset}

K ^u

^{and thenull}multi-index element(

k = 0

^).

Fortherstorderterms,i.e. if

card(u) = 1

^{, weanwrite}

f u (ξ u ) = X

k∈K u ⁰

β k

¯ p u

p(ξ u ¯ ) Ψ k (ξ) − f 0 ,

⁽³⁹⁾

obviouslystillholds

f 0 = β 0

^.

To omputethe onditional varianerelative to eah funtion

f ^u

^{the rst}

stepis to raise to theseond powertheequation (39)and thenintegrate over

thespae

Ξ ^u ⊂ Ξ

^{withtheweightfuntion}

p(ξ u )

^.

Theequation(39)raisedtotheseondpoweris

f u ² (ξ ^u ) = X

k∈K u ⁰

β _k ² p ¯ ² u

p ² (ξ u ) Ψ ² _k (ξ) + 2 X

i∈K ⁰ u

X

j≥i+1 j∈K u ⁰

β i β j

¯ p ² u

p ² (ξ u ) Ψ i (ξ)Ψ j (ξ)

− 2β 0

X

k∈K u ⁰

β k

¯ p u

p(ξ u ) Ψ k (ξ) + β ₀ ²

(40)

Thelast equationneedtobeintegratedandthen, in thefollowing,weanalyze

termbytermtheintegrationoftherighthandsideofthe(40) . Therstterm

oftherighthandsideintegratedanbewritten as

X

k∈K u ⁰

β ² _k Z

Ξ u

¯ p ² u

p ² (ξ u ) Ψ ² _k (ξ) p(ξ u )dξ u = X

k∈K ⁰ u

β ² _k Z

Ξ u

¯ p ² u

p(ξ u ) Ψ ² _k (ξ) dξ u .

⁽⁴¹⁾

Theseondtermanbetreatedin ananalogueway

2 X

i∈K u ⁰

X

j≥i+1 j∈K ⁰ u

β i β j

Z

Ξ u

¯ p ² u

p ² (ξ u ) Ψ i (ξ)Ψ j (ξ) p(ξ u )dξ u =

2 X

i∈K u ⁰

X

j≥i+1 j∈K ⁰ u

β i β j

Z

Ξ u

¯ p ² u

p(ξ u ) Ψ i (ξ)Ψ j (ξ) dξ u .

(42)

Thethirdtermanbedramatiallysimpliedas

2β 0

X

k∈K u ⁰

β k

Z

Ξ u

¯ p u

p(ξ u ) Ψ k (ξ) p(ξ u )dξ u = 2β ₀ ²

Z

Ξ u

¯ p ^u

p(ξ u ) p(ξ u )dξ u + 2β 0

X

k∈K u

β k

Z

Ξ u

¯ p ^u

p(ξ u ) Ψ k (ξ) p(ξ u )dξ u = 2β ² ₀

(43)

Thenal formof theonditionalvariane,fortherstorderterms,is then

obtaineddiretlybysummingupthese simpliedterms

D u = X

k∈K u ⁰

β _k ² Z

Ξ u

¯ p ² u

p(ξ u ) Ψ ² _k (ξ) dξ u +

2 X

i∈K u ⁰

X

j≥i+1 j∈K u

β i β j

Z

Ξ u

¯ p ² u

p(ξ u ) Ψ i (ξ)Ψ j (ξ) dξ u − β ₀ ² .

(44)

5 Numerial results

In this setion some numerial results are reported for model problems with

ustom dened pdfs. All theresults are provided with aomparison between

thenPCandtheolloationapproah. Weremarkherethatweexpetabetter

behaviorin termofonvergenefrom theolloationapproah,buttheimpor-

tane of the present nPC approah is to provide ametamodel of thesolution

andaframeworktoomputehighordermomentsonditionalstatistis. Exam-

pleswith dimensionsuptothree areherereportedandtheeetiveness ofthe

strategyisveriedwithrespettotheanalytialsolution.

Threedierentkindofustompdfsareemployedinthiswork(forastohas-

tiparameterdened in

[−1, 1]

^):

ˆ alinearpdf

p 1 (ξ) = 1/2 + 1/3ξ

ˆ aquadratipdf

p 2 (ξ) = ξ ² + 1/2ξ + 1/6

ˆ anonpolynomialpdf

p 3 (ξ) = 1/2 + 1/3 sin (πξ)

^.

Intheaseofmultidimensionalproblemsthepdfisobtainedbytensorization

ofequalpdfsforeah diretionofthestohastispae.

Therstmodelproblemisamonodimensionalfuntion

f (ξ) = sin (πξ) +e ^{ξ 2}

dened on

Ξ = [−1, 1]

^withaprobabilitydistributionequalto

p 1 (ξ)

^.

Theresultsobtainedfortherstmodelproblemintermofperentageerror

with respet the analytial results are reported in gure 2. The olloation

approahandthenonpolynomialapproahshowthesamerateofonvergene.

As evidentalowererrorisreahedbytheolloationapproahevenif,in this

ase,afullyonvergedsolutionanbeobtainedfortheproblemin bothases.

N

s ta ti s ti c s p e rc e n ta g e e rr o rs

10 20 30 40 50

10 ^-8 10 ^-6 10 ^-4 10 ^-2 10 ⁰

10 ² err_mean (PC)

err_var (PC) err_var (Collocation) err_skew (PC) err_skew (Collocation) err_kurt (PC) err_kurt (Collocation)

Figure2: Comparisonbetweentheolloationapproahandthenonpolynomial

approahforthemonodimensionalproblem.

Inthefollowingsomemultidimensionalasesarereported withdimensions

uptothree. Twokindoffuntion areemployed

ˆ apolynomialfuntion

f pol (ξ) = Q d

i=1 (ξ i /2 + 1)

ˆ anonpolynomialfuntion

f np (ξ) = Q d

i=1 sin(πξ i )

^.

Inpartiular theresultsfor thepolynomialfuntion

f pol

^{in dimensiontwo}

with distributions (for eah dimension)

p 1 (ξ)

^,

p 2 (ξ)

^and

p 3 (ξ)

^{are reported,}

respetively,ingures3,4and5.

The results for the polynomial funtion

f pol

^{in dimension three with dis-}

tributions (for eah dimension)

p 1 (ξ)

^and

p 2 (ξ)

^{are reported,} respetively, in gures6and7.

For the non polynomial funtion only the ase with stohasti dimension

equal to twois analyzed. In this asethe three probability funtions are em-

ployed and the perentage errorsare reported respetively in thegures 8, 9

and10.

Theresultsobtainedareinaordtoourpreditions: thesolutiononverges

slowlyforhighermomentswithrespetto themeanandvarianeandtheon-

N

s ta ti s ti c s p e rc e n ta g e e rr o rs

0 100 200 300 400

10 ^-8 10 ^-6 10 ^-4 10 ^-2 10 ⁰ 10 ²

10 ⁴ err_mean (PC)

err_var (PC) err_var (Collocation) err_skew (PC) err_skew (Collocation) err_kurt (PC) err_kurt (Collocation)

Figure3: Comparisonbetweentheolloationapproahandthenonpolynomial

forthepolynomialfuntion

f pol

^withprobabilitydistribution

p 1

⁽

d = 2

^).

N

s ta ti s ti c s p e rc e n ta g e e rr o rs

0 100 200 300 400

10 ^-7 10 ^-5 10 ^-3 10 ^-1 10 ¹ 10 ³ 10 ⁵

err_mean (PC) err_var (PC) err_var (Collocation) err_skew (PC) err_skew (Collocation) err_kurt (PC) err_kurt (Collocation)

Figure4: Comparisonbetweentheolloationapproahandthenonpolynomial

forthepolynomialfuntion

f pol

^withprobabilitydistribution

p 2

⁽

d = 2

^).

vergene ismore sti for problemswith higher stohasti dimension and with

nonpolynomialfuntion orpdf.

Atually thehomemade ode we employedis asequential oneand is very

time demanding to obtain a fully onverged solution for more sti problems,

but we aspet to improvethe present resultswith a parallel implementation.

N

s ta ti s ti c s p e rc e n ta g e e rr o rs

0 100 200 300 400

10 ^-8 10 ^-6 10 ^-4 10 ^-2 10 ⁰ 10 ² 10 ⁴ 10 ⁶

err_mean (PC) err_var (PC) err_var (Collocation) err_skew (PC) err_skew (Collocation) err_kurt (PC) err_kurt (Collocation)

Figure5: Comparisonbetweentheolloationapproahandthenonpolynomial

forthepolynomialfuntion

f pol

^withprobabilitydistribution

p 3

⁽

d = 2

^).

N

s ta ti s ti c s p e rc e n ta g e e rr o rs

0 500 1000 1500

10 ^-9 10 ^-7 10 ^-5 10 ^-3 10 ^-1 10 ¹ 10 ³ 10 ⁵

err_mean (PC) err_var (PC) err_var (Collocation) err_skew (PC) err_skew (Collocation) err_kurt (PC) err_kurt (Collocation)

Figure6: Comparisonbetweentheolloationapproahandthenonpolynomial

forthepolynomialfuntion

f pol

^withprobabilitydistribution

p 1

⁽

d = 3

^).

Howeverasageneralruleweannotify,thatasweexpeted,asimpleolloation

ismoreeientintermofsimulationsinpratiallyalltheases. Howeverthe

presentapproahretainitsinterestinthepossibilityto obtainametamodelof

thesolutionandanestimationoftheonditionalstatistis.

N

s ta ti s ti c s p e rc e n ta g e e rr o rs

0 500 1000 1500

10 ^-8 10 ^-6 10 ^-4 10 ^-2 10 ⁰ 10 ² 10 ⁴ 10 ⁶

err_mean (PC) err_var (PC) err_var (Collocation) err_skew (PC) err_skew (Collocation) err_kurt (PC) err_kurt (Collocation)

Figure7: Comparisonbetweentheolloationapproahandthenonpolynomial

forthepolynomialfuntion

f pol

^withprobabilitydistribution

p 2

⁽

d = 3

^).

N

s ta ti s ti c s p e rc e n ta g e e rr o rs

0 100 200 300 400

10 ^-8 10 ^-6 10 ^-4 10 ^-2 10 ⁰ 10 ² 10 ⁴

err_mean (PC) err_var (PC) err_var (Collocation) err_skew (PC) err_skew (Collocation) err_kurt (PC) err_kurt (Collocation)

Figure8: Comparisonbetweentheolloationapproahandthenonpolynomial

forthepolynomialfuntion

f np

^withprobabilitydistribution

p 1

⁽

d = 2

^).

6 Conlusions and perspetive

Inthepresentworkanovelapproahhasbeenpresentedtoextendthepolyno-

mialhaosexpansiontotheaseofnonlassialdenedpdfs,i.e. pdfsthatfall

outsidetheso-alledWiener-Askeysheme. Thestrategyis basedonthelas-

N

s ta ti s ti c s p e rc e n ta g e e rr o rs

0 100 200 300 400

10 ^-9 10 ^-7 10 ^-5 10 ^-3 10 ^-1 10 ¹ 10 ³ 10 ⁵

err_mean (PC) err_var (PC) err_var (Collocation) err_skew (PC) err_skew (Collocation) err_kurt (PC) err_kurt (Collocation)

Figure9: Comparisonbetweentheolloationapproahandthenonpolynomial

forthepolynomialfuntion

f np

^withprobabilitydistribution

p 2

⁽

d = 2

^).

N

s ta ti s ti c s p e rc e n ta g e e rr o rs

0 100 200 300 400

10 ^-8 10 ^-6 10 ^-4 10 ^-2 10 ⁰ 10 ²

10 ⁴ err_mean (PC)

err_var (PC) err_var (Collocation) err_skew (PC) err_skew (Collocation) err_kurt (PC) err_kurt (Collocation)

Figure 10: Comparison betweentheolloationapproahandthenon polyno-

mialforthepolynomialfuntion

f np

^withprobabilitydistribution

p 3

⁽

d = 2

^).

sialPCapproahbutuniformequivalentdistributionsareemployedtore-map

the originalproblem in anuniform oneirrespetiveless of thetrueprobability

distribution. Thisapproaheventuallydegradestheonvergeneofthelassial

approah,in termofnumberofsimulationsrequiredtoestimatethestatistis,

ifalassialpdfisemployed. Thisisdue toaquadratureruleoptimalonlyin

theaseofpolynomialfuntionswithuniformdistribution. Howeverthisdraw-

bakisnotmuhinuentin realappliationsasesbeausethemodelfuntion

is,often, notpolynomial. Anotherdrawbaksouldemergeifoneomparethe

presentstrategytoabrutalforeolloationstrategyin whiheverystatistial

moments is deomposed in expetanies of thefuntion and all the funtions

itselfraisedto apowerequaltothedegreeofthemaximumstatistialmoment

required. Thislastapproahallowsto reduedramatiallythenumberofinte-

grals to beomputed. Anywaythe present strategy is motivated beausethe

advantagedespite to thehigheromputational ostis to provide, at thesame

time,aompletemetamodelofthemodelfuntionandaknownstrutureofthe

funtiononwhihispossibletoomputeonditionalstatistis. Thispossiblyis

notompletely explored in thisworkand onlythe rstorder onditionalvari-

anesare expliitlyshown. Atthepresenttime onlyanhomemadesequential

ode is at our disposal and this limits the possibility to ompute high order

statistis formore omplexproblems in higher dimension thantwo. Toallow

more real appliation ases, in whih thesingle ost of eah omputation an

be muh more expensive, we expet to really inrease the eieny oupling

the present strategywith aSmolyak algorithmto omputethe set ofquadra-

turepointsonwhihthesimulationsmustbeperformed. Intheaseofsmooth

modelfuntionsandprobabilitydistribution,forhighdimensionproblems,the

ouplingbetweenournovelnPCapproahandthesparsegridisstraightforward.

7 Aknowledgements

Remi Abgrallhas been partially supportedby theERC Advaned GrantAD-

DECCON.226316,whileGianluaGeraihasbeenfullysupportedbytheERC

AdvanedGrantADDECCON.226316. GianluaGeraialsoaknowledgethe

supportoftheAssoiatedTeam'AQUARIUS'whihprovidedthenanialsup-

portforafruitfulvisitattheStanfordUniversityattheendof2011.

Referenes

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[2℄ Olivier Le Maître and Omar M. Knio. Spetral Methods for Uner-

taintyQuantiation: WithAppliationstoComputationalFluidDynamis.

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[3℄ JasmineFooandGeorgeEmKarniadakis. Multi-elementprobabilistiol-

loation method in high dimensions. Journal of Computational Physis,

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[4℄ ThierryCrestaux,OlivierLeMaître,andJean-MarMartinez. Polynomial

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Safety,94(7):11611172,July2009.

[5℄ R. Abgrall,P.M.Congedo, G.Gerai,andG. Iaarino. Computation and

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351, Cours de la Libération Bâtiment A 29

Domaine de Voluceau - Rocquencourt

BP 105 - 78153 Le Chesnay Cedex

inria.fr