A simple, flexible and generic deterministic approach to uncertainty quantifications in non linear problems: application to fluid flow problems

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A simple, flexible and generic deterministic approach to

uncertainty quantifications in non linear problems:

application to fluid flow problems

Remi Abgrall

To cite this version:

Remi Abgrall. A simple, flexible and generic deterministic approach to uncertainty quantifications

in non linear problems: application to fluid flow problems. [Research Report] 2008, pp.26.

�inria-00325315�

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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

A simple, flexible and generic deterministic approach

to uncertainty quantifications in non linear

problems: application to fluid flow problems

Rémi Abgrall

N° ????

Centre de recherche INRIA Bordeaux – Sud Ouest

un ertainty quanti ations in non linear problems: appli ation

to uid ow problems

RémiAbgrall

ThèmeNUMSystèmesnumériques

Équipes-ProjetsS AlApplix

Rapportdere her he n°???? Septembre200723pages

Abstra t: Thispaperdealswiththe omputation ofsomestatisti sof thesolutionsof linearandnon

linearPDEs by mean of amethod that is simple and exible. A parti ular emphasis is given on non

linearhyperboli typeequationssu hastheBurgerequationandtheEulerequations.

GivenaPDEandstartingfromades riptionofthesolutionintermofaspa evariableanda(family)

of random variables that may be orrelated, the solution is numeri ally des ribed by its onditional

expe tan ies of point values or ell averages. This is done via a tessellation of the random spa e as

in nite volume methods for the spa e variables. Then, using these onditional expe tan ies and the

geometri aldes riptionofthetessellation,apie ewisepolynomialapproximationintherandomvariables

is omputedusing are onstru tionmethod that is standardforhigh order nitevolumespa e, ex ept

that the measure is no longer the standard Lebesgue measure but the probability measure. Starting

fromagivens hemeforthedeterministi versionofthePDE,weusethisre onstru tiontoformulatea

s hemeonthenumeri alapproximationofthesolution.

This method enables maximum exibility in term of the PDE and the probability measure. In

parti ular, the s heme is non intrusive, an handle any type of probability measure, even with Dira

terms. Themethodis illustratedonODEs,ellipti andhyperboli problems,linearandnonlinear.

Key-words: Un ertaintyquanti ation, determinsti methods, non linearPDEs, Burgersand Euler

in ertitudes pour les problèmes nn linéaires : appli ation à la

mé anique des uides.

Résumé : On propose une méthode simple et exible permettant la détermination de paramètres

statistiquesdessolutionsdeproblèmesauxdérivéespartielleslinéairesounonlinéaires.

Etantdonné une EDP, une des riptionspatiale dela solutionet une famillede variablesaléatoires

(quipeuventêtre orrélées),lasolutionestdé ritenumériquementparlesespéran es onditionnellesde

sesvaleurspon tuelles oude sesvaleursmoyennes dansdes ellules de ontrle. Pourdéterminer une

approximationde esespéran e onditionnelles,onintroduitunetriangulationstru turéedel'espa edes

variablesaléatoires.

Ensuite,enemployantlesespéran es onditionnellesetlades riptiongéométriquedelatriangulation.

une approximation polynmiale parmor eau est onstruite, omme ela est ourammentfait dans les

méthodesdetypevolumenid'ordreélevé,endehorsdufaitquelare onstru tionemployéei in'utilise

pas la mesure de Lebesgue, mais la mesure de probabilité. Partant d'une approximation de l'EDP

déterministe,onmontrealors omment onstruireuns hémaportantsurlesespéran es onditionnelles.

Laméthodeest illustréesur plusieursproblémeslinéeaires etnon linéaires,y omprisleséquations

d'Eulerdelamé aniquedesuides.

Mots- lés : Quanti ation desin ertitudes, méthodedéterministes, EDPnon linéaires,équation de

1 Introdu tion

Weareinterestedin solvinglinearandnonlinearPDE,su has

∂u

∂t

+

∂f (u)

∂x

= ν

∂

2

_u

∂x

2

+ S(x)

t > 0, x

∈ R

Initial and/orboundary onditions

(1)

with

ν

≥ 0

and

S

isasour eterm. Examplesaregivenbyatransportequationwith

f (u) = au, S = 0

, theBurgersequationwith

f (u) =

1

2

u

2

,theheatequationwhere

f = 0

orevenaLapla eequationwhere

f = 0

and atime independent solution. Of ourse, in ea h ase, the boundary onditions haveto be

adaptedto the aseunderstudy.

Inthis paper, weassume in addition that theinitial ondition is arandomvariable

u

0

:= u

0

(x, ω)

where

ω

∈ Ω

aprobabilisti spa e. Weassumethat

u( . , ω)

hasaknowndistributionlaw

dµ

whi hmay ormaynothaveadensity. Examplesare givenbytheuniform distribution,theGaussiandistribution,

ofa ombinationofapdf withadensitywithDira -likedistributions. Inthat asethere isnodensity.

Inthiswork,weassumetoknowthepdf. Theproblemof knowingthedistribution lawofthesolution

of (1)isadi ultproblemingeneralwhi hisstillopenuptoourknowledge.

There are many situations where one wantsto estimate somestatisti s on the solutionof a PDE.

Consider the ow around an air raft for example. The boundary onditions (inow ma h number,

Reynoldsnumber, somegeometri al parameters)may onlybeknownapproximatelyeither in anozzle

oworatrueight. Inhyper-soni s,theequationof stateorthevis ousmodel playanimportantrole

and they are sometimes known veryapproximately. The sameis true formultiphase ows. Onemay

alsowishtoextrapolate experimentalresultswhi harepartialyknowntoow onditionsthatarenot

ontainedin the experimental database : whatis the onden e onemay have? The questionisnot

onlyto knowthe sensitivity of thesolutionof (1), but alsoto to understand theimportan e (i.e. the

weight)of thesevariations. Inother terms,assumingthelikely-hoodofrelativevariations,how anwe

weighttheirinuen eonthesolution?

Theaim of this paper is to propose a general method whi h enables to ompute (approximations

of) the statisti s of theexa t solution with the smallestpossiblemodi ationof an existing ode. In

parti ularweareinterestedin developinggeneralpurposemethods abletoeasily handle, withlittle or

no ode/s hememodi ation,thefollowinglist:

1. thepdfisgeneralandmay hangeeitherin timeorthroughsomeoptimisationloopforexample,

2. the pdf may ormay havea density, may ormay be not ompa tly supported, or anbe known

onlytroughanhistogram,

3. Theinitialsolutionmaydependonseveral orrelatedorun orrelatedrandomvariables.

4. nomodi ationofthe odehastobedonewhenone hangesthepdf,

5. aslittleaspossiblemodi ationofanexistingdeterministi ode/methodisneeded.

Namely,when oneapproximates(1)ormore omplex onservationsystems,the odingeortisput on

thespatialdis retisation,theapproximationoftheuxandthetimeapproximation.

The problem (1) is a very rude approximation of the above mentioned problem. Our aim is to

developageneralmethodologythat ouldeasilybeextended tothesemore omplexphysi alproblems.

Inmostengineeringsituations,thenumeri almethod isatmostse ond ordera urate. Sayingthis,

wehaveinmindthe aseofanonlinearproblemofhyperboli typeorpossiblywithse ondorderterms

but whi h roleissigni antonlyin asmallpartofthe omputationaldomain. Theprototypeexample

is again (1) with

ν << 1

. In engineering appli ations, this is the Navier Stokesequations. In these ases, sin edeterministi methods aregenerally se ond ordera urate in time and spa e, ourbelief is

that there is noneed to have amethod ableto ompute statisti al quantities with an extremely high

a ura y. The best would be to have amethod where the dominant sour e of error omes from the

deterministi method. Theaim of thispaperis to propose amethod that is ableto ombine all these

Thepaperisorganisedasfollow. Firstwereviewseveralexistingte hniques. Inase ondse tion,from

several omputationalremarks,weproposeageneralframeworkto a hieveourgoal. Thisframeworkis

illustratedbyseveralexamples,rangingfromstandardODEs,toEulerequations,viaanellipti problem

and several s alar hyperboli problems. Every time this is possible, error with respe t to the exa t

solutionaregiven.

2 Review of existing te hniques

Inmany ases,thedenitionofthephysi alproblemisnotfullyknown. Thismaybethe aseforseveral

reasonsin luding:

ˆ The geometry may be known only partially. Imagine that the body surfa e is rough, one an

ertainlyparametrizetheroughnessbysomerandomparametrisation

ˆ The boundary onditionsmaybepartially knownonly, forexamplein the aseof u tuations of

someparameters,

ˆ Some onstants in the model an be un ertain, think for example of a turbulen e model orthe

parametrizationoftheequationofstate. Thisis averyimportantpra ti alproblemforindustry.

Inea h ase,evenifthemodel, hen ethenumeri almethod ...,suers from de ien ies, thereis still

aneedto omputeandsimulate !

In order to ta kle this issues, there are urrently several te hniques available in the engineering

ommunity,and thisisaverya tiveresear htopi .

Onete hniquerelieson polynomial haos expansion. Assumingthat therandominputsdatawhi h

dependson spa e

x

∈ A ⊂ R

d

andaarandom parameter,saytheboundary onditionsto x ideas,is

denedonaprobabilisti spa e

(Ω,

A, P )

andhasanitevarian e,we andenethe ovarian ematrix

C(x, y) = E(X(x, .)X(y, .)),

for

x, y

∈ A.

If

f

k

isthe

k

theigenfun tion

Z

A

C(x, y)f

k

(y)dy = λ

k

f

k

(x),

one anwritetheKarhunen-Loèveexpansionof

X

,

X(x, ω) =

∞

X

k=0

pλ

k

f

k

(x)ζ

k

(ω)

(2)

where the

ζ

k

are un orrelated Gaussian random variables. Then, following [1℄, one an expand the solutionof (1)as

u(t, x, ω) = a

0

Γ

0

+

∞

X

i

1

=1

a

i

1

Γ

1

(ζ

1

(ω)) +

∞

X

i

1

=1

∞

X

i

2

=1

a

i

1

i

2

Γ

2

(ζ

1

(ω), ζ

2

(ω))

+ . . .

+

∞

X

i

1

=1

∞

X

i

1

=1

. . .

∞

X

i

k

=1

a

i

1

i

2

...i

k

Γ

k

(ζ

1

(ω), ζ

2

(ω), . . . , ζ

k

(ω))

+ . . .

(3)

Thefun tions

Γ

k

aredenedby

Γ

k

(ζ

1

, ζ

2

, . . . , ζ

k

) = (

−1)

k

e

ζ·ζ

T

∂

k

e

−ζ·ζ

T

∂ζ

1

. . . ∂ζ

k

.

Theideais, aftertrun ationbothin therandominputand(3),to introdu ethisrelationinto (1),then

to useaspe tralmethod (be ause ofthe form ofthe

Γ

k

). There areother versionsof this polynomial haos,see forexample[2,3℄.

Inouropinion,there areatleastthree drawba kstothisapproa h. First,it isnot learat allwhat

should betherighttrun ationlevelintheexpansion(3),seeforexample[4℄. Se ond,ifonehasagood

numeri almethod to solveoneproblem, thenumeri alstrategy has to be revisited from A to Z to go

to another one,whi his not a eptablefrom an engineering pointof view. Itis also not lear how to

handledis ontinuitiesintheformulation. Thelastoneisthat ifone hangesthestru tureoftheinput

randomfun tion,everything hasto berestarted froms rat h. This isthe asein parti ularwhennew

informationsareintrodu edto thesystem.

The se ond problem of the previous approa h, that the method is intrusive, an be ta kled by a

method whi h isin betweenthespe tralexpansionthat hasbeensket hedaboveandtheMonteCarlo

method. One hoosesagood setofrandomrealisationsandonerunthebaselinenumeri als hemefor

these random parameters. Sin e theoutput of the whole omputation is to evaluate expe tation of a

fun tionalfofthethesolution,saythepressuredistribution toxideas,thesefun tionaldependon

ζ

1

, ...,

ζ

N

. Therandomparametersare hosensu hthattheexpe tan y

E(f ) =

Z

Ω

f (ζ

1

, . . . , ζ

N

)dµ

anbeevaluatedeasilywithagooda ura y. Thisamountstondquadraturepointsforthisintegral.

These quadrature pointsare related in generalto zeros ofsomeorthogonal polynomials. The urse of

dimensionality anbeta kledbymeanoftheSmolyakquadratureformula,forexample. Thispathhas

beenexplored byseveralresear hers,seeforexample[5℄.

Inouropinion,oneoftheweaknessofthiste hniqueisthatiftheprobabilitydensityfun tionsarenot

smoothenoughthismayo urinsome ombustionproblems,see[4,6℄forexample,the onvergen e

oftheintegralmaybeveryslow.

In both ases, an other major drawba kis the following: the pdf is in generalnot known, so that

thewholepro ess ollapses. Thenumeri alpro eduremaybeonepartofamoregeneralloopinwhi h

alearningpro essisimplemented,viasomeoptimisationloopforexample. Clearly,on etheexpansion

(2)hasbeen hosen,thereisnospa eforanylearningpro esssothat theexpe tedresultsofthewhole

methodology anbedisappointing. How anwe onstru tanumeri almethod,abletohandletrueuid

problems,forwhi halearningpro ess anbeimplemented?

3 Prin iples of the method

3.1 Some omputational remarks

Let

dµ

aprobabilitymeasureand

X

arandomvariabledenedontheprobabilityspa e

(Ω, dµ)

. Assume wehaveade ompositionof

Ω

bynonoverlappingsubsets

Ω

i

,

i = 1, N

of stri tlypositivemeasure:

Ω =

_∪

N

i=1

Ω

i

.

We are given the onditional expe tan ies

E(X

|Ω

i

)

. Can we estimate for agiven

f

,

E(f (X))

? We assume

X = (X

1

, . . . , X

n

)

Theideaisthefollowing: Forea h

Ω

i

,wewishtoevaluateapolynomial

P

i

∈ R

n

_[x

1

, . . . , x

n

]

ofdegree

n

su h that

E(X

|Ω

j

) =

R

R

n

1

Ω

j

(x

1

, . . . , x

n

)P (x

1

, . . . , x

n

)d˜

µ

µ(Ω

i

)

for

j

∈ S

i

(4)

where

d˜

µ

istheimageof

dµ

and

S

i

isasten ilasso iatedto

Ω

i

. 1

This problem is reminis ent of what is done in nite volume s hemes to ompute a polynomial

re onstru tioninordertoin reasethea ura yoftheuxevaluationthankstheMUSCLextrapolation.

Amongthemanyreferen esthat havedealt withthis problem,with theLebesguemeasure

dx

1

. . . dx

n

, onemayquote[7℄andforgeneralmeshes,onemayquote[8,9℄. Asystemati methodfor omputingthe

solutionofproblem(4)isgivenin[10℄.

1

forexample

dµ

isthesumofaGaussianandaDira at

x

0

,

Z

R

n

P

(x)d˜

µ

= α

√

1

2πσ

Z

R

P

(x)e

−

(x−m)2

2σ

dx

_{+ (1 − α)P (x}

0

)

6 Abgrall Assumethatthesten il

S

i

isdened,thete hni al onditionthatensureauniquesolutionto prob-lem (4) isthat the Vandermondelikedeterminant(given hereforonerandom variable forthe sakeof

simpli ity)

∆

i

= det



E(x

l

|Ω

j

)



0≤l≤n,j∈S

i

.

isnonzero. Inthe aseofseveralrandomvariable, theexponent

l

aboveisrepla edbyamultiindex. On ethesolutionof (4)isknown,we anestimate

E(f (X))

≈

N

X

j=1

Z

R

n

1

Ω

j

(x

1

, . . . , x

n

)f



P (x

1

, . . . , x

n

)



d˜

µ.

Wehavethefollowingapproximationresults: if

f

∈ C

p

_(R

n

₎

with

p

≥ n

then









E(f (X))

₋

N

X

j=1

Z

R

n

1

Ω

j

(x

1

, . . . , x

n

)f



P (x

1

, . . . , x

n

)



d˜

µ

_{| ≤ C(S) max}

j

µ(Ω

j

)

p+1

p



for a set of regular sten il whi h proof is straightforward generalisation of the approximation results

ontainedin[11℄.

Inallthe pra ti alillustrations,wewill useonlyoneortwosour esofun ertaintyeventhoughthe

method anbeusedforanynumberofun ertainparameters,thisleadingtootherknownproblemssu h

thatthe urseofdimensionality. Thespa e

Ω

issubdividedintononoverlappingmeasurablesubsets. In the aseof onesour eof un ertainty,the subsets anbe identied, viathemeasure

dµ

, to

N

intervals of

R

whi h aredenoted by

[ω

j

, ω

j+1

]

. The aseof multiple sour es anbe onsidered by tensorisation of theprobabilisti mesh. Thisformalismenablesto onsider orrelatedrandom variables,asweshow

laterinthetext.

Let us des ribe in details what is done for one sour e of un ertainties. In the ell

[ω

i

, ω

i+1

]

, the polynomial

P

i+1/2

is fully des ribed by a sten il

S

i+1/2

=

{i + 1/2, i

1

+ 1/2, . . .

}

su h that in the ell

[ω

j

, ω

j+1

]

with

j + 1/2

∈ S

i+1/2

wehave

E(P

i+1/2

|[ω

j

, ω

j+1

]) = E(u

|[ω

j

, ω

j+1

]).

Itiseasytoseethatthereisauniquesolutiontothatproblemprovidedthattheelementsof

{[ω

j

, ω

j+1

]

}

j+1/2

∈

S

i+1/2

donot overlap,whi h is the ase. In thenumeri alexamples, we onsider three re onstru tion me hanisms:

ˆ arstorderre onstru tion: wesimplytake

S

i+1/2

=

{i + 1/2}

andthere onstru tionispie e-wise onstant,

ˆ a enteredre onstru tion: thesten ilis

S

i+1/2

=

{i − 1/2, i + 1/2, i + 3/2}

andthere onstru tion is pie ewise quadrati . Attheboundaryof

Ω

, weusethe redu edsten ils

S

1/2

=

{1/2, 3/2}

for therst ell

[ω

0

, ω

1

]

and

S

N −1/2

=

{N − 1/2, N − 3/2}

for thelast ell

[ω

N −1

, ω

N

]

,i.e. weusea linearre onstru tionattheboundaries.

ˆ An ENO re onstru tion : for the ell

[ω

i

, ω

i+1

]

, we rst evaluate two polynomials of degree 1. Therst one,

p

−

i

, is onstru tedusing the ells

{[ω

i−1

, ω

i

], [ω

i

, ω

i+1

]

}

andthe se ondone,

p

+

i

,on

{[ω

i

, ω

i+1

], [ω

i+1

, ω

i+2

]

}

. We anwrite(with

ω

i+1/2

=

ω

i

+ω

i+1

2

)

p

+

_i

(ξ) = a

+

_i

(ξ

_{− ω}

i+1/2

) + b

+

i

and

p

−

i

(ξ) = a

−

i

(ξ

− ω

i+1/2

) + b

−

i

.

We hoose theleast os illatory one,i.e. theonewhi h realisestheos illation

min(

|a

+

i

|, |a

−

i

|)

. In

that ase,wetakearstorderre onstru tionontheboundaryof

Ω

.

Other hoi esarepossiblesu hasWENO-likeinterpolants. Again,the aseofmultiplesour eof

3.2 A general strategy

LetusstartfromaPDEofthetype

L(u, ω) = 0

(5)

dened in adomain

K

of

R

d

, subje tedto boundary onditions. Sin e thedis ussionof this se tionis

formal,weputthedierentboundary onditionsoftheprobleminthesymbol

L

. Theterm

ω

isarandom parameter,i.eanelementofaset

Ω

equippedwithaprobabilitymeasure

dµ

. In(5) , theun ertaintyis weakly oupledwiththePDE,i.e.

ω

doesnotdependonanyspa evariables. However,themeasure

dµ

maydependonsomespa evariable. Theexampleswehaveinmindaresu hthatforagivenrealisation

ω

0

∈ Ω

,

L(u, ω

0

) = 0

isastandard PDE, su h astheLapla e equation,Burgersequation, theNavier

Stokesequations,et . Tomakethingsevenmore lear,andtogiveanexample,letus onsidertheheat

equation

∂T

∂t

=

div

(κ

∇T ) + S(t, x),

t > 0, x

∈ K ⊂ R

d

withDiri hletboundary onditions

T = g

on

∂K

andinitial onditions

T (x, 0) = T

0

(x).

Inthisexample,

κ

,thesour eterm

S

,theboundary ondition

g

,thedomain

K

andtheinitial ondition

T

0

mayberandom. Foranyrealisationof

Ω

,weareableto solvetheheat equationbysomenumeri al method. Whatwearelookingforis,forexample,statisti sontheapproximatesolution

T

when

ω

follows agivenprobabilitylaw.

Wearegivenanumeri almethod forsolving

L(u, ω

0

) = 0

,say

L

h

(u

h

, ω

0

) = 0.

forany

ω

0

∈ Ω

Thisgivesbirth to amethod for solving (5) that we denote

L

h

(u

h

, ω) = 0

. On e this is done, we havetodis retise theprobabilityspa e

Ω

: we onstru tapartition of

Ω

, i.e. aset of

Ω

j

,

j = 1, . . . , N

thataremutuallyindependent

µ(Ω

i

∩ Ω

j

) = 0

forany

i

6= j

andthat over

Ω

Ω =

∪

N

i=1

Ω

i

.

Weassume

µ(Ω

i

) > 0

forany

i

. Ourproblemistoestimate

E(u

h

|Ω

j

)

from

L(u, ω) = 0

. Forexample,ifaniterativete hniqueisusedforsolvingthedeterministi problem, say

u

n+1

_h

=

J (u

n

h

),

thisleadsto

u

n+1

_h

(ω) =

J (u

n

h

, ω),

sothat

E(u

n+1

_h

|Ω

j

) = E(

J (u

n

h

)

|Ω

j

).

Intheexampleswehaveinmind,theoperator

J

isasu essionofadditions,multipli ationsandfun tion evaluations. Theaverage onditionalexpe tan ies,aswehaveexplainedinthepreviousse tion,enableto

omputeapproximationsoftheaverage onditionalexpe tan iesofanyfun tional,sothattheevaluation

of

E(

J (u

n

h

)

|Ω

j

)

anbedonein pra ti e: weareableto onstru tasequen e

E(u

n+1

h

|Ω

j

)

n≥0

. If this sequen e onvergesin somesense,thelimitisthesoughtsolution.

This example is also useful for larifying what we are not looking for. It may well be that the

fun tional ominginto

J

depend onseveralinstan eof

u

h

,forexamplethevalueof

u

h

at severalmesh pointlo ations. Inourmethod,weneedthattheprobabilitylawbethesamealloverthe omputational

domain

K

, orwewouldneedjoint probabilitiesbetweenthevariousvariables omingintoplay. If this isdoablein theory,wedonotbelieveitisdoablein pra ti e. Moreover,foralltheexampleswehavein

mind,itisreasonabletoassumethattheprobabilitylawisthesamealloverthe omputationaldomain.

Inthenext se tions, we provide examplesof realisationsofthis programon ellipti , paraboli and

4 Example of an ODE

OurrstexampleisasimpleODEequationwithinitial ondition,

du

dt

= f (u, t)

u(x, t = 0, ω) = u

0

(x, ω)

(6)

where

f

is assumedto be smoothenoughforhavingauniquesolution. Here,weassumethat

f

is

C

1

,

butthisassumptionis ertainlytoostrongand ouldbelowerbyadeeperanalysis,Thisisnotourpoint

here.

The equation (6)is dis retised,for any

ω

, by anODE solver. Tomakethings simple,but without lossofgenerality,assumethatweusetherstorderEulerforwardmethod

u

n+1

_{(ω) = u}

n

_(ω)

− ∆t f(u

n

_{(ω), t}

n

).

Thenwehave,forany

Ω

i

E(u

n+1

|Ω

i

) = E(u

n

|Ω

i

))

− ∆t E(f(u

n

, t

n

)

|Ω

i

).

(7)

Theproblemistoevaluate

E(f (u

n

_{, t}

n

)

|Ω

i

)

. This anbedoneviaanumeri alquadraturethankstothe re onstru tionwehavedevelopedinse tion3.1. Letusgiveanexample,say

f (u, t) =



u(u

₋

1

2

)(1

− u)

if

u

∈ [

1

4

,

3

4

]

0

else. (8) Inthatexample,

u

_∞

= lim

t→+∞

u(t) =







u

0

if

u

0

<

1

4

1

2

if

1

4

≤ u

0

<

3

4

u

0

if

u

0

>

3

4

.

Fromthis, wesee easilythatif

u

0

israndomwithprobabilitylaw

dµ

,therepartitionfun tionof

u

∞

is

Φ(u) := P (ω

_|u

_∞

(ω)

_{≤ u) =}















P (s

_{≤ u)}

if

u <

1

4

P (s

_≤

1

4

)

if

1

4

≤ u <

1

2

P (s

_≤

3

4

)

if

1

2

≤ u <

3

4

P (s

_{≤ u)}

if

u

≥

3

4

Take the s heme is (7) with a third order re onstru tion (with a entered sten il) ex ept on the

boundariesof

Ω

where,if

Ω

0

and

Ω

N

aretheboundary ells,wetakethefollowingsten ils

ˆ for

Ω

0

,

S = {0, 1}

,

ˆ for

Ω

N

,

S = {N − 1, N}

.

The resultsare independent of the high order re onstru tion formula be ausethe solution limitvalue

u

_∞

=

1

2

isstable.

A thirdorderquadratureGaussianformulaisusedinthe aseofaprobabilitywithadensity

Z

b

a

f (x)dx

_≈

b

− 1

2



f

a + b

2

+ θ(b

− a)

+ f

a + b

2

− θ(b − a)



,

θ =

_√

1

3

.

An optimal 6th order Gauss quadratureformula ould havebeenused, but sin ethe time stepping is

onlyrstorder,and sin ethenumeri alexampleswe onsideraresteady,there isnoneedtodealwith

optimally order quadrature formula. When there is no density, for example if

dµ = f dx + Cδ

a

, the regularpartof

µ

isdealt withthepreviousformula,andthesingularonebyanadho one.

Inordertoillustrate themethod, we onsiderthreepdfs. Here,weset

dµ = f (x)dx

ˆ auniformdistribution:

Simple andexibleUn ertaintyQuanti ationfor non linear problems 9 ˆ AGaussian distributionon

[

−2, 2]

with

0

meanandvarian e

σ = 1

,

f (x) =

e

−x

2

_/2

Z

3

−3

e

−s

2

/2

_ds

,

ˆ APoissondistributionon

[0, 2]

,

f (x) = 1

[0,2]

e

−x

1

_{− e}

−2

,

,

Intheseexample,anyofthethree re onstru tionmethodspresentedinse tion3.1worksne. Wehave

hosenthe enteredoneforthenumeri alillustrationsin eitisapriorithemosta urateone.

Weshowhowthemethodapproximatesthevalues

P (s

≤

1

4

)

and

P (s

≤

1

2

)

. Wehave hosenavery rudewayoftherepartitionfun tion oftherandomvariable

u

∞

,

Ψ(U

_∞

)

≈

X

j

P (s

∈ [ω

j

, ω

j+1

]

su hthat

u

∞

(s)

≤ U

∞

).

Thisexplainsthestair aselikebehaviorofthe urvesofFigure2wherewehavedisplayedtheresultsfor

thesethreepdfs. Thismethodisa uratehoweverwhen

1/4

and

1/2

aremeshpointsintheprobability spa e,inwhi h asetheonlyapproximationholdsonthequadraturedeningtheterms

P (s

∈ [ω

j

, ω

j+1

])

. Weseeongure1thatthemethodhasthea ura yofthequadratureformula(fourthordera urate)

when the probability mesh meet this ondition; this test has been ondu ted with the Gaussian pdf.

Otherresultsobtainedfortheotherpdfsand101pointsaredisplayedinTable1.

0.01

0.1

-14

-12

-10

-8

P1

P2

slope 4

Figure1: Errorin theevaluationof

Ψ(U

∞

)

when

U

∞

∈]

1

4

,

1

2

[

and

U

∞

∈]

1

2

,

3

4

[

foroptimalmeshes.

Theerrorbetweentheexa tand numeri alresultsfor

P

1

= Ψ(

1

4

)

and

P

2

= Ψ(

3

4

)

in the aseofthe GaussiandistributionisdisplayedinFigure 1.

At rstglan e, it might look strangethat a entered re onstru tion works well fora problemthat

admits dis ontinuous results. It iswell known that su h re onstru tion suers from a Gibbslike

phe-nomena. Howeverhere,theproblemabitspe ial. Thetwosolutions

u

∞

= 0

and

u

∞

= 1

areunstable and thewaywehavepro eed avoidthem. Wehavekeptonly thethe stable one

u

∞

=

1

2

. More over, if

u

∈ [

1

4

,

3

4

]

wehave

u

∞

= u

0

and theinitial onditionislinear. Thusthere onstru tionis exa there. Ifanyos illationdevelop(i.e. whenin thetransient,weare outare outof

[

1

4

,

3

4

]

),thesolutionwill be attra tedtothe loseststablelimitsolution,i.e.

u

∞

=

1

2

. twopossible ases

5 Example of an ellipti problem

Asanillustration,we onsider thesimpleproblem

(κ(x, ω)u

′

₎

′

_{= 0}

_{, x}

_{∈]0, 1[}

10 Abgrall

-1

0

1

2

0

0.2

0.4

0.6

0.8

1

uniform

Gaussian

Exponential

Figure2: Repartitionfun tionfortheEDO(8)andthethreepdf.

Ψ(U

_∞

)

Uniform Gauss Poisson

U

_∞

∈]

1

4

,

1

2

[

1

2

erf

(1/4)

−

erf

(

−1)

erf

(1)

−

erf

(

−1)

1

_{− e}

−1/2

1

_{− e}

−2

exa t

≈ 0.64458450997090083670

≈ 0.2558207969

omputed 0.25

0.644584509951823

0.246768643147764

U

_∞

_∈]

1

2

,

3

4

[

3

4

erf

(3/4)

−

erf

(

−1)

erf

(1)

−

erf

(

−1)

1

_{− e}

−3/4

1

− e

−2

exa t

≈ 0.90043482545390904212

0.6102173907

Computed

0.75

0.900434825432325

0.615653168991007

Table1: Numeri al valuesfound for

Ψ

at the riti al points. There are101points in the probability mesh.

where

ω

∈ Ω

israndomwithagivenpdf,and

κ(x, ω) > κ

0

> 0

on

]0, 1[

×Ω

whi hsolutionis

u(x, ω) =

R

x

0

1

κ(x,ω)

dx

R

1

0

1

κ(x,ω)

dx

.

Equation(9)isapproximatedonthemesh

x

i

= i∆x

,

i = 0, . . . , N

by(themesh isuniform)

κ

i+1/2

(u

i+1

− u

i

)

− κ

_i−1/2

(u

i

− u

i−1

) = 0

1

≤ i ≤ N − 1

u

0

= 0, u

N

= 1

thatis

u

i

=

κ

i+1/2

u

i+1

+ κ

i−1/2

u

i−1

κ

i+1/2

+ κ

i−1/2

u

0

= 0, u

N

= 1,

(10)

and

κ

j+1/2

:= κ(x

j+1/2

, ω)

.

Therestofthemethodissimilar,andwehaveusedthesamequadratureformulaasin theprevious

paragraph.

Inthenumeri alexamples,wehave hosen

κ(x, ω) = 4(x

− 0.5)

2

_{+ 0.33 cos}

2πω

4

(x − 0.5) + 0.01

Here

κ

≥ 0.003

. ThepdfisaGaussiandistributionwithmean

0.5

andvarian e

0.5

.

Ingure3, weshowthe resultobtainedfor

101

pointsin thespa e dire tion and

21

points in the probabilitydire tion. Againa enteredre onstru tionisused.

Simple andexibleUn ertaintyQuanti ationfor non linear problems 11

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

mean

mean-sigma/2

mean+sigma/2

Figure 3: Mean and varian e for

101

points in the spa e dire tion and

21

points in the probability dire tion.

Theexa tsolutionis

u(x, ω) =

arctan

x−ϕ

δ

+ arctan

ϕ

δ

arctan

1−ϕ

_δ

+ arctan

ϕ

δ

with

ϕ = 0.5

−

0.33

₈

cos(2πω),

δ =

r

0.0025

−

0.35

₈

2

cos

2

2πω

4

.

sothatitiseasyto estimatethe

L

∞

and

L

2

errorsofthemeanand varian e. This isdoneongure4.

Theresultsarese ondordera uratewithrespe ttothespa evariable. Weseealsothattheresultsare

almost independentof thedis retisationin the probabilitydire tion. Converged resultsareobtained

alreadywith9 ellsintheprobabilitydire tion.

0.1

1e-05

0.0001

0.001

mean L2

mean Linf

variance L2

variance Linf

0.01

0.0001

0.01

mean L2

Mean Linf

variance L2

variance Linf

slope 2

(a) (b)

Figure 4: Errors in the mean and varian e. (a) represents the error for axed spa e dis retisation

(

∆x = 10

−2

)andavaryingprobabilitydis retisation(from10to80)points. (b)representstheerrorfor

axedprobabilitydis retisation(50points)and avaryingspa edis retisation(from

20

to

101

points). Theslope

−2

isrepresented. Theresultsareobtainedwith a enteredre ontru tion.

6 Example of the onve tion and Burgers equations

OurnextexampleistheBurgersequation(1). Inordertoillustratethestrategy,westartfromaMUSCL

typepredi tor orre torse ond orders heme.

∂u

∂t

+

∂f (u)

∂x

= 0

t > 0, x

∈ R

u(x, t) = u

0

(x)

x

∈ R

(11)

with periodi boundary onditions on

[0, 2π]

. For the onve tion problem, we take

f (u) = u

and for the Burgers equation, we take

f (u) = u

2

_/2

. The interval

[0, 2π]

is subdivided into equally spa ed subintervals

[x

i−1/2

, x

i+1/2

]

where

x

j+1/2

=

x

j

+x

j+1

2

with

x

j

= j∆x

.

Astandard onservativeformulationfor(11)writes, initsrstorderversion,

u

n+1

i

= u

n

i

− λ



ˆ

f (u

n

i

, u

n

i+1

)

− ˆ

f (u

n

i−1

, u

n

i

)



(12)

with

λ = ∆t/∆x

. These ond orderpredi tor orre tors hemeweuseis

u

n+1/2

_i

= u

n

i

−

λ

2



ˆ

f (u

n,L

_i+1/2

, u

n,R

_i+1/2

)

_{− ˆ}

f (u

n,L

_i−1/2

, u

n,R

_i−1/2

)



(13a)

u

n+1

i

=

u

n

i

+ u

n+1/2

i

2

− λ



ˆ

f (u

n+1/2,L

_i+1/2

, u

n+1/2,R

_i+1/2

)

− ˆ

f (u

n+1/2,L

_i−1/2

, u

n+1/2,R

_i−1/2

)



(13b) where

u

R

j+1/2

= u

j

+ δ

j

/2,

u

L

_j−1/2

= u

j

− δ

j

/2

and

δ

j

=

L(u

j+1

− u

j

, u

j

− u

j−1

)

and

L

isastandardlimiter. Inthispaper,wehave hosenthesuperbee limiter,

L(a, b) = max 0, min(2a, b), min(a, 2b)

.

Theux istheMurmanRoeux withHarten'sentropyxfortheBurgersequation,

ˆ

f (a, b) =

1

2



f (a) + f (b)

_{− λ(a, b)(b − a)}



(14)

where,if

f

˜

′

(a, b)

istheRoeaverage,

λ(a, b) =







| ˜

f

′

_{(a, b)}

_|

if

| ˜

f

′

(a, b)

| > ε

| ˜

f

′

_{(a, b)}

_|

2

_{+ ε}

2

2ε

else.

Here,wehavetaken

ε = 0.01

.

Wehavealsoruna asewhere

S(x) = (sin

2

_(x))

_′

. Here,theux in(14)ismodiedbyrepla ing

f (a)

and

f (b)

by

f (a)

_{− sin}

2

(x

a

),

f (b)

− sin

2

(x

b

)

where

x

a

and

x

b

arethephysi al lo ationsoftheunknown

a

and

b

,i.e.

ˆ

f (a, b) =

1

2



f (a) + f (b)

_{− λ(a, b)(b − a)}



−

1

₂



sin

2

(x

a

) + sin

2

(x

b

)



.

(15)

Assume now that the initial ondition, though still periodi of period

2π

depends on a random parameter. Ea hof (12) ,(13a)and (13b)issimilar,sowe on entrateon(12). Wehaveagain, forany

physi al ell

[x

i=1/2

, x

i+1/2

]

andany

Ω

j

,

E(u

n+1

_i

_|Ω

j

) = E(u

n

i

|Ω

j

)

− λ



E ˆ

f (u

n,L

_i+1/2

, u

n,R

_i+1/2

)

_|Ω

j

− E ˆ

f (u

n,L

_i−1/2

, u

n,R

_i−1/2

)

|Ω

j



Simple andexibleUn ertaintyQuanti ationfor non linear problems 13 This amountsto omputing theux expe tan ies,

E ˆ

f (u

n,L

i+1/2

, u

n,R

i+1/2

)

|Ω

j

. Forthis, again, we

re on-stru t the variables

u

n,L

i+1/2

and

u

n,R

i+1/2

with the te hnique of se tion 3.1. In the numeri al examples,

wehaveeitherused athird ordera urate entered re onstru tionorthese ond orderENOoneinthe

probabilitydire tion. As expe ted, the entered re onstru tiongenerates (slight)os illations. Weonly

displaytheresultswiththeENOone. Intheprobabilitydimension,weuseaGaussianquadrature

for-mulawithtwopoints. Hen e,ifwehave

N

prob

ellsintheprobabilitydire tion,weneed

2N

prob

solution evaluations. The time step,i.e.

λ

is hosenbya worst ase s enario,but this strategymight be over pessimisti . Furtherstudiesare ertainlyneeded.

6.1 Conve tion problem

Theexampleisthe onve tionequation(velo ityofunity)withaninitial ondition

u

0

. Sin etheexa t solution is

u(x, t) = u

0

(x

− t)

it is easy to evaluate the error on the mean and the varian e. In the example, thepdf is

N (1, 1)

. In gure5, wehave displayedthe resultsfor 11, 21 and 41points in the probability spa e. The results are se ond order a urate in spa e. We have also displayed the same

100

1e-05

0.0001

0.001

0.01

Mean Linf

Mean L2

Variance Linf

Variance L2

slope 2

100

1e-05

0.0001

0.001

0.01

0.1

Mean Linf error

Mean L2 error

Variance Linf error

Variance L2 error

slope 2

11pointsinprobspa e 21pointsinprobspa e

100

1e-05

0.0001

0.001

0.01

0.1

Mean Linf

Mean L2

Variance Linf

Variance l2

slope 2

41pointsin probspa e

Figure5: Errorsinmeanandvarian eforthe onve tionproblemwiththepdf

N (1, 1)

,theCFLnumber

is

0.75

. Thenaltimeis

t = 1.5

andthedomainis

[0, 2π]

. Theslopelimiteris enter intheprobability

spa e.

resultsbutforaGaussianlawwiththesamemeanand thevarian e

σ = 0.1

. Inthat ase,thegradient ofthepdfislargerandthenthequadratureformula(twopointGaussian)islesse ient. This ouldbe

improvedbyan adapted mesh, i.e. amesh where the measure,withrespe tto the probabilitylaw, of

ea hprobability ellwouldbethesame[12℄. However,theresultsofFigure6indi atethesametypeof

errors. This is onrmedbygure7where thesame asearererunwithaxednumberofmeshpoints

and

11

to

41

points in the probability spa e. Note that the sameexample gives, in the deterministi ase, thefollowingerrors:

0.799 10

−2

in themax norm and

0.7919 10

−3

for the

L

2

norm. This shows

100

1e-06

1e-05

0.0001

0.001

0.01

Mean Linf

Mean L2

Variance Linf

Variance L2

slope 2

100

1e-06

1e-05

0.0001

0.001

0.01

Mean Linf

Mean L2

Variance Linf

Variance L2

Slope 2

11pointsinprobspa e 21pointsinprobspa e

100

1e-06

1e-05

0.0001

0.001

0.01

Mean Linf

Mean L2

Variance Linf

Variance L2

Slope 2

41pointsin probspa e

Figure 6: Errors in mean and varian e for the onve tion problem with the pdf

N (1, 0.1)

, the CFL numberis

0.75

. Thenaltimeis

t = 1.5

andthedomain is

[0, 2π]

. Theslopelimiter is enteredin the probabilityspa e.

20

30

40

1e-05

0.0001

0.001

0.01

Mean Linf

Mean L2

Variance Linf

Variance L2

20

30

40

0.0001

0.001

0.01

Mean Linf

Mean L2

Variance Linf

Variance L2

σ = 0.1

σ = 1

Figure7: Errorsinmeanandvarian eforthe onve tionproblem withthepdf

N (1, σ)

,

σ = 0.1

and

1

.

Thenaltimeis

t = 1.5

andthedomainis

[0, 2π]

,theCFLnumberis

0.75

andthereare

51

meshpoints.

Theslopelimiteris enteredin theprobabilityspa e.

6.2 Burgers equation

Theinitial onditionis

u

0

(x, ω) =

|a(ω)| sin(2x − a(ω))

with

a(x) = x

. This resultsin a adis ontinuous solutionwith adis ontinuity lo alised at

x

ω

= a(ω)

.

Simple andexibleUn ertaintyQuanti ationfor non linear problems 15 1. AGaussian pdf:

N (m, σ)

with

m = 1

and

σ = 0.1

or

1

onditionedby

x

∈ [−3, 3]

.

2. Adis ontinuouspdfdened by

dµ =

f

w

dx

with

f (x) =















0

if

x

≤ −1

0.1

if

x

∈] − 1, 0.5[

1

if

x

∈ [0.5, 1[

0

if

x

≥ 1

and

w = 0.65

(i.e. theweightingfa torso thattheintegralof

f /w

isequalto

1

.)

OnFigure8,wehaverepresentedforthetwotypesofpdfs. Theaimofthispi tureistoshowthehighly

os illatorybehaviorofthesolution. Togetthem,wehadtousetheENOlikelimiter: the enteredone

produ es os illationsasexpe ted. Inthe aseof theGaussian pdfs, therealisationsare thesame,but

their weights aredierent. Figure 9is moreinteresting. We show, forea h example, themean, mean

0

0.5

1

1.5

2

2.5

3

3.5

-3

-2

-1

0

1

2

3

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

"solution.dat"

0

0.5

1

1.5

2

2.5

3

3.5

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

"solution.dat"

withtheGaussianpdfs withthedis ontinuouspdf

Figure8: Plotofea hrealisationfortheGaussianpdfsandthedis ontinuousone.

±σ/2

oftheinitialsolutionsandthe omputedsolutionsattime

1.5

. The omputationshavebeendone with

101

spa e ells and

21

ellsin theprobability dire tion. Clearly the solutionsare verydierent. Oneshould not be surprised by thehighly os illatory behaviorofthe Gaussian solutionwith

σ = 1

. A lose look at Figure 8plus the understanding that the pdf is notverypeaky in that ase helps to

understandthattheweightofthedis ontinuitiesforamplitudes(andphases)near

±3

playanimportant roleintheevaluationofthemeans.

6.3 Burgers equation with sour e term

Thisexampleistakenfrom [5℄. Theinitial onditionis

u(x, 0) = β sin x

withperiodi boundary onditionson

[0, π]

. Theexa tsteadysolutionis

u

_∞

(x, β) = lim

t→+∞

u(x, β, t) =



u

+

_{= sin x}

if

0

≤ x ≤ X

s

u

−

₌

_{− sin x}

if

X

s

≤ x ≤ π

wherethesho klo ationis

X

s

=



arcsin

p1 − β

2

if

− 1 < β ≤ 0

π

_{− arcsin}

_{p1 − β}

2

if

0 < β

≤ 1

0

1

2

3

-1

0

1

Mean

Mean-sigma/2

Mean+sigma/2

Initial mean

Initial mean +sigma/2

0

1

2

3

-1

0

1

Mean

Mean-sigma/2

Mean+sigma/2

Initial mean

Initial mean-sigma/2

Initial mean +sigma/2

σ = 0.1

σ = 1

0

1

2

3

-0.4

-0.2

0

0.2

0.4

Mean

mean-sigma/2

mean+sigma/2

Initial mean

Initial mean-sigma/2

Initial mean+sigma/2

Dis ontinuouspdf

Figure 9: Meansandmean

±

σ

2

forea hpdf.

In[5℄is onsideredthe aseof

β

randomwhere

α =

β

1

_{− β}

2

is Gaussianwithmean

m

andvarian e

σ

. Wehave

β =







−1 +

√

1 + α

2

2α

if

α

6= 0

0

else.

We see that

β

dened as this is alwaysin

[

−1, 1]

, soa sho k always exists. Thedensity of the sho k lo ationis

p(x) =







1

σ

√

2π

1 + β

2

(1

_{− β}

2

₎

2

e

−[(x−m)

2

_/2σ

2

_]

sin(x)

if

x

∈ [0, π],

0

else

Inthenumeri alse tion,wearegoingto evaluatetherepartitionfun tion

x

7→ P

σ,m

(X

x

≤ x)

. An easy al ulationshowsthat

P

σ,m

(x) = P (X

s

≤ x) =

1

σ

√

2π

Z

−

_{sin2 x}

cosx

−∞

e

−

(x−µ)

2σ2

dx =







erf

(

−

cosx

sin2 x

−m

σ

√

2

)

if

0

≤ x ≤

π

2

1

2

+

erf

(

−

cosx

sin2 x

−m

σ

√

2

)

if

π

2

≤ x ≤ π

Inorder to show theexibility of themethod, we also onsider thesum ofthe previouspdf anda

Dira measure. Morepre isely,

dµ =

1

I

1

σ

√

2π

e

−[(x−m)

2

/2σ

2

_]

1

[A,B]

dx + θδ

ω

c

!

(17a)

where

ω

c

∈]A, B[

,

θ

≥ 0

,

I

isthenormalizingfa tor

I =

Z

B

A

1

σ

√

2π

e

−[(x−m)

2

_/2σ

2

_]

dx + θ.

(17b)

Simple andexibleUn ertaintyQuanti ationfor non linear problems 17 and

δ

ω

c

istheDira distributionat

ω = ω

c

. Inthat ase,therepartitionfun tionis

P

µ

(x) = P

σ,m

(x) + θ1

[Y,B]

where

Y

isthesho klo ationfor

α = x

c

,i.e.

Y = π

_{− arcsin}



p1 − β

2



,

x

c

=

β

1

_{− β}

2

.

Thesimulationisinitialised with,in ea h ell,theexpe tan y ofthespatialaveragedsolution. The

solutiondevelopssho ks. Sin ethemethod isanite volumeone,these sho ksareatbestknownwith

ana ura yof

O(∆x)

. Wehaveadoptedthefollowingpro eduretolo alizethesho kposition: forea h ell

[ω

j−1/2

, ω

j+1/2

]

,wedeterminethe ell

]x

i

j

−1/2

, x

i

j

+1/2

[

su hthat











E(u

i+1

|Ω

j

)

− E(u

i

|Ω

j

)

∆x











ismaximal. If thiso ursat twodierentlo ations,we hoosethesmallestindex. On etheindex

i

j

is known,we ompute

p(x

i

j

) = P (x

≤ x

i

j

)

Notethatformany

j

,

|i

j

− i

j+1

| > 1

: thismeansthatthereareholesinthenumberingsin ethesho k

dete tionpro edure maywellde ide that for

j

and

j + 1

,the sho k hasthesamelo ation. Seegure 10 foran illustration. It order to llthese gaps and to beableto draw

x

i

7→ p(x

i

)

, wemakealinear interpolationbetweentwo onse utivegaps. Thenumeri alrepartitionfun tionis omparedtotheexa t

0

0

1

1

0

0

0

1

1

1

0

0

1

1

00

00

11

11

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

0

0

1

1

xi

x

ω

Figure 10: Illustrationof thesho klo ationme hanism. Thetheoreti alsho klo ationisrepresented

bythedotted urve. Thebla kspotsarethesho klo ations. Considerthethi kverti alline : thereis

ajumpof

2∆x

whengoingto

ω

j

to

ω

j+1

sothatthere isno

ω

l

forwhi h

x

i

orrespondstoasho kfor therealisationsin

[ω

l

, ω

l+1

]

.

repartitionone,andtotherepartitionfun tion omputedfortheexa tsho klo ations orrespondingto

theevents

ω

l

+ω

l+1

2

,

l = 1,

· · ·

.

Inthe rstset of example,wehave onsidered aGaussian distribution with

m = 1

and

σ = 1

for

101

and

151

mesh points in spa e, and

5

and

11

pointsin the probability dire tion. This orresponds to

100

and

150

'spa e ellsand

4

and

10

probability ells. Thepoint repartitionis uniform in both dire tions. The resultsaredisplayedin Figure 11. A good agreementis obtained: rememberthat the

sho klo ationsareat mostknownwithan

O(∆x)

error. A loseinspe tionofthegureindi atesthat for

x

≈ π

,thenumeri alrepartitionfun tion hasadis ontinuitywhi h isnotexistingin theexa tone.

18 Abgrall

1.5

2

2.5

3

0

0.2

0.4

0.6

0.8

1

Numerics

Exact using space mesh

Exact using prob mesh

2

3

0

0.2

0.4

0.6

0.8

1

Numerics

Exact using space mesh

Exact using prob mesh

spa e: 101,prob: 5 spa e: 101,prob: 11

2

3

0

0.2

0.4

0.6

0.8

1

Numerics

Exact using space mesh

Exact using prob mesh

2

3

0

0.2

0.4

0.6

0.8

1

Numerics

Exact using space ,esh

exact using prob mesh

spa e: 151,prob: 5 spa e: 151,prob: 11

Figure11: SolutionforthedoublenozzleproblemobtainedwiththeGaussianlaw. Numeri :

distribu-tionobtainedfrom thesolver. Exa t : exa tdistribution. Exa tfrom prob mesh: distribution at the

points

ω

j+1/2

.

Theexplanationisthefollowing: ourvariablesaretheexpe tan iesoftheaveragedvariables:

E(u

i

|Ω

j

)

, whi h anbethoughasthevalueof

u

i

at

ω

j+1/2

=

ω

j

+ω

j+1

2

. Hen e,thesho klo ationforthisrandom

parameterdoesnot orrespondtothe randomparameter

α

forwhi h

x

l

= arcsin



p1 − β

2

_(α)



. The

dieren e isthemost visiblefor therstrow. Thisis whywehave alsoplotted theexa tvalue ofthe

repartitionfun tionforthe

ω

j+1/2

. Wealsoseeajump,andtheagreementisnowverygood.

TheFigure12representsthesametypeofresultsforthesingularpdf(17) . Here

θ = 1

and

ω

c

= 0.5

. This orresponds to the sho k lo ation

x

c

≈ 1.997874914

. The same omments as in the previous examples anbegiven.

7 Example of the Euler equations

Themethod iseasilyextendedtotheEulerequations. Thebases hemeisthese ondRoes hemeusing

thesuperbee limiteron the hara teristi variables. Weusethewaveinterpretationof theRoes heme

to onstru tthese ondorders heme(see[13℄). Even-thoughthes hemeusethe onservativevariables

W = (ρ, ρu, E)

for the time evolution, the main variables are the density

ρ

, the velo ity

u

and the

pressure

p

. Thetotalenergyisrelatedto thesevariable viaanequationof state. Here,wehave hosen aperfe tgasEOS,

E =

p

γ

_{− 1}

+

1

2

ρu

2

_,

wheretheratioofspe ifheats

γ

maybenonuniform. Inthat ase,weusetheversionoftheRoes heme developedin [14℄. Examplesofthistypehavebeensu essfullyrun,butarenotreportedhere.

Simple andexibleUn ertaintyQuanti ationfor non linear problems 19

2

0

0.2

0.4

0.6

0.8

1

Numerics

Exact with space mesh

Exact with prob mesh

1.4

1.6

1.8

2

2.2

2.4

0

0.2

0.4

0.6

0.8

1

Numerics

Exact with space mesh

Exact with prob mesh

spa e: 101,prob: 11 spa e: 201,prob: 11

Figure 12: Solution forthedoublenozzle problemobtainedwiththelaw(17). Numeri : distribution

obtainedfromthesolver. Exa t: exa tdistribution. Exa tfromprobmesh: distributionatthepoints

ω

j+1/2

.

Inorderto showtheversatilityofourmethod,wehaverun themethodontwo lassesofexamples,

namelyasho ktubelikeproblem, andtheintera tionofadensitysinewavewithasho k(asproposed

byShuandOsher).

Theun ertaintyparametersarenowtwodimensional,andintheexamplesweshow,wehave hosen

aGaussian typelawwheretheun ertaintyare orrelated. Thepdfis

f (ω

1

, ω

2

) = Ke

−

ω2

_{1 −(ω2 −1)}

2

−ω1(ω2 −1)

2

,

(18)

and

K

isanormalizing oe ientand

(ω

1

, ω

2

)

∈ [−3, 3]

2

.

7.1 Sho k tube like test ases

Theinitial onditionsare

ˆ If

x

≤ 0.5

,

ρ(x) = ρ

L

(1 + 0.2 sin(ω

1

))

u(x) = u

L

p(x) = p

L

(1 + 0.2 sin(ω

2

))

ˆ else

ρ(x) = ρ

R

(1 + 0.2 sin(ω

1

))

u(x) = u

L

p(x) = p

R

(1 + 0.2 sin(ω

2

))

with

γ = 1.4

. Thedensityof the randomvariables

ω

1

and

ω

2

is givenby (18). TheCFL onditionis

setto

0.75

. TheresultsaredisplayedinFigure14. Againseveralresolutionintheprobabilitydire tions havebeenrun. Again,weseethattheresults,onthat ase,areindistinguishable.

7.2 Sho k-turbulen e intera tion

Theinitial onditionsare, with

ρ

L

= 3.857143

,

u

L

= 2.629369

,

p

L

= 10.333333

and

u

R

= 0.

,

p

R

= 1.

, and

s

loc

=

−4

,

ˆ If

x

≤ s

loc

,

ρ(x) = ρ

L

u(x) = u

L

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Mean

Mean-sigma/2

Mean+sigma/2

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Mean

Mean-sigma/2

Mean+sigma/2

Pressure Density

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

Mean

Mean-sigma/2

Mean+sigma/2

0.6

0.8

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

Mean 10x10

Mean-sigma/2 10x10

Mean+sigma/2 10x10

Mean 20x20

Mean-sigma/2 20x20

Mean+sigma/2 20x20

Velo ity Comparison

Figure 13: Sod tube ase with random densities and pressure. Two orrelated random variables are

used.

ˆ else

ρ(x) = 0.5 (1 + 0.2 sin(5x))

u(x) = u

R

p(x) = p

R

(1.0 + 0.2 sin(ω

1

+ ω

2

))

On gure14, thesimulation is donewith

400

points in the spa edire tion and

10

× 10

or

20

× 20

in the probability spa e. Againthe same on lusion holds: there is little dependen y on the probability

dire tionresolution.

8 CPU onsiderations

Forseveralofthepreviousproblems,wegiveontable2theCPU ostobtainedonaMa BookProrunning

at 2.4Ghz with2Goof ramand theIntel(10.1) ompiler (nooption). No parti ularoptimisationhas

beenperformed. ThistableshowthattheCPUisratherlow.

9 Con lusions

Wehavedes ribedandillustratedageneralmethodthatenableto omputesomestatisti sonthesolution

ofaPDEorandODE,linearand nonlinear. Thesour eofun ertaintymaybemultidimensional. The

te hniquerelies on are onstru tion method in the random variable. This re onstru tion te hnique is

standardinnitevolumes heme,ex eptthat themeasureisnolongertheLebesguemeasurebutisthe

probability law. This method enable to onsider random variable that depend onpossibly orrelated

randomvariables. Thesolutiondes riptionusestheexpe tan iesofagivenfun tion onditionedbythe

0

0.2

0.4

0.6

0.8

1

0.001

0.01

0.1

Pressure 10x10

Pressure 20x20

0

0.2

0.4

0.6

0.8

1

0.001

0.01

0.1

Density 10x10

Density 20x20

Pressure Density

0

0.2

0.4

0.6

0.8

1

0.001

0.01

0.1

Velocity 10x10

Velocity 20x20

A A A A A A A A A A A A A A A A A

A

A

A

A

A A A A A A A A A A A A A A A A A

0.5

0.52

0.54

0.56

0.58

0

0.1

0.2

0.3

0.4

0.5

P O2 10x10

Rho O2 10x10

U O2 10x10

P O2 20x20

Rho O2 20x20

U O2 20x20

A A

Velo ity Varian e

Figure 14: Shu andOsher asewitharandom post sho kstate. Two orrelatedrandom variablesare

used.

Problem numberofpoints(spa e) numberofpoints(prob) CPU

BurgersNozzle 101 11 10s

BurgersNozzle 201 11 43s

Burgers 101 11 0.8s

Burgers 201 11 3s

Euler2(Shu-Osher) 200 20x20 152s

Table2: CPU ostforseveralproblems. Thestopping riteriaare: iterative onvergen e

10

−6

forthe

steadynozzle;Burgers: naltime

t = 1

;Eulerequationswith2un ertainties: naltime: 1. Thepdfs areGaussian.

Given one problem and an integration method, we have shown how to onstru t as heme whi h

enableto approximatethese onditional expe tan ies.

The method is illustrated onseveral types of problems, linear and non linear, in luding the Euler

equations. Themethodis heapandexible. Itsa ura yislinkedtothea ura yofthere onstru tion

andthedeterministi solutionmethod. Weshownumeri allythatthemainsour esoferrorsarestillin

thedeterministi s heme. Correlatedsour eofun ertainties aneasilybeimplementedinthisframework,

aswellasverygeneralpdfs.

In a future work, we will extend this to the Navier Stokes equations, see how the ase of many

randomvariables anbehandledinthisframework.Notethat[15℄,usingarelatedmethod,hasalready

Referen es

[1℄ R.H.CameronandW.T.Martin.Thebehaviorofmeasureandmeasurabilityunder hangeofs ale

inWienerspa e. Bull. Am.Math. So ., 53:130137,1947.

[2℄ X. Wan and G. E. Karniadakis. BeyondWiener-Askeyexpansions: handlingarbitrary PDFs. J.

S i.Comput.,27(1-3):455464,2006.

[3℄ X.WanandG.E.Karniadakis.Multi-elementgeneralizedpolynomial haosforarbitraryprobability

measures. SIAMJ.S i. Comput.,28(3):901928,2006.

[4℄ B. J. Debuss here, H. N. Najm, P. P. Pébay, O. M. Knio, R. G. Ghanem, and O. P. Le Maître.

Numeri al hallengesintheuseofpolynomial haosrepresentationsforsto hasti pro esses.SIAM

J.S i. Comput.,26(2):698719,2004.

[5℄ D.XiuandJ.S.Hesthaven. High-order ollo ationmethodsfordierentialequationswithrandom

inputs. SIAMJ. S i. Comput.,27(3):11181139,2005.

[6℄ O.P.LeMaître,H.N.Najm,R.G.Ghanem,andO.M.Knio.Multi-resolutionanalysisofWiener-type

un ertaintypropagations hemes. J. Comput.Phys.,197(2):502531,2004.

[7℄ A.Harten,B.Engquist, S.Osher,and S.Chakravarthy. Uniformlyhigh ordera urateessentially

non-os illatorys hemes.III. J.Comput. Phys.,71:231303,1987.

[8℄ Timothy J. Barth and Paul O. Frederi kson. Higherorder solution of theeuler equations on

un-stru turedgridsusingquadrati re onstru tion. AIAApaper90-0013,January1990.

[9℄ R. Abgrall. On essentially non-os illatory s hemes on unstru tured meshes: Analysis and

imple-mentation. J. Comput.Phys.,114(1):4558,1994.

[10℄ R. Abgralland ThomasSonar. On theuse of Mühlba h expansionsin the re overystep of ENO

methods. Numer.Math., 76(1):125,1997.

[11℄ P.LesaintandP.A.Raviart. Onaniteelementmethodforsolvingtheneutrontransportequation.

Math.Aspe tsniteElem. partialDier.Equat.,Pro .Symp.Madison1974,89-123(1974).,1974.

[12℄ T.J.Barth. Private ommuni ation,may2008.

[13℄ R.J.LeVeque. Finitevolume methods forhyperboli problems. CambridgeUniversityPress,2002.

[14℄ R. Abgralland Smadar Karni. Ghost-uids for the poor: asingle uid algorithm formultiuids.

InHyperboli problems: theory,numeri s,appli ations,Vol.I,II(Magdeburg, 2000),volume141of

Internat.Ser.Numer. Math., 140,pages110.Birkhäuser,Basel,2001.

[15℄ T.J. Barth. Deterministi propagation of model parameter un ertainties in ompressible

navier-stokes al ulations. 6thInternationalConferen e onEngineering and ComputationalTe hnology,

Contents

1 Introdu tion 3

2 Review of existingte hniques 4

3 Prin iplesofthe method 5

3.1 Some omputationalremarks . . . 5

3.2 Ageneralstrategy . . . 7

4 Exampleof an ODE 8

5 Exampleof an ellipti problem 9

6 Exampleof the onve tion and Burgersequations 12

6.1 Conve tionproblem . . . 13

6.2 Burgersequation . . . 14

6.3 Burgersequationwithsour eterm . . . 15

7 Exampleof the Eulerequations 18

7.1 Sho ktubeliketest ases . . . 19

7.2 Sho k-turbulen eintera tion . . . 19

8 CPU onsiderations 20