Variance reduction in stochastic homogenization : proof of concept, using antithetic variables

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of concept, using antithetic variables

Ronan Costaouec, Claude Le Bris, Frédéric Legoll

To cite this version:

Ronan Costaouec, Claude Le Bris, Frédéric Legoll. Variance reduction in stochastic homogenization : proof of concept, using antithetic variables. [Research Report] RR-7207, Inria. 2010, pp.1-21. �inria- 00457946�

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a p p o r t

d e r e c h e r c h e

Thème NUM

Variance reduction in stochastic homogenization:

proof of concept, using antithetic variables

Ronan Costaouec — Claude Le Bris — Frédéric Legoll

N° 7207

February 2010

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RonanCostaoue

∗†

, ClaudeLe Bris

∗†

, FrédériLegoll

‡†

ThèmeNUMSystèmesnumériques

ProjetMICMAC

Rapportdereherhe n°7207February201021pages

Abstrat: We showthat we an redue the variane in asimple problem of stohasti

homogenization using the lassialtehniqueof antitheti variables. The setting, and the

presentation, aredeliberately kept elementary. Wepointout the main issues, show some

illustrative results, and demonstrate, both theoretially and numerially, the eieny of

theapproahonsimpleases.

Key-words: stohastihomogenization,varianeredution,antithetivariables

∗

CERMICS,EoleNationaledesPontsetChaussées,6et8avenueBlaisePasal,CitéDesartes,77455

Marne-la-ValléeCedex2,Frane. Contat: {ostaour,lebris}ermis.enp.fr

†

INRIA Roquenourt, MICMAC team-projet, Domaine de Volueau, B.P.105, 78153 LeChesnay

Cedex,Frane.

‡

Institut Navier, LAMI, EoleNationale desPonts et Chaussées, 6 et 8 avenue Blaise Pasal, Cité

Desartes,77455Marne-la-ValléeCedex2,Frane.Contat: legolllami.enp.fr

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Résumé: Dansetravail,nousmettonsenoeuvreunetehniquederédutiondevariane

dans le adre de l'homogénéisation stohastique. Plus préisément, nous montrons qu'il

est possiblede réduirelavarianedela matriehomogénéiséealuléenumériquement,en

utilisantlatehniquedesvariablesantithétiques. Nousavonsvolontairementhoisidenous

plaer dans un adre de travail simple, an d'identier les prinipales diultés. Nous

démontrons,àlafoisthéoriquementet numériquement,l'eaitéde l'approhe, dansdes

assimples.

Mots-lés : homogénéisationstohastique,rédutiondevariane,variablesantithétiques

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1 Introdution

Several settingsin homogenizationrequirethesolutionoforretorproblemsposedonthe

entirespaeR^d. Inpratie,trunationsofthese problemsoverboundeddomains areon- sidered and the homogenizedoeients are obtained in the limit of largedomains. The

questionarisestoaeleratesuhomputations. Inthedeterministiase,aelerationteh-

niquesreminisentfromsignallteringhavebeenintroduedin[5℄. Theworkhassinethen

beensigniantlyimprovedin[12℄. In[5℄,itwasshownthataelerationtehniqueseient

fordeterministiproblems donotneessarilyperformwellin thestohastiframework. In

thelatterase,themaindiultyisrelatedtotheintrinsinoisepresentin thesimulation.

Thehallenge is onsequentlynotthat muh to improvethe rateof onvergene,whih is

intrinsiallythat of the entral limit theorem, but rather to redue the variane, thereby

improvingthe prefator of the onvergene givenby the entral limit theorem. Although

very well investigated in other appliation elds suh as nanial mathematis, variane

redutiontehniques seemto havenot beenapplied to the ontext of stohasti homoge-

nization. Thepurpose ofthepresentontributionis topresentarstattempt inreduing

thevarianeinstohastihomogenization.Forthispurpose,weonsiderasimplesituation,

and a simple variane redution tehnique. The probability theoreti arguments we will

makeuseofareelementary. Theequationunder onsiderationisasimpleelliptiequation

in divergeneform, withasalaroeient. The oeientisassumed to onsistof inde-

pendent,identiallydistributedrandomvariablessetonasimplemesh(see(2)below). The

tehniqueusedforvarianeredutionisthatofantithetivariables. Oursettingisaademi

innature,somewhatfarfromphysiallyrelevantases,andelementary. Manymorediult

situations ouldbeaddressed: othertypesof stationaryergodi oeients,matrixrather

than salaroeients, other typesof equations, other tehniques for varianeredution,

...The present ontribution is aproof of onept: variane redutionan be ahieved in

stohastihomogenization. Futureworks[3,4,11℄willprovidemoredetailsonthenumeris

andthetheory,andalsoaddresssomeofthemanypossibleextensionsmentionedabove.

2 Stohasti homogenization theory

Althoughwewish to keepthe mathematialformalismaslimitedaspossiblein ourexpo-

sition, weneed to introdue the basisetting of stohasti homogenization (see[16℄ for a

similarpresentationand relatedissues). Throughoutthis artile,(Ω,F,P)^is ^aprobability spaeandwedenotebyE(X) =R

ΩX(ω)dP(ω)^theêxpetation^valueôfâny^random^variable X ∈L¹(Ω, dP)^. ^We^next^xd∈N^∗ (theambientphysialdimension),andassumethatthe group(Z^d,+)âtsônΩ^. ^We^denote^by(τk)k∈Z^d^thisâtion,ândâssume^thatît^preserves^the

measureP,thatis,forallk∈Z^d andallA∈ F^,P(τkA) =P(A)^. ^Weâssume^that^theâtion τ îsêrgodi,^that îs,îfA∈ F îs ^suh^that τkA=A^forânyk∈Z^d, thenP(A) = 0ôr^1. În

addition, wedene thefollowingnotionof stationarity(see [7℄): any F ∈L¹_loc R^d, L¹(Ω)

issaidtobestationary if,forallk∈Z^d,

F(x+k, ω) =F(x, τkω), ⁽¹⁾

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almosteverywhereinxândâlmost^surely. În^this^setting,^theêrgodi^theorem^[15, ^17℄ ân

bestatedasfollows: LetF ∈L^∞ R^d, L¹(Ω)

beastationary randomvariable inthe above

sense. Fork= (k1, k2, . . . kd)∈Z^d,weset |k|^∞= sup

1≤i≤d|ki|^. ^Then 1

(2N+ 1)^d X

|k|∞≤N

F(x, τkω)_N→∞−→ E(F(x,·)) ⁱⁿL^∞(R^d), ^almost ^surely.

Thisimpliesthat (denoting byQ^the ^unit^ubeⁱⁿR^d)

Fx ε, ω _∗

−⇀

ε→0

E Z

Q

F(x,·)dx

in L^∞(R^d), ^almost ^sur^ely.

Besides tehnialities, the purpose of the above setting is simply to formalizethat, even

thoughrealizationsmayvary, thefuntion F ^at^pointx∈R^d andthefuntion F ^at ^point x+k^, k ∈ Z^d, share thesame law. In thehomogenization ontext wenow turn to, this meansthattheloal,mirosopienvironment(enoded intheoeienta^,^see⁽³⁾^below)

is everywhere the sameon average. From this, homogenized, marosopi properties will

follow.

We now x an open, regular, bounded subset D ôf R^d, an L² ^funtion f ônD^, ând â

randomfuntiona âssumed^stationaryⁱⁿ^the^sense ⁽¹⁾^dened âbove. ^Weâlsoâssumeaîs

bounded, positive andalmost surelybounded awayfrom zero. Forsimpliity,wetakea ^a

randompieewiseonstantfuntionoftheform:

a(x, ω) = X

k∈Z^d

1Q+k(x)ak(ω), ⁽²⁾

where Q îs ^the ûnit ûbe ôf R^d and (ak(ω))_k∈Zd ^denotes â ^family ôf î.i.d. ^random ^vari-

ables. The standard results of stohasti homogenization [2, 14℄ apply to the boundary

valueproblem





−^div ax

ε, ω

∇u^ε

= f ⁱⁿ D, u^ε = 0 ^on ∂D.

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These resultsstate that,in thelimitε−→0^,^thehomogenized problemobtainedfrom (3)

reads: (

−^div(A^⋆∇u^⋆) = f ⁱⁿ D,

u^⋆ = 0 ^on ∂D. ⁽⁴⁾

ThehomogenizedmatrixA^⋆ ^is^dened^as [A^⋆]ij =E

Z

Q

(ei+∇wei(y,·))^Ta(y,·) ej+∇wej(y,·) dy

, ⁽⁵⁾

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where,foranyp∈R^d,wpîs^the^solution^(uniqueûp^to^theâdditionôfâ^(random)ônstant)

in

w∈L²_loc(R^d, L²(Ω)), ∇w∈L²_unif(R^d, L²(Ω)) ^to











−^div[a(y, ω) (p+∇wp(y, ω))] = 0 ^a.s. ^onR^d,

∇wp ^is^stationaryⁱⁿ ^the^sense ^of⁽¹⁾, E

Z

Q∇wp(y,·)dy

= 0,

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wherewehaveusedthenotationL²_unif ^for^theûniformL²^spae,^thatîs^the^spaeôf^funtions

forwhih, say,theL² ^normônâ^ballôfûnit ^sizeîs^boundedâboveindependentlyfromthe enteroftheball.

Thesolutionu^ε^to⁽³⁾îs^known^toônverge^to^the^solutionu^⋆^to⁽⁴⁾ⁱⁿ^variousappropriate senses. The tensor and funtion A^⋆ ând u^⋆ âre deterministi quantities, although they originatefrom a series of random problems. This is a onsequeneof the ergodi setting

desribedabove,whihallowsrandommirosopiquantitiestoaverageoutindeterministi

marosopiquantities. NotehoweverthattheomputationofA^⋆^requires^the^omputation

oftheso-alledorretorfuntions wp^, ^whih^are^random.

Theaboveresultgeneralizesthatofthelassialperiodisetting(seee.g. [2,9℄) where,

insteadof being stationary ergodi, the funtion a ⁱⁿ ⁽³⁾ ^is ^periodi. ^Then, ^although ^the

homogenizedproblem anbeexpressed similarly, the ruial diereneis that (at least in

this simplelinearase)the orretorprobleman, in theperiodiase, bereduedto the

equation −^div[a(y) (p+∇wp(y))] = 0 ^set ôn^the ^periodi êllQ = [0,1]^d^, ând^not ôn^the

entirespaeR^d asin (6). Correspondingly,thetermsofthehomogenizedtensorin (5)are simple deterministi integralson Q^. În ^the ^random âse, êquation ⁽⁶⁾ îs intrinsially set on the entire spae and the numerial approximation of the solution wp ^to ^the ôrretor

problem(6)isthemainomputationalhallenge. Problem(6)isinpratietrunatedona

boundeddomainQN = [−N, N]^d ândûsually^supplied^with^periodi ^boundaryônditions:

( −^div a(·, ω) p+∇w_p^N(·, ω)

= 0 ^on QN,

w_p^N ^isQN^-periodi. ⁽⁷⁾

Correspondingly,weset:

[A^⋆_N]_ij(ω) = 1

|QN| Z

QN

ei+∇w^N_e_i(y, ω)T

a(y, ω)

ej+∇w_e^N_j(y, ω)

dy. ⁽⁸⁾

In the limit of large domains QN^, ^the homogenized tensor (5) is reovered. In addition, therate of onvergene withwhih the trunated valuesapproahthe exathomogenized

value A^⋆ ^an ^be ^assessed theoretially. We refer to [8, 18℄ for the proof of all the above statements. As will be seenbelow, the varianeof the random variables involvedplays a

roleintheapproximationproedure. Reduingthisvarianeistheproblemwenowonsider.

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3 Variane redution

3.1 Classial Monte Carlo method

Asmentionedabove,thelargesize (largeN⁾^limitôf^theôeient⁽⁸⁾ôbtainedûsing^the

solutionofthetrunatedorretorproblem(7)givesthevalueofthehomogenizedoeient

(5). Formally,thisisaonvergeneofthetypeA^⋆_N(ω)−→A^⋆ ^asN −→+∞^almost^surely

inΩ^. ^The^pratial^approah^to^this^problem^is^theMonte-Carloapproah. Wenowbriey investigatetheroleofthevarianein theproblem.

Tostartwith, webriey onsidertheone-dimensionalsetting. Althoughthis settingis

verypartiular(andsometimesmisleadingbeauseoversimplied),italsoallowstoalready

understandthebasifeaturesoftheproblemandthebottomlineoftheapproah,withthe

eonomyofmanyunneessarytehnialities.

Intheone-dimensionalsetting, thedenition (2)reads

a(x, ω) =X

k∈Z

1[k,k+1[(x)ak(ω) ⁽⁹⁾

with (ak(ω))_k∈_Z â ^family ôf î.i.d. ^random ^variables. Ît îs êasily ^seen ^that ^the ^trunated

orretorproblem(7)an beexpliitlysolvedandleadstothevalue

a^⋆_N(ω) = 1 2N

NX−1 k=−N

1 ak(ω)

!⁻¹

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oftheapproximationforthehomogenizedtensor(here,asalaroeientofourse). Inthe

limitoflargeN^,îtâlmost^surelyônverges^to^the^valueôf^theêxat homogenizedoeient

a^⋆=E 1

a0

−1

. ⁽¹¹⁾

This exat value is readily obtained expliitly solving (5)-(6). The simplest possible ar-

gument onsists now in onsidering (a^⋆_N(ω))⁻¹ = 1 2N

NX−1 k=−N

1

ak(ω) ^and ^remark ^that ^the

rateof onvergeneof this quantityto (a^⋆)⁻¹ îs êvidently^given ^by ^theêntral^limit ^theo-

rem, where the variane of the randomvariable (ak(ω))⁻¹ ^plays^a ^ruial ^role. ^Although

orret, this argument exploits toomuh the verypeuliar nature of theone-dimensional

setting (we havetaken the inverse of the oeient and reasted it asa sum, afat that

is not possible otherwise than in one dimension). An argument with slightly more gen-

erality onsists in onsidering a^⋆_N(ω) îtself ând ^not îts înverse, ând, ûsing êlementary

alulus, showingthat it also onverges to a^⋆ ^with â ^rate ôf ônvergene ^where ^the ^vari-

ane of a0(ω) âgain ^plays ^the ^ruial ^role. Îndeed, ône ^may ^for înstane ^remark ^that

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E 1

2N

PN−1 k=−N 1

ak

−1

−E

1 a0

−1

2!

may be bounded from above (using a simple al-

mostsureupperboundofak(ω)⁾^byE

1 2N

PN−1 k=−N

1 a_k

−E

1 a0

2

uptoanirrelevant

multipliative onstantand that the latterquantity, one easily omputed, is of the form

1 2N Var

1 a0

. Again,thevarianeoftherandomoeientplaysarole.

Indimensionshigherthanone,thesituationisonsiderablymoreintriateandtherate

ofonvergenewithwhihtheoeientarisingfromthetrunatedomputationonverges

toits limitisnotsosimpleto evaluate. Thisis thepurpose, under appropriateonditions

(alledmixingonditions andwhihareindeedmetinourpresentsetting),ofthework[8℄.

Thenumerialpratieisasfollows. AsetofM independentrealizationsoftherandom oeienta^are ^onsidered. ^The orrespondingtrunated problems(7) aresolved, andan empirialmeanofthetrunatedoeients(8)isinferred. Thisempirialmeanonlyagrees

withthetheoretialvalueofthetrunatedoeientwithinamarginoferrorwhihisgiven

bytheentrallimittheorem(intermsofM^). ^The^varianeôf^theôeients^thereforeâgain

playsarole, asaprefator. Forasuiently largetrunation size N^, ^this^trunated^value

is admitted to be the exatvalue of the oeients. The error made is ontroled by the

estimationsofthetheoretialwork[8℄. Of ourse,theoverallomputation desribedabove

isexpensive,beauseeahrealizationrequiresanewsolutiontothed-dimensionalboundary valueproblem(7)ofpresumablylargeasizesineN îs^taken^large. ^Thereîs^thereforeâ^huge

interestin reduingtheostof theomputation, or,otherwisestated, in reahing abetter

auray ata givenomputational ost. Sine thevarianeof thetrunated homogenized

tensorisanimportantingredient,reduingthevarianebeomesahallengingandsensitive

issue.

Moreexpliitly,let(a^m(x, ω))_1≤m≤M ^denoteM independentandidentiallydistributed underlyingrandomelds. Wedeneafamily A^⋆,m_N

1≤m≤M ^of^i.i.d. homogenizedmatries by,forany1≤i, j≤d^,

A^⋆,m_N

ij(ω) = 1

|QN| Z

QN

ei+∇w^N,m_e_i (·, ω)^T

a^m(·, ω)

ej+∇w^N,m_e_j (·, ω) ,

wherew^N,m_e_j îs ^the^solutionôf^the ôrretor^problem âssoiated^to a^m^. ^Then^we^dene^for

eah omponentofA^⋆_N ^the^empirial^mean^and^variane µM

[A^⋆_N]_ij

= 1

M XM m=1

A^⋆,m_N

ij,

σM

[A^⋆_N]_ij

= 1

M−1 XM m=1

A^⋆,m_N

ij−µM

[A^⋆_N]_ij2

.

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SinethematriesA^⋆,m_N âreî.i.d.,^the^strong^lawôf^large^numbersâpplies:

µM

[A^⋆_N]_ij

(ω)_M→+∞−→ E [A^⋆_N]_ij

almostsurely.

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Theentrallimittheoremthenyields

√M µM

[A^⋆_N]_ij

−E

[A^⋆_N]_ij _L

M−→→+∞

r Var

[A^⋆_N]_ij

N(0,1), ⁽¹³⁾

wheretheonvergeneholdsin law,andN(0,1)^denotes^the^standard^gaussian ^law. ^Intro-

duing its 95 perent quantile,it is standard to onsider that the exat meanE [A^⋆_N]_ij

isequalto µM

[A^⋆_N]_ij

within amarginof error1.96 r

Var

[A^⋆_N]_ij

√M

. Theexatvariane

Var

[A^⋆_N]_ij

beingunknowninpratie,itisustomarytoreplaeitbytheempirialvari-

anegivenin (12)above. Itis thereforeonsideredthat theexpetation E [A^⋆_N]_ij

liesin

theinterval





µM

[A^⋆_N]_ij

−1.96 r

σM

[A^⋆_N]_ij

√M , µM

[A^⋆_N]_ij + 1.96

r σM

[A^⋆_N]_ij

√M





. ⁽¹⁴⁾

ThevalueµM

[A^⋆_N]_ij

isthus,forbothM ândN ^suiently^large,âdoptedâs^theâpprox-

imationoftheexatvalue[A^⋆]_ij^.

Of ourse, atensorialargumentouldbeapplied here, notonsidering separately eah

entry of the matrix but treating the matrix as a whole. The approah developed above,

omponentbyomponent,issuientforthesimpleasesonsideredinthepresentwork.

3.2 Antithetivariable for stohasti homogenization

WeknowfromtheprevioussetionthatonstrutingempirialmeansapproximatingE(A^⋆_N)

withasmallervarianeatthesameomputationalostisofhighinterest. Wenowdesribe

apossibleapproahtoahievethisgoal.

Ingenerality,x M = 2M^. ^Suppose^that ^we^haveMî.i.d. ôpies(a^m(x, ω))_1≤m≤M ôf a(x, ω)^. ^Construt^nextMî.i.d. ântitheti ^randomêlds

b^m(x, ω) =T(a^m(x, ω)), 1≤m≤ M,

from the(a^m(x, ω))_1≤m≤M^. ^The^map T ^transforms^the^randomêld a^m întoânother, ^so-

alled antitheti, eld b^m^. Êxpliit êxamples ôf ^suh T âre ^given ⁱⁿ ^the ^sequel ^(see ⁽²⁰⁾

andSetion4below). Thetransformationisperformedinsuh awaythat,foreahm^,b^m

should have the samelawas a^m^, ^namely ^the ^law ^of ^the ^oeient a^. ^Somewhat ^vaguely

stated,iftheoeienta^wasôbtainedⁱⁿ âôin^tossing^game^(usingâ^fairôin),^then b^m

wouldbeheadeahtimea^m^is^tail^and^vie^versa. ^We^refer^the^reader^to^Figure¹^below^for