On ANOVA expansions and strategies for choosing the anchor poi

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AppliedMathematicsandComputation217(2010)3274–3285
ContentslistsavailableatScienceDirect
Applied Mathematics and Computation
journal homepage: www.elsevier.com/locate/amc
On ANOVA expansions and strategies for choosing the anchor point
⇑
Zhen Gaoa,b, Jan S. Hesthavenb,
aResearchCenterforAppliedMathematics,OceanUniversityofChina,Qingdao266071,PRChina
bDivisionofAppliedMathematics,BrownUniversity,Providence,RI02912,USA
a r t i c l e i n f o a b s t r a c t
Keywords: TheclassicLebesgueANOVAexpansionoffersanelegantwaytorepresentfunctionsthat
ANOVAexpansion dependonahigh-dimensionalsetofparametersanditoftenenablesasubstantialreduc-
AnchoredANOVAexpansion tionintheevaluationcostofsuchfunctionsoncetheANOVArepresentationisconstructed.
Sparsegrid Unfortunately,theconstructionoftheexpansionitselfisexpensiveduetotheneedtoeval-
High-dimensionalintegrals
uatehigh-dimensionalintegrals.Awayaroundthisistoconsideranalternativeformula-
tion, known as the anchored ANOVA expansion. This formulation requires no integrals
buthasanaccuracythatdependssensitivelyonthechoiceofaspecialparameter,known
astheanchorpoint.
Wepresentacomparativestudyofseveralstrategiesforthechoiceofthisanchorpoint
andarguethattheoptimalchoiceofthisanchorpointisthecenterpointofasparsegrid
quadrature.Thischoiceinducesnoadditionalcostand,asweshallshow,resultsinanat-
uraltruncationoftheANOVAexpansion.Theefficiencyandaccuracyisillustratedthrough
severalstandardbenchmarksandthischoiceisshowntooutperformthealternativesover
arangeofapplications.
(cid:2)2010ElsevierInc.Allrightsreserved.
1.Introduction
Theanalysis-of-variation–ANOVA–expansionprovidesanelegantwaytorepresentfunctionsthatdependonahigh-
dimensional set of parameters. As such it has been used in numerous applications during the last decade to represent
andefficientlymanipulatehigh-dimensionalproblemsandtoenableonetotakeadvantageoftheinherentlow-dimensional
interdependence,oftenfoundinmanysuchproblems.In[6]itwasexploredinthecontextofhigh-dimensionalintegration
methods, in [1] it was demonstrated in relation with parameterized partial differential equations and in [2] the ANOVA
expansionwasutilizedtodevelopasensitivityindextoenabletheeffectivereductionofparametricdimensionalitywithout
impactingtheaccuracyofthepredictedoutputfunction.
However, the classic ANOVA expansion is projection based and this construction requires the use of high-dimensional
integration,renderingthisconstructionveryexpensive.Toaddressthischallenge,analternativeformulation,namedthean-
choredorDiracANOVAexpansion,hasbeenproposed[6].Itwasalsoconsideredin[4]underthenameCUT-HDMR.Itrelies
onexpressingafunctionu(a)asasuperpositionofitsvaluesalonglines,planesandhyperplanepassingthroughananchor
pointb= (b ,...,b ).Ascanbeexpected,thechoiceofthisanchorpointiscloselytiedtotheoverallefficiencyandaccuracy
1 p
of the expansion and making this choice correctly becomes a key element of the formulation. Unfortunately, there is no
knownrigorousresultofhowtomakethischoiceinaoptimalwayforgeneralfunctions.
Recently,anumberoftechniquesformakingthischoicehavebeenproposed.Astraightforwardchoiceistouseananchor
pointchosenrandomlyinthehigh-dimensionalspace.Whileusedwidelyitcannotbeexpectedtoyieldanoptimalchoice.
⇑
Correspondingauthor.
E-mailaddresses:[email protected](Z.Gao),[email protected](J.S.Hesthaven).
0096-3003/$-seefrontmatter(cid:2)2010ElsevierInc.Allrightsreserved.
doi:10.1016/j.amc.2010.08.061
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Z.Gao,J.S.Hesthaven/AppliedMathematicsandComputation217(2010)3274–3285 3275
In[13],itissuggestedthattheoptimalanchorpointisfoundasthetrialpointwhoseoutputisclosesttothemeanofthe
function,beingcomputedfromamoderatenumberofquasi-randomsamples.Arigorousalternative,basedonideasofopti-
malweightsinquasiMonteCarlomethods,isproposedin[16]andshowntoyieldgoodresults.Unfortunately,thisapproach
isonlyrigorousforfunctionsthatallowadimensionalvariableseparation.Bothoftheselattermethodsrequiresomecom-
putationalworktoidentifytheanchorpoint.
In this work, we propose to use the center point of a particular sparse grid quadrature as the anchor point and offer a
comparativestudywiththepreviouslyintroducedmethodsmentionedabove.Anargumentforthisnewapproachisbased
onthestructureoftheSmolyaksparsegridwhichiscloselyrelatedtotheanchoredANOVAexpansion[6].Thisanchorpoint
canbecomputedaminimalcostandweshowthatitsuseleadstoaaverynaturaltruncationoftheanchoredANOVAexpan-
sionwhenoneiscomputingintegralsoftheexpansion.Whilemostpastworkhaveassumedthattheparametersareuni-
formlydistributedrandomvariables,wealsodiscusstheuseofthisapproachwhenappliedtocaseswheretheparameters
aremoregeneralnon-uniformlydistributedrandomvariables.
What remains of the paper is organized as follows. Section 2 introduces the ANOVA expansion based on the Lebesgue
measureandtheDiracmeasure,respectively.WealsodiscussthestructureoftheSmolyaksparsegridinthispart.Section3
introducesfourstrategiesforthechoiceoftheanchorpointandinSection4wedemonstratetheefficiencyandaccuracyof
theproposedanchorpointthroughseveralexamples.Section5containsafewconcludingremarks.
2.TheANOVAexpansion
We begin by introducing the ANOVA expansion and its two different representations based on different product mea-
sures. Without loss of generality, we take the integration domain D to be [0,1]p, and u2L2ðDÞ. Take t to be any subset
ofcoordinateindicesP¼f1;...;pgwithjtjdenotingthecardinalityoft.Letalsoa denotethejtj-vectorthatcontainsthe
t
components of the vector a2[0,1]jtj indexed by t and take Ajtj to denote the jtj-dimensional unit hypercube defined as
theprojectionofthep-dimensionalunithypercubeApontothehypercubeindexedbyt.Assumedltobeaprobabilitymea-
sureonAp.ThenucanbeexpressedasanANOVAexpansion[4,13]
X
uðaÞ¼u þ uðaÞ; ð1Þ
0 t t
t#P
whereuðaÞ; t#Pisdefinedrecursivelythrough
t t
Z
X
uðaÞ¼ uðaÞdlða Þ(cid:2) u ða Þ(cid:2)u ; ð2Þ
t t Pnt w w 0
Ap(cid:2)jtj w(cid:3)t
startingwith
Z Z
u ¼ uðaÞdlðaÞ; uðaÞdlða Þ¼uðaÞ: ð3Þ
0 ;
Ap A0
Heredlða Þindicatesintegrationoverallcoordinatesexceptindicescontainingt.ThetotalnumberoftermsintheANOVA
Pnt
expansionis2p.
The ANOVA expansion is a finite and exact expansion of a general high-dimensional function [4,13]. Furthermore, the
individualtermsintheexpansionaremutuallyorthogonal,i.e.
Z
uðaÞu ða ÞdlðaÞ¼d ð4Þ
t t w w tw
Ap
and,asanaturalconsequenceofthis,eachtermexceptu hasazeromean
0
Z
uðaÞdlðaÞ¼0; jtj>0: ð5Þ
t t
Ap
The computational realization of the ANOVA expansion is achieved through the recursive expression, (2), and the use of
orthogonality(4).
2.1.TheLebesgueexpansion
IntheclassicANOVAexpansion,oneassumesdltobeaLebesguemeasureinEq.(1)andEqs.(2)and(3)yielditsreal-
izationthroughhigh-dimensionalintegration.
LetusdefinethetruncatedANOVAexpansionofordersas
X
uða;sÞ¼u þ uðaÞ; ð6Þ
0 t t
t#P;jtj6s
whereu(a)andu aredefinedabove.
t t 0
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Theconceptofaneffectivedimensionofaparticularintegrandwasintroducedin[7,8]andalsodiscussedin[9]asaway
toreflectandutilizetheobservationthatmanyhigh-dimensionalfunctionsareeffectivelylow-dimensional.Itwasalsoob-
servedthattheANOVAexpansionwasparticularlywellsuitedfrombringingoutthishiddenlowdimensionalnature.
Theeffectivedimensionisthesmallestintegerp suchthat
s
X
V ðuÞPqVðuÞ; ð7Þ
t
0<jtj6ps
whereq61.HereV(u)andV(u)aredefinedby
t
Z
V ðuÞ¼ ðuðaÞÞ2da; VðuÞ¼XV ðuÞ; ð8Þ
t t t t
Ap jtj>0
andcanberecognizedasameasureofthevariabilityofuwhenconsideringagivensett.
TherelationshipbetweentheaccuracyofthetruncatedANOVAexpansionandthesuperpositiondimensionismadeclear
throughthefollowingresult[11,12,15].
Theorem1. Assumethatthefunctionu(a)hassuperpositiondimensionp basedonqandletu(a;p)denotethetruncatedANOVA
s s
expansionoforderp.Then
s
Errða;pÞ61(cid:2)q;
s
whereErr(a,p)isthenormalizedapproximationerrordefinedby
s
1 Z
Errða;pÞ¼ ½uðaÞ(cid:2)uða;pÞ(cid:4)2da:
s VðuÞ Ap s
Thisshowsthatifthesuperpositiondimensionissmall,p (cid:5)p,thefunctioncanbewellapproximatedbyjustafewterms
s
intheANOVAexpansion.Thisallowsonetoreducethecostofcomputingtheexpansionandreducethecostofthesubse-
quentevaluationoftheexpansion.
2.1.1.Sparsesmolyakgrids
Tocontrolthecomputationalcostofevaluatingtherequiredhigh-dimensionalintegrals,Eqs.(2)and(3),ahigh-dimen-
sionalefficientquadratureruleneedtobeconsidered.HereweusesparsegridmethodsbasedontheSmolyakconstruction
[10].Theseallowonetoconstructsparsemultivariatequadratureformulasbasedonsparsetensorproductsofone-dimen-
sionalquadratureformulas.
Considerthenumericalintegrationofafunctionu(a)overap-dimensionalunithypercubeAp=[0,1]p,
Z
I½u(cid:4):¼ uðaÞda: ð9Þ
Ap
Tointroducethealgorithm,wechooseaone-dimensionalquadratureformulaforaunivariatefunctionuas
n1
Q1u¼Xl xu(cid:2)c1(cid:3); ð10Þ
l i i
i¼1
wherex representtheintegrationweightsandc1 reflectthequadraturepoints.
i i
Nowdefineasequence
(cid:4) (cid:5)
M1u¼ Q1(cid:2)Q1 u ð11Þ
i i i(cid:2)1
withQ1u¼0andfori2N .Smolyak’salgorithmforthep-dimensionalquadratureformulaisthengivenas
0 +
Qpu¼ X (cid:4)M1 (cid:6)(cid:7)(cid:7)(cid:7)(cid:6)M1 (cid:5)u; ð12Þ
l k1 kp
jkj16lþp(cid:2)1
forl2nandk¼ðk ;...;k Þ2np.Analternativeformofthislastexpressionis
1 p
Qplu¼ X ð(cid:2)1Þlþp(cid:2)jkj1(cid:2)1(cid:6)jkpj(cid:2)(cid:2)1l(cid:7)(cid:4)Q1k1 (cid:6)(cid:7)(cid:7)(cid:7)(cid:6)Q1kp(cid:5)u: ð13Þ
l6jkj16lþp(cid:2)1 1
Forotherequivalentexpressions,see[14].
Equation(13)clearlyonlydependsonfunctionvaluesatafinitenumberofpoints.Tohighlightthestructureofthequad-
raturepoints,let
(cid:8) (cid:9)
cki ¼ cki;...;cki (cid:3)½0;1(cid:4); ð14Þ
1 n1
l
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Z.Gao,J.S.Hesthaven/AppliedMathematicsandComputation217(2010)3274–3285 3277
denote the one-dimensional quadrature grid corresponding to Q1u;16k 6p. The tensor product in Eq. (13) depends on
ck1 (cid:8)(cid:7)(cid:7)(cid:7)(cid:8)ckp andtheunionofthese ki i
Xp ¼ [ (cid:2)ck1 (cid:8)(cid:7)(cid:7)(cid:7)(cid:8)ckp(cid:3) ð15Þ
l
l6jkj16lþp(cid:2)1
iscalledthesparsegrid,usedtoevaluate(13).Ifckisanestedset,Xp (cid:3)Xp andEq.(15)simplifies
l lþ1
Xp ¼ [ (cid:2)ck1 (cid:8)(cid:7)(cid:7)(cid:7)(cid:8)ckp(cid:3); ð16Þ
l
jkj1¼lþp(cid:2)1
which is more compact than Eq. (15). In this work we use a sparse grid based on the Gauss–Patterson quadrature points
whenpossible.Thisishierarchicalandthemostefficientapproachwhenoneconsidersattainableaccuracyforagivencom-
putationalcost[3,5].
ToillustratetheefficiencyandaccuracyoftheLebesgueANOVAexpansionandtheconceptoftheeffectivedimension,we
considerap-dimensionaloscillatoryfunction,
!
p
X
u ðaÞ¼cos 2px þ ca ; ð17Þ
1 1 i i
i¼1
proposed in [18,19] as a suitable test function for high-dimensional integration schemes. Both c and x are generated as
i 1
randomnumbersandweconsiderp=10asatestcase.
Figure1showstheaccuracyandthecomputationalcostoftheLebesgueANOVAexpansionmeasuredinboththeL norm
2
andtheL norm.Clearly,the4th-ordertruncatedexpansionrepresentsthefunctionwelldowntoanaccuracybelow10(cid:2)10.
1
However,thisaccuracycomesatconsiderablecomputationalcostduetotheevaluationofthehigh-dimensionalintegrals.
2.2.TheDiracexpansion
NowassumethatdlisaDiracmeasurelocatedattheanchorpointb=(b ,b ,...,b )2[0,1]p.Thisleadstowhatisknown
1 2 p
astheanchoredortheDiracANOVAexpansion.
TherecursiveformulaEq.(2)andtheinitialformulaEq.(3)nowtakestheforms
X
uðaÞ¼uðb ;...;b ;a ;b ;...;b ;a ;b ;...;b ;a ;b ;...;b Þ(cid:2) u ða Þ(cid:2)u ; ð18Þ
t t 1 i1(cid:2)1 1 i1þ1 i2(cid:2)1 2 i2þ1 ijtj(cid:2)1 jtj ijtjþ1 p w w 0
w(cid:3)t
and
u ¼uðb ;b ...b Þ: ð19Þ
0 1 2 p
ThecomputationalrealizationoftheanchoredANOVAexpansionisconsiderablymoreefficientthantheLebesgueANOVA
expansionasthereisnoneedtoevaluatehigh-dimensionalintegralsinEqs.(18)and(19).
LetusagainconsidertheexampleinEq.(17).InFig.2weillustratethaterrorsareagainreducedtobelow10(cid:2)12withthe
4th-order anchored ANOVA expansion with the anchor point taken to be (0,0,...,0). With a comparable accuracy, the
102 102
100 100
10−2 10−2
10−4 10−4
10−6 10−6
10−8 10−8
10−10 10−10
10−12 L2 error 10−12 L2 error
L∞ error L∞ error
10−14 10−14
1 2 3 4 5 6 0 1000 2000 3000 4000 5000 6000 7000
order of ANOVA expansion computational time
Fig.1. Ontheleft,weshowtheL andtheL errorsofthe6th-ordertruncatedLebesgueANOVAexpansionwithincreasingnumberofterms.Theright
2 1
showstheassociatedcomputationaltime.
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102 102
100 100
10−2 10−2
10−4 10−4
10−6 10−6
10−8 10−8
10−10 L2 error 10−10 L2 error
10−12 L∞ error 10−12 L∞ error
1 2 3 4 5 6 0 2 4 6 8 10 12
order of ANOVA expansion computational time
Fig.2. Ontheleft,weshowtheL andtheL errorsofthe6th-ordertruncatedanchoredANOVAexpansionwithanincreasingnumberofterms.Theright
2 1
showstheassociatedcomputationaltime.
anchoredANOVAexpansionisachievedatafractionofthetimerequiredfortheclassicANOVAexpansion.Forhigherdimen-
sionalproblems,thegaincanbeexpectedtobeevenmoresignificant.
3.Strategiesforchoosingtheanchorpoint
AkeyelementintheanchoredANOVAexpansionisthechoiceoftheanchorpointasthisdirectlyimpactstheaccuracyof
theANOVAexpansionandthetruncationdimensionand,hence,thetotalcostofevaluatingtheexpansion.
Anumberofstrategieshavebeenproposedforchoosingtheanchorpointandwewillbrieflysummarizethesebelowbe-
forearguingforanalternativeapproach.
Asimpleapproachistorandomlychooseapointastheanchorpoint.Thisisclearlystraightforwardandwithnegligible
cost.However,therearenoguaranteesfortheaccuracyofthisapproachand,asweshallsee,itgenerallyleadstoanANOVA
expansionofpoorquality,
In[13]itwassuggestedtochoseananchorpointbasedonamoderatenumberoffunctionevaluationstoestimatethe
mean,denotedu(cid:2), througha numberofquasi-randomtrialpoints in[0,1]p. The anchorpointis thenchosento be thetrial
pointwhoseoutputisclosesttothemeanofthefunction.Thisguaranteesthatthezeroordertermapproximatesthefunc-
tionasaccurateaspossiblebutdoesnotofferanyguaranteesforthequalityofthehigherorderterms.Whilethereisacost
associatedwiththecomputationoftheanchorpointthroughthesampling,anobviousadvantageisthatthisgeneralizesto
thecaseofnon-uniformlydistributedparameters.Inthe followingweshallrefer tothemeananchorpointasone chosen
usingthisapproach.
In[16],analternativeapproachforchoosingtheanchorpointforamorerestrictedclassofproblemofthetype
p
Y
uðaÞ¼ uðaÞ; ð20Þ
j j
j¼1
wasdeveloped.Thistechnique,basedonanalysisborrowedfromquasiMonteCarlomethods,isexpressedbydefiningthe
dimensionalweightsc,j=1,...,p,as
j
ku (cid:2)uðbÞk
c ¼ j j j 1; uðbÞ–0: ð21Þ
j juðbÞj
j j
whereb= (b ,b ,...,b )istheanchorpoint.Withthegoaltominimizec,[16]provesthefollowingresult.
1 2 p j
Lemma2. Assumethattheanchored-ANOVAexpansionistruncatedatorderv~ andthatp satisfies
v~
!
p p
X X Y Y
c ¼ð1(cid:2)p Þ ð1þcÞ(cid:2)1 : ð22Þ
j v~ j
m¼v~þ1jSj¼m j2S j¼1
Then,therelativeerrorinL canbeestimatedas
1
ku(cid:2)PjSj6v~uSkL1 6ð1(cid:2)p Þ Yp ð1þcÞ(cid:2)1! Yp jujðbjÞj!: ð23Þ
kuk v~ j kuk
L1 j¼1 j¼1 j L1
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Furthermore,forone-signedfunctionswiththeanchorpointb= (b ,b ,...,b )selectedas
1 2 p
1
uðbÞ¼ ðmaxuðaÞþminuðaÞÞ; ð24Þ
j j 2 ½0;1(cid:4) j j ½0;1(cid:4) j j
thecorrespondingc minimizestheweightsinEq.(21).
j
Thismethodislimitedtofunctionswithseparatedvariables,(20),andthecomputationof(24),albeitone-dimensionalin
nature,canbecostly.Whiledevelopedforuniformlydistributedparameters,thisapproachcanbeextendedtoincludemore
generaldistributionsbygeneralizingthetheorywithappropriateL norms.Inthefollowing,weshallrefertotheextremum
1
anchorpointasonechosenusingthisapproach.
3.1.Anchorpointascenterofsparsegridquadrature
Anintuitivealternativeistosimplychoosethecentroidpointintheparameterspace.Foruniformlydistributedparam-
etersthiscanbeexpectedtoworkwell.However,forthemoregeneralsituationwithnon-uniformlydistributedvariables,it
isreasonabletogeneralizethischoiceoftheanchorpointtothatofthecentroidofthelowestdimensionaltensorialGaussian
quadratureinthep-dimensionalspace.Thequadratureshouldbechosentoreflectthepropermeasureassociatedwiththe
non-uniformlydistributedparameter.
Assimpleaschoosingtheanchorasthecentroidofthetensorialquadratureis,itsutilizationishighlightedwhenrecalling
thatoneoftenseekstobeabletoeffectivelycomputemomentsoftheANOVAexpandedfunctionusingsparsegrids.Aswe
shall show in the following theorem, there is a strong connection between between the anchored ANOVA expansion, the
sparsegridSmolyakconstruction,andtheanchorpointbasedonthecentroid.
Theorem3. Letu(a)beap-dimensionalintegrablefunctionwhichisrepresentedbytheanchoredANOVAexpansionlocatedatthe
anchorpointb= (b ,...,b ),chosentobethecentroidoftheSmolyaksparsegrid.Then,alltermsoforderl+16porhigherare
1 p
identicallyzerowhenevaluatedatthep-dimensionall+1levelsparseSmolyakgrid.
Proof. Letdl,i=1,2,...,pbeaDiracmeasureonAianddefinetheaveragingoperator
i
Z
ðCuÞðaÞ¼ uða ;...;a ÞdlðaÞ¼uða ;...;b;...;a Þ: ð25Þ
i 1 p i i 1 i p
Ai
Lettheidentitybedecomposedas
Y Y X Y X Y Y
I¼ ðC þðI(cid:2)CÞÞ¼ C þ ðI(cid:2)CÞ C þ ðI(cid:2)CÞðI(cid:2)CÞ C þ(cid:7)(cid:7)(cid:7)þ ðI(cid:2)CÞ: ð26Þ
i i i i j i j k i
i i i i–j i<j k–i;j i
Eachtermof(1)isgeneratedbyeachofthecomponentsofthisdecomposition(26)[17],
Y
u ¼ Cu;
0 i
i
Y
u ¼ðI(cid:2)CÞ Cu;
1 i j
i–j
. . .
. . .
. . .
Y
ul ¼ðI(cid:2)CL1ÞðI(cid:2)CL2Þ...ðI(cid:2)CLlÞ CMu; ð27Þ
M–L
Y
u ¼ðI(cid:2)C ÞðI(cid:2)C Þ...ðI(cid:2)C Þ C u;
lþ1 ðLþ1Þ1 ðLþ1Þ2 ðLþ1Þlþ1 N
N–ðLþ1Þ
. . .
. . .
. . .
Y
u ¼ ðI(cid:2)CÞ:
p i
i
Withoutlossofgenerality,weconsiderthefirsttermofthel+1ordertermoftheanchoredANOVAexpansion,
Y
u ðaÞ¼ðI(cid:2)C ÞðI(cid:2)C Þ...ðI(cid:2)C Þ C uðaÞ
lþ1 ðLþ1Þ1 ðLþ1Þ2 ðLþ1Þlþ1 N
N–ðLþ1Þ
Z
¼ðI(cid:2)C ÞðI(cid:2)C Þ...ðI(cid:2)C Þ uða ;...;a Þdlða Þ
ðLþ1Þ1 ðLþ1Þ2 ðLþ1Þlþ1 Ap(cid:2)n 1 p N
¼ðI(cid:2)C ÞðI(cid:2)C Þ...ðI(cid:2)C Þuða ;...;a ;b ;...;b Þ: ð28Þ
ðLþ1Þ1 ðLþ1Þ2 ðLþ1Þlþ1 1 lþ1 lþ2 p
wheren=p(cid:2)l(cid:2)1.ObservethatEq.(28)containsatmostl+1variables(a ,...,a ).
1 l+1
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Thel+1levelsparsegridisgivenby(15)
[
Xp ¼ ðck1 (cid:8)(cid:7)(cid:7)(cid:7)(cid:8)ckpÞ: ð29Þ
lþ1
ðlþ1Þ6jkj16lþp
Assumenowthattherearel+1variablesthatarenotsolelydefinedatthecentroid.Then
k P2;i¼1;...;lþ1
i
k ¼1;j¼lþ2;...;p
j
Xlþ1 Xp
jkj ¼ k þ k;
1 i j
i¼1 j¼lþ2
ð30Þ
Xlþ1 Xp
P 2þ 1;
i¼1 j¼lþ2
¼2(cid:8)ðlþ1Þþp(cid:2)ðlþ1Þ;
¼pþlþ1:
whichcontradictsjkj 6lþpin(29).Therefore,atleastforonewehavek =1,i=1,...,l+1,i.e.,c =b mustbethecentroid
1 i i i
fora.
i
Withoutlossofgenerality,leta bethisone.Eq.(28)becomes
l+1
u ðaÞ¼ðI(cid:2)C Þ...ðI(cid:2)C Þuða ;...a;b ;b ;...;b Þ;
lþ1 ðLþ1Þ1 ðLþ1Þlþ1 1 l lþ1 lþ2 p
¼ðI(cid:2)C Þ...ðI(cid:2)C Þ½Iuða ;...a;b ;b ;...;b Þ;
ðLþ1Þ1 ðLþ1Þl 1 l lþ1 lþ2 p
(cid:2)C uða ;...a;b ;b ;...;b Þ(cid:4); ð31Þ
ðLþ1Þlþ1 1 l lþ1 lþ2 p
¼ðI(cid:2)C Þ...ðI(cid:2)C Þ½uða ;...a;b ;b ;...;b Þ(cid:2)uða ;...a;b ;b ;...;b Þ(cid:4);
ðLþ1Þ1 ðLþ1Þl 1 l lþ1 lþ2 p 1 l lþ1 lþ2 p
¼0:
Itisnotdifficulttoconcludethatallm>l+1ordertermsoftheexpansionarezerobyrepeatingthisargument.Thiscom-
pletestheproof. h
ApartfrommakingtheconnectionbetweentheANOVAexpansionandtheSmolyaksparsegridclear,animportantimpli-
cationofthisresultfollowsforevaluationofthemomentsoftheanchoredANOVAexpansionsinceonecandecideexactly
howmanylevelsofthesparsegridismeaningfulforanexpansionofacertainlength.Note,however,thattheaboveresult
doesnotofferanymeasureoftheaccuracyoftheexpansionand,hence,theresultingmoment.
4.Numericalexamples
Inthefollowingweconsideracomparativestudyofthedifferentapproachesforchoosingtheanchorpoint.Wedothis
usingstandardhigh-dimensionaltestfunctionsandalsoofferadirectcomparisonoftheaccuracyoftheanchoredANOVA
expansiontothatoftheLebesgueANOVAexpansionforahigh-dimensionalsystemofordinarydifferentialequations.
4.1.Integrationofhigh-dimensionalfunctions
TomeasuretheaccuracyoftheANOVAexpansionwedefineameasureofrelativeerrorofanintegralas
jR uðaÞda(cid:2)R u ðaÞdaj
(cid:2)tr ¼ Ap jR uðaÞAdpajtr ; ð32Þ
Ap
whereu (a)isthetruncatedANOVAexpansion.
tr
Weconsidertheclassictestfunctions[18,19]andoneadditionaltestexample:
(cid:9) ProductPeakfunction:u ðaÞ¼Qp ðc(cid:2)2þða (cid:2)nÞ2Þ(cid:2)1,
2 i¼1 i i i
(cid:9) CornerPeakfunction:u ðaÞ¼(cid:2)1þPp ca(cid:3)(cid:2)ðpþ1Þ,
3 i¼1 i i
(cid:4) (cid:5)
(cid:9) Gaussianfunction:u ðaÞ¼exp (cid:2)Pp c2ða (cid:2)nÞ2 ,
4 i¼1 i i i
(cid:9) Continuousfunction:u ðaÞ¼exp(cid:2)(cid:2)Pp cja (cid:2)n(cid:3)j,
5 i¼1 i i i
(cid:9) Quadraturetestexample:u6ðaÞ¼(cid:4)1þ1p(cid:5)pQpi¼1ðaiÞ1p.
wheretheparametersc= (c ,...,c )andn= (n ,...,n )aregeneratedrandomly.Theparameternactsasashiftparameter
1 p 1 p
andtheparameterscareconstrained.See[18,19]forthedetails.Recallthatthetestfunctionu isdefinedinEq.(17).
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Z.Gao,J.S.Hesthaven/AppliedMathematicsandComputation217(2010)3274–3285 3281
4.1.1.Uniformlydistributedvariables
Inthefirstsetoftests,weassumethatallvariables,a,i=1,2,...,10,areuniformlydistributedrandomvariablesdefined
i
on[0,1]p.Weusea10-dimensional7-levelsparsegridsbasedontheone-dimensionalGauss–Pattersonquadraturepointsto
computetheintegralsandconsiderthistobetheexactsolution.6-levelsparsegridsareusedtointegratetheanchoredAN-
OVAexpansionbasedondifferentchoicesoftheanchorpoint.
InFig.3weillustratetherelativeerroroftheintegralsrecoveredwithdifferentchoicesoftheanchorpoint.Notethatfor
most cases, the accuracy reaches 10(cid:2)6 with the exception of the fifth test function where all choices lead to less accurate
(i)101 (ii)100
center points center points
100 mean points 10−1 mean points
random points random points
10−1 10−2 extremum points
e error 1100−−32 ve error 1100−−43
relativ 10−4 relati 10−5
10−5 10−6
10−6 10−7
10−7 10−8
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
order of ANOVA expansion order of ANOVA expansion
(iii)100 (iv)100
center points center points
10−1 mraenadno mpo pinotisnts 10−1 mraenadno mpo pinotisnts
extremum points
10−2
10−2
error error 10−3
e 10−3 e
v v
ati ati 10−4
el el
r r
10−4
10−5
10−5 10−6
10−6 10−7
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
order of ANOVA expansion order of ANOVA expansion
(v)101 (vi)100
100 10−1
10−1 10−2
error 10−2 error 10−3
e e
v v
ati 10−3 ati 10−4
el el
r r
10−4 center points 10−5
center points
mean points
mean points
10−5 random points 10−6 random points
extremum points extremum points
10−6 10−7
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
order of ANOVA expansion order of ANOVA expansion
Fig.3. Relativeerrorsofthetestfunctionscomputedusingdifferentstrategiesforchoosingtheanchorpoint.Allvariablesareassumedtobeuniformly
distributed.(i)u ,(ii)u ,(iii)u ,(iv)u ,(v)u ,(vi)u .
1 2 3 4 5 6
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3282 Z.Gao,J.S.Hesthaven/AppliedMathematicsandComputation217(2010)3274–3285
result.Thisisassociatedwiththisparticulartestfunction,iscausedbyalimitedsmoothnessintherandomvariablesandhas
alsobeenreportedbyotherauthors[20].
Whiletherearedifferencesamongtheresults,thechoiceofthecentroidastheanchorappearstobesuperiortothealter-
nativetechniquesinallcases.WealsonotethattheresultsconfirmtheresultinTheorem3,i.e.,witha6-levelsparsegridwe
shouldnotexpectanyadditionalimprovementsintheaccuracyoftheintegralswhenusingmorethan5termsintheANOVA
expansion,
(i) 102 (ii) 101
center points center points
101 mean points 100 mean points
random points random points
100
10−1
or 10−1 or
err err 10−2
e 10−2 e
v v
ati ati 10−3
el 10−3 el
r r
10−4
10−4
10−5 10−5
10−6 10−6
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
order of ANOVA expansion order of ANOVA expansion
(iii) 102 (iv) 101
center points center points
101 mean points 100 mean points
random points random points
100
10−1
or 10−1 or
err err 10−2
e 10−2 e
v v
ati ati 10−3
el 10−3 el
r r
10−4
10−4
10−5 10−5
10−6 10−6
0 1 2 3 4 5 6 7 8 0 1 2order o3f ANO4VA exp5ansion6 7 8
order of ANOVA expansion
(v) 101 (vi) 101
center points center points
mean points 100 mean points
random points
random points
100
10−1
or or
err err 10−2
e 10−1 e
v v
ati ati 10−3
el el
r r
10−4
10−2
10−5
10−3 10−6
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
order of ANOVA expansion order of ANOVA expansion
Fig. 4. Relative errors of the test functions computed using different strategies for choosing the anchor point. All variables are assumed to be beta-
distributedwithc=1/2,s=1/3.(i)u ,(ii)u ,(iii)u ,(iv)u ,(v)u ,(vi)u .
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