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AppendixAEssentialElementsofProbabilityTheoryandRandomProcessesProbabilitytheoryisamathematicalconstructiontodescribephenomenawhoseoutcomesoccurnon-deterministically.
Dependingonthecontext,theprobabilityoftheoutcomeinarandomphenomenoncanbeinterpretedeitherasitslong-runfre-quencyofoccurrenceorasameasureofitssubjectiveuncertainty.
ThesetwopointsofviewcorrespondtothefrequentistandBayesianinterpretationsofprobabilities.
Thisdistinctionisnotnecessaryhere,andwerestrictthepresentationtomaterialsusedinthismonograph.
Inparticular,weonlyintroducenotionsofreal-valuedran-domvariables.
Furtherelementsoftheprobabilitytheorycanbefoundinstandardreferences;seeforinstance[78,104,137,181].
ThisappendixfollowstheexpositioninChaps.
1and2ofthebookStochasticcalculus:applicationsinscienceandengineeringbyM.
Grigoriu[94].
Thediscus-sionbelowdistillsessentialelementsfromthisreference.
A.
1ProbabilityTheoryA.
1.
1MeasurableSpaceWeconsiderarandomphenomenonorexperiment.
Wedenotethesetofallpos-sibleoutcomesoftheexperiment.
Thesetiscalledthesamplespace,whileθdenotesanelementof.
Thesamplespacemayhaveaniteorinnite(countableoruncountable)numberofelements.
Letbeannon-emptycollectionofsubsetsof.
Thecollectioniscalledaσ-eldonif∈A∈A∈Ai∈I∈∪i∈IAi∈whereAisthecomplementofAinandIisacountableset.
AnelementA∈iscalledaneventor-measurablesubsetof;itisacollectionofoutcomes.
ThepairO.
P.
LeMatre,O.
M.
Knio,SpectralMethodsforUncertaintyQuantication,ScienticComputation,DOI10.
1007/978-90-481-3520-2,SpringerScience+BusinessMediaB.
V.
2010483484AEssentialElementsofProbabilityTheoryandRandomProcesses(,)iscalledameasurablespace.
ABorelσ-eldisgeneratedbythecollectionofallopensetsofatopologicalset.
ThemembersofaBorelσ-eldarecalledBorelsets.
AnimportantexampleofBorelσ-eldisB(R),thecollectionofallopenintervalsofR.
ThedenitionofB(R)extendstointervals;forinstanceB([a,b])denotestheBorelσ-eldontheinterval[a,b].
Denotingσ(A)thesmallerσ-eldgeneratedbyacollectionAofsubsetsof,wehaveB(R)=σ((a,b),∞≤a≤b≤+∞).
A.
1.
2ProbabilityMeasureAsetfunctionμ:→[0,∞]isameasureonifitiscountablyadditive:μ∞i=1Ai=∞i=1μ(Ai),Ai∈,Ai∩Ai=j=.
Thetriple(,,μ)iscalledameasurespace.
Anitemeasureisameasuresuchthatμ(AnitemeasurePsuchthatP:→[0,1]P()=1iscalledaprobabilitymeasureorsimplyaprobability.
A.
1.
3ProbabilitySpaceThetriple(,,P)iscalledaprobabilityspace.
Aprobabilityspace(,,P)suchthatforA,B∈withABandP(B)=0P(A)=0issaidcomplete.
Aprobabilityspacecanalwaysbecompletedsoweshallimplicitlyassumecom-pleteprobabilityspaces.
Wethenhavethefollowingpropertiesofaprobabilityspace(,,P)forA,B∈:P(A)≤P(B)forAB,P(A)=1P(A),P(A∪B)=P(A)+P(B)P(A∩B).
Considertheprobabilityspaces(k,k,Pk)fork=1,.
.
.
,n.
Fromthiscollec-tion,wedenetheproductprobabilityspace(,,P)as=1*···*n,=1*···*n,P=P1*···*Pn.
A.
2MeasurableFunctions485Thisdenitionalsoholdsforn=∞.
Whenever,thespaces(k,k,Pk)areallidentical,theproductsamplespace,σ-eldandprobabilitymeasurewillbedenotedn,nandPnrespectively.
Let(,,P)beaprobabilityspaceandletB∈beagiveneventsuchthatP(B)>0.
Wedeneanewprobabilitymeasure,calledtheprobabilityconditionalonBasP(A∈|B)≡P(A∩B)P(B).
Fromthisdenition,ifthesetofeventsAi∈isapartitionofthesamplespace(i.
e.
ifAi∩Aj=i=andiAi=),thenP(B)=iP(B∩Ai)=iP(B|Ai)P(Ai),(lawoftotalprobability)andP(Aj|B)=P(Aj)P(B|Aj)P(B)=P(Aj)P(B|Aj)iP(Ai)P(B|Ai),P(B)>0(Bayesformula).
A.
2MeasurableFunctionsA.
2.
1InducedProbabilityConsidertwomeasurablespaces(,)and(,)andafunctionh:→withdomainandrange.
Thefunctionhissaidmeasurablefrom(,)to(,)ifforanyeventBinh1(B)≡{θ:h(θ)∈B}∈.
Ifhismeasurablefrom(,)to(,)and(,,P)isaprobabilitythenQ:→[0,1]denedbyQ(B∈)≡P(h1(B))isaprobabilitymeasureon(,)calledtheprobabilityinducedbyh,orsim-plythedistributionofh.
A.
2.
2RandomVariablesConsideraprobabilityspace(,,P)andafunctionX:→Rmeasurablefrom(,)to(R,B).
Then,XisaR-valuedrandomvariable,sometimesdenoted486AEssentialElementsofProbabilityTheoryandRandomProcessesX(θ).
ThedistributionofXistheprobabilitymeasureinducedbythemappingX:→Ronthemeasurablespace(R,B),denedbyQ(B)=P(X1(B)),B∈B.
ThisdenitionimpliesthatXisarandomvariableifandonlyifX1((∞,x])∈,x∈R.
Inotherwords,aR-valuedrandomvariableisafunctionwhichmapsthesamplespacetoR.
ThepreviousdenitionofarandomvariablecanbeextendedtoRd-valuedfunc-tionsXmeasurablefrom(,)to(Rd,Bd).
IfallthecoordinatesofXarerandomvariables,thenXiscalledarandomvector.
A.
2.
3MeasurableTransformationsThismonographisconcernedwithnumericaltechniquesforthecharacterizationoftheoutputofphysicalmodelsinvolvingrandominput.
Wearethendealingwithtransformationsofrandomvariables.
DenotingXtheRd-valuedrandomvectorrep-resentingthemodelinput,weareforinstanceinterestedinamodeloutputY=g(X)whereg:Rd→R.
Themodeloutputinthendenedbythemapping(,)X→(Rd,Bd)g→(R,B).
ItcanbeshownthatifXandgaremeasurablefunctionsfrom(,)to(Rd,Bd)andfrom(Rd,Bd)to(R,B),respectively,thenthecomposedmappinggX:R,B)ismeasurable.
Asaresult,themodeloutputisaR-valuedrandomvariable.
A.
3IntegrationandExpectationOperatorsA.
3.
1IntegrabilityForXR,B),theintegralofXwithrespecttoPovertheeventA∈isIA(θ)X(θ)dP(θ)=AX(θ)dP(θ),whereIAistheindicatorfunctionofA.
Ifthisintegralexistsandisnite,XissaidtobeP-integrableoverA.
ArandomvectorXisP-integrableoverA,ifallofitscomponentsareindividuallyP-integrableoverA.
LetXandYbetworandomvariablesdenedonaprobabilityspace(,,P).
Wehavethefollowingproperties:A.
3IntegrationandExpectationOperators487ForA∈,A|X|dP0,X∈L2,X=0,X,Y=Y,X,X,Y∈L2,X+Y,Z=X,Z+Y,Z,X,Y,Z∈L2,λX,Y=λX,Y,X,Y∈L2,λ∈R,andd:L2*L2→[0,+∞)denedbyd(X,Y)=XYL2isametriconL2sinced(X,Y)=0iifX=Ya.
s.
,d(X,Y)=d(Y,X),X,Y∈L2,d(X,Y)≤d(X,Z)+d(Z,Y),X,Y,Z∈L2.
Finally,L2equippedwiththeinnerproduct·,·andtheL2-normisaHilbertspace.
Asaresult,ifX∈L2(,,P)andisasub-σ-eldof,thenthereisauniquerandomvariableX∈L2(,,P)suchthatXXL2=minXZL2:Z∈L2(,,P),A.
4RandomVariables489andXX,Z=0,Z∈L2(,,P).
TherandomvariableXistheorthogonalprojectionofXonL2(,,P),orbestmeansquareestimatorofX,andhasthesmallestmeansquareerrorofallmembersofL2(,,P).
A.
4RandomVariablesWerecall(seeA.
2.
2)thataR-valuedrandomvariabledenedonaprobabilityspace(,,P)isameasurablefunctionfrom(,)to(R,B(R)).
A.
4.
1DistributionFunctionofaRandomVariableThecumulativedistributionfunction,orsimplydistributionfunction,ofarandomvariableXdenedonaprobabilityspace(,,P)isdenedbyFX(x)=PX1((∞,x])=P({θ:X(θ)≤x})=P(X≤x).
Thedistributionfunctionisright-continuous,increasing,withrange[0,1].
Inaddi-tion,limx→+∞FX(x)=1,limx→∞FX(x)=0,P(a0.
Convergenceindistribution,Xnd→Xiflimn→∞FXn(x)=FX(x),x∈R.
ConvergenceinLp,Xnm.
p.
→Xiflimn→∞E|XnX|p=0.
Wewillmostlyusethemeansquareconvergence,i.
e.
convergenceinL2,anddenoteitasXnm.
s.
→X.
A.
5RandomVectorsConsideraprobabilityspace(,,P);themeasurablefunctionXRd,Bd),d>1,denesarandomvectorinRdorRd-valuedrandomvariable,withtheinducedprobabilitymeasureQ(B)=P(X∈B)=P(X1(B)).
A.
5.
1JointDistributionandDensityFunctionsThejointdistributionfunctionofXisthedirectextensionofthedenitionoftherandomvariabledistributionfunction:FX(x)=Pdi=1{Xi≤xi},x=(x1,.
.
.
,xd)∈Rd.
ThejointdistributionfunctionofarandomvectorXhasthefollowingproperties:limxk→∞FX(x)=0,k=1,.
.
.
,d,492AEssentialElementsofProbabilityTheoryandRandomProcessesxk→FX(x)isincreasing,k=1,.
.
.
,d,xk→FX(x)isright-continuous,k=1,.
.
.
,d.
Inaddition,limxk→+∞FX(x)=FX|k(x|k),1≤k≤d,isthejointdistributionoftheRd1-valuedrandomvectorX|k=(X1,.
.
.
,Xk1,Xk+1,.
.
.
,Xd).
ThejointdensityfunctionoftherandomvectorX,ifitexists,isgivenby:fX(x)=dFX(x)x1···xd.
Theprobabilitythattherandomvectortakesvalueswithinthedomain(x1,x1+dx1]*(xd,xd+dxd],forgiveninnitesimalvectordx=(dx1,.
.
.
,dxd)isrelatedtothejointdensitybyPdi=1{Xi∈(xi,xi+dxi]}≈fX(x)dx.
Fromthejointdistributionanddensity,onecanderiveexpressionsforthejointdensityordistributionofasubsetofcoordinatesofX.
Suchreductionisknownasmarginalization.
Forinstance,thejointdistributionofX|kisFX|k(x|k)=FX(x1,.
.
.
,xk1,∞,xk+1,.
.
.
,xd),whereasfX|k(x|k)=+∞∞fX(x1,.
.
.
,xd)dxk.
Inparticular,FXi(xi)=FX(xi,isthemarginaldistribu-tionofXiandfXi(xi)=Rd1fX(x)dx|iisthemarginaldensityofXi.
Anotherusefulexpressionconcernstheconditionaldensityofasubsetofco-ordinatesofX.
ConsiderforinstancethatthecoordinatesofXhavebeensplitintoA.
5RandomVectors493twosubsetscorrespondingtotwovectorsX1∈R0ThenthejointdensityofX1conditionedbyX2=x2isfX1|X2(x1,x2)=fX(x1,x2)fX2(x2).
Thejointdensityoftransformedrandomvectorscanalsobederived.
Consideraonetoonemappingh:x∈Rd→y=h(x)∈Rd,anddenetherandomvectorY=h(X).
ThejointdensitiesfYandfXarerelatedbyfY(y)=fXh1(y)yh1,whereh1istheinversemappingandyh1itsJacobianmatrix.
Thisrelationcanbeintuitivelyderivedfromtheprincipleofprobabilityconservation.
Indeed,denotingDyaneighborhoodofyandDxitsimagebytheinversetransformationh1,theequalityP(Y∈Dy)=P(X∈Dx)leadstoP(X∈Dx)=DxfX(η)dη=DyfX(h1(ξ))ξh1dξ=DyfY(ξ)dξ=P(Y∈Dy).
A.
5.
2IndependenceofRandomVariablesConsiderasetofrandomvariablesXi,i∈I,denedonaprobabilityspace(,,P).
TheserandomvariablesareindependentifandonlyifP(Xi≤xi,i∈J)=i∈JP(Xi≤xi),JI,xi∈R.
ForniteindexsetI={1,.
.
.
,d},thecollectionofrandomvariablescanbeseenasarandomvectorXofRd,andtheindependenceimpliesafactorizedstructureforthejointdistributionFX,FX(x1,.
.
.
,xd)=di=1FXi(xi),andjointdensityfX,fX(x1,.
.
.
,xd)=di=1fXi(xi).
494AEssentialElementsofProbabilityTheoryandRandomProcessesA.
5.
3MomentsofaRandomVectorConsiderarandomvectorXdandanintegermulti-indexl=(l1,.
.
.
,ld)∈Nd.
Wedene|l|=ili,andthefunctiongl:x∈Rd→g(x)=i=di=1xlii.
(A.
1)Thefunctiongl(X)isaR-valuedrandomvariable.
IfX∈L|l|,i.
e.
foreachofthevectorcoordinateXi∈L|l|,thenthemomentsoforder|l|ofXaregivenbyml(X)=Egl(X).
(A.
2)IfthecoordinatesofXaremutuallyindependent,themomentsmlcanbeexpressedas:ml(X)=Ei=di=1Xlii=i=di=1EXlii=i=di=1mli(Xi).
(A.
3)OfparticularimportancearethemeanμiofXiwhichcorrespondstotherst-ordermomentwithlj=δij,andthecorrelationrijof(Xi,Xj)whichisthesecond-ordermomentwithalllk=0exceptli=lj=1.
Combiningtheserstandsecondmoments,weobtainthecovariancecijof(Xi,Xj),cij=E(Xiμi)(Xjμj)=rijμiμj,(A.
4)andthevarianceσ2iofXi:σ2i=E(Xiμi)2=riiμ2i≥0.
(A.
5)Ifcij=0fori=j,thenXiandXjaresaidtobeuncorrelated.
Ifrij=0fori=j,XiandXjaresaidtobeorthogonal.
Therstandsecond-ordermomentsofarandomvectorcanbesummarizedusingthemeanvectormandcorrelationrorcovariancecmatrices:m=E[X],r=[rij]=EXXt,c=[cij]=E(Xm)(Xm)t.
(A.
6)NotethatifXiandXjareindependent,wehavecij=E(Xiμi)(Xjμj)=E[Xiμi]EXjμj=0.
(A.
7)Inotherwords,independentrandomvariablesareuncorrelated.
However,thecon-verseisusuallynottrue:uncorrelatedrandomvariablesarenotindependentingen-eral.
Also,randomvectorswithmutuallyuncorrelatedcoordinateshavediagonalcorrelationmatrices.
A.
6StochasticProcesses495A.
5.
4GaussianVectorThecaseofGaussianvectorisofparticularimportanceinprobabilitytheoryandinPolynomialChaosexpansionsonHermitebases.
AGaussianvectorofRdwithmeanmandcovariancematrixchasforjointdensityfunction:fX(x)=1(2π)d/2√|c|exp12(xm)tc1(xm),(A.
8)where|c|isthedeterminantofthecorrelationmatrix.
AGaussianvectoristhencompletelyspeciedbyitssecond-orderpropertiesandweshallwriteXN(m,c).
(A.
9)AnimportantpropertyofGaussianvectoristhatifcisdiagonal(itscoordinateareuncorrelated)thentheXiaremutuallyindependent.
Indeed,fX(x)=1(2π)d/2√|c|exp12di=1σ1i(xiμi)2=di=112πσ2iexp12σ2i(xiμi)2,(A.
10)becausecii=σ2iand|c|=ciifordiagonalmatrices.
Also,ifXandYaretwoRd-valuedGaussianvectors,then(αX+βY)isaGaussianrandomvector.
A.
6StochasticProcessesA.
6.
1MotivationandBasicDenitionsInthismonograph,weareinterestedinthecomputationofthestochasticsolutionsofequations,essentiallyODEsorPDEs.
Thereforethesesolutionsarefunctionsoftimeand/orspace.
Then,thenotionofrandomvariableandrandomvectorneedbeextendedtoincorporateadependenceontimeor/andspacecoordinates.
ConsiderafunctionX:T*→Rdwitharguments–time–t∈T(arealin-terval)and–event–θ∈.
IfX(t)isaRd-valuedrandomvectorontheprobabilityspace(,,P)forallt∈T,thenXiscalledaRd-valuedstochasticprocessorvectorstochasticprocess.
ThefunctionX(·,θ):t∈T→Rdforgivenθ∈iscalledasamplepathorsimplyarealizationofX.
Conversely,thedenitionshowsthatX(t,·):θ∈→Rdisarandomvectoron(,,P).
Thedenitioncanbeextendedtorandomvectorsindexedbyaspatialcoordinate.
Forinstance,considery∈DRn,thenX:D*→X∈RdiscalledaRd-valued496AEssentialElementsofProbabilityTheoryandRandomProcessesstochasticeldifX(y,·)isarandomvectorforeachy∈D.
ThefunctionX(·,θ)iscalledarealizationofthestochasticeld.
Thetwodenitionscanbecombinedandresultsinspace-timestochasticpro-cesses:X:T*D*→Rd,X(t,y,P)t∈T,y∈D.
(A.
11)Therearenoticeabledifferencesbetweenstochasticprocessesandstochasticelds,arisingfromorientednatureoftime.
However,thesedifferenceswillnotap-pearinthepresentmonograph,andweshallcallstochasticprocesses,orrandomprocess,anytypeofrandomvariableorvectorindexedbyadeterministicsetofcoordinatesinD.
Weshalloftenrelyonthelooseterminologyofrandomprocess.
Consequently,weshalldenoteDthedomainofindexationofthestochasticprocess,makingnodistinctionbetweenspaceandtimecoordinatesiny∈Rs.
Inthefollowing,werestrictourselftostochasticprocessesX:D*→Rdde-nedonaprobabilityspace(,,P),withDR,i.
e.
n=1.
Thedenitionsandconceptscanbeextendedtostochasticeldsn≥1,subjecttoappropriateresolutionofsometechnicaldifcultiesandnotationissues.
A.
6.
2PropertiesofStochasticProcessesIfXissuchthatlims→tP(X(s)X(t)0,(A.
12)andanyt∈D,theprocessissaidcontinuousinprobability.
Ifforgivenintegerp≥1,lims→tEX(s)X(t)p=0,t∈D,(A.
13)theprocessissaidcontinuousinthep-thmean,andcontinuousinthemeansquaresenseinthecaseofp=2.
IfPθ:lims→tX(s,θ)X(t,θ)=0=0,t∈D,(A.
14)theprocessissaidalmostsurelycontinuous.
Inthepreviousdenitionsofcontinuity,thenorm·istheclassicalEuclideannormforvectors.
Thecontinuitycanalsoberestrictedtoparticulart∈Dorsubin-tervals.
A.
6StochasticProcesses497A.
6.
2.
1FiniteDimensionalDistributionsandDensitiesLetm≥1beaninteger,andasetofmdistinctpointstiinD.
Thenitedimensionaldistribution(fdd)ofordermofXisdenedby:Fmx(1),.
.
.
,x(m);t1,.
.
.
,tm=P∩mi=1{X(ti)∈*dk=1(∞,x(i)k]},(A.
15)wherex(i)=(x(i)1,.
.
.
,x(i)d)∈Rd.
Thus,thenitedimensionaldistributions,Fm,areprobabilitymeasuresinducedbyXonthemeasurablespace(Rm*d,Bm*d).
Therst-orderfddofXattiscalledthemarginaldistribution.
ItisinfactthedistributionoftherandomvectorX(t,·).
Ford=1,them-thorderfdd'sareFm(x(1),.
.
.
,x(m);t1,.
.
.
,tm)=PX(t1)≤x(1),.
.
.
,X(tm)≤x(m),(A.
16)fromwhichthenitedimensionaldensitiesofXcanbeobtainedbydifferentiation:fmx(1),.
.
.
,x(m),t1,.
.
.
,tm=mx(1)···x(m)Fm(x(1),.
.
.
,x(m);t1,.
.
.
,tm).
(A.
17)ThemarginaldensityofXattimetcorrespondstotherst-orderfddf1(x;t).
TheRd-valuedstochasticprocessissaidstationaryinthestrictsenseifallitsfdd'sareinvariantunderarbitraryshiftτofalltheti→ti+τ.
IfXisstationary,themarginaldistributionofXist-invariant.
AstochasticprocessXisaGaussianprocessifallitsfdd'sareGaussian.
A.
6.
3SecondMomentPropertiesAsecond-orderstochasticprocessXhasallitscomponentsXi∈L2(,,P)forallt∈D.
Forasecond-orderrandomprocess,wedenethemeanfunction:x(t)=E[X]thecorrelationfunctionr(t,s)=E[X(t,·)X(s,·)t]thecovariancefunctionc(t,s)=E[(X(t,·)x(t))(X(s,·)x(s))t]Thesecondmomentpropertiesofastochasticprocessaregivenbythecouple(x,r)or(x,c).
Forastationaryprocess,themeanfunctionxisconstantandthecorrelationandcovariancefunctionsdependonlyonthelagτ=ts.
Conversely,aprocesswithconstantmeanfunctionandcorrelationandcovariancefunctionsoftheformr(t,s)=r(ts)andc(t,s)=c(ts),issaidweaklystationary.
Aweaklystationaryprocessisnon-necessarilystrictlystationary.
Thecorrelationfunctionhasthefollowingremarkableproperties:rii(t,s)=rii(s,t).
Iftheprocessisweaklystationaryrii(τ)=rii(τ)whereτ=ts.
498AEssentialElementsofProbabilityTheoryandRandomProcessesrii(t,s)2≤rii(t,t)rii(s,s).
Iftheprocessisweaklystationary,|rii(τ)|≤rii(0).
rii(t,s)ispositivedenite.
Xismeansquarecontinuousattifandonlyifrii(t,s)iscontinuousatt=s=tfor1≤i≤d.
Fori=j,rij(t,s)=rji(s,t).
Iftheprocessisweaklystationary,rij(τ)=rji(τ).
|rij(s,t)|2≤rii(t,t)rjj(s,s).
Iftheprocessisweaklystationary,|rij(τ)|2≤rii(0)rjj(0).
AppendixBOrthogonalPolynomialsByasystemoforthogonalpolynomials,wedenoteaset{Pn(x)}∞n=1,wherePn(x)isapolynomialofdegreen.
ThepolynomialsaredeemedorthogonalinthesensethattheinnerproductPnPmvanisheswhenevern=m.
Thevariablexmaybeeithercontinuousordiscrete.
Intheformercase,theinnerproductisdenedas:PnPm=baPn(x)Pm(x)w(x)dx(B.
1)whereasinthelatteritisgivenby:PnPm=Mi=1Pn(xi)Pm(xi)w(xi).
(B.
2)Inbothcases,w(x)isapositiveweightfunction.
Notethatthelimitsofintegrationaandbin(B.
1),andthelimitMin(B.
2)maybeeitherniteorinnite.
WearespecicallyinterestedinorthogonalfamiliesofpolynomialsthatformabasisofthefunctionspaceL2w={f:ff<∞}.
Withinthisgeneralsetting,wefurtherfocusourattentiononfamiliesforwhichexpansionsoftheform:f(x)=Nn=1anPn(x)result,forsuitablefunctionsf,inspectrathatdecayexponentiallyasN→∞.
Inthisappendix,weprovideabriefoutlineoffamiliesoforthogonalpolynomialsthatarefrequentlyusedinPCexpansions.
O.
P.
LeMatre,O.
M.
Knio,SpectralMethodsforUncertaintyQuantication,ScienticComputation,DOI10.
1007/978-90-481-3520-2,SpringerScience+BusinessMediaB.
V.
2010499500BOrthogonalPolynomialsB.
1ClassicalFamiliesofContinuousOrthogonalPolynomialsInthissection,weshallfocusonthecaseof"continuous"polynomialsdenedforontheintervala≤x≤b.
WeshalldenotebyhnthesquareoftheL2normofPn,i.
e.
hn=PnPn=αbaP2n(x)w(x)dx.
(B.
3)NotethatforapplicationstoPCexpansions,itisconvenienttoadopttheconventionthatthespaceL2whasmeasureone,oralternativelythatthenormoftheindicatorfunctionoftheentirespaceisequaltoone,i.
e.
baw(x)dx=1.
(B.
4)Thisamountstoaconstantscalingofweightsfunctionsthatarenormallyutilizedinthedenitionoforthogonalpolynomials.
Wealsorecall[1]thatorthogonalpolynomials:(a)satisfythedifferentialequa-tion:g2(x)Pn+g1(x)P+anPn=0(B.
5)whereg1andg2areindependentofnandthean'sareconstantsthatdependonnonly;and(b)canbegeneratedusingRodrigues'formula:Pn(x)=1enw(x)dndxnw(x)g(x)n(B.
6)wheregisapolynomialinxthatisindependentofnandtheen'sarearbitrarynor-malizationfactorsthatdependonnonly.
NotethatRodrigues'formulaimmediatelyshowsthataconstantmultiplicativescalingofwhasnoimpactonthecorrespondingrepresentationornormalization.
B.
1.
1LegendrePolynomialsTheLegendrepolynomials,{Len(x),n=0,1,.
.
.
},areanorthogonalbasisofL2w[1,1]withrespecttotheweightfunctionw(x)=1/2forallx∈[1,1].
ThearetypicallynormalizedsothatLen(1)=1,inwhichcasetheyaregivenby[26]:Len(x)=12n[n/2]l=0(1)lnl2n2lnxn2l.
(B.
7)Here[n/2]denotestheintegralpartofn/2.
Thepolynomialsareevenwhennisevenandoddwhennisodd.
B.
1ClassicalFamiliesofContinuousOrthogonalPolynomials501TheLegendrepolynomialssatisfytherecurrencerelation:Len+1(x)=2n+1n+1xLen(x)nn+1Len1(x)(B.
8)withLe0(x)=1andLe1(x)=x.
TherstsevenLegendrepolynomialsaregivenby[1,26]:Le0(x)=1,Le1(x)=x,Le2(x)=123x21,Le3(x)=125x33x,Le4(x)=1835x430x2+3,Le5(x)=1863x570x3+15x,Le6(x)=1162313x6315x4+105x25.
Withthedenitionofwandthenormalizationconventionspeciedabove,thesquarenormsoftheLegendrepolynomialsaregivenby[1]:hn=Len,Len=11Le2n(x)w(x)dx=12n+1.
(B.
9)Theysatisfy(B.
5)with[1]:g2(x)=1x2,g1(x)=2x,andan=n(n+1),(B.
10)and(B.
6)with:g(x)=1x2,anden=(1)n2nn!
.
(B.
11)B.
1.
2HermitePolynomialsTherearetwowidespreaddenitionsoftheHermitepolynomials,accordingtowhethertheweightfunctionsaregivenby:w(x)=1√2πexpx22(B.
12)502BOrthogonalPolynomialsorw(x)=1√πexpx2.
(B.
13)Inbothcases,xisdenedovertherealline,i.
e.
a=∞andb=∞.
Todistinguishbetweenthetwofamilies,weshallusetheconventionalnotationHen(x)toindicatethat(B.
12)isused,andHn(x)toindicatethat(B.
13)isadopted.
NotethatinPCexpansions,werelyexclusivelyonHen,butforcompletenessandcomputationalconvenience(seeAppendixC)weaddressbothcases.
Forbothdenitionsofw,theHermitepolynomialsareconventionallynormal-izedsothatthefactorsappearinginRodrigues'formula(B.
6)aregivenby[1]:en=(1)n.
(B.
14)Thefunctionsg(x)appearingin(B.
6)arealsoidentical,g(x)=1.
ItfollowsthattheHermitefamiliesaredenedaccordingto[111]:Hen(x)=1(1)nexpx2/2dndxnexpx2/2,(B.
15)He(x)=1(1)nexpx2dndxnexpx2.
(B.
16)Theyhavethefollowingexplicitrepresentations:Hen(x)=n!
[n/2]m=0(1)m1m!
2m(n2m)!
xn2m,(B.
17)Hn(x)=n!
[n/2]m=0(1)m1m!
(n2m)!
(2x)n2m.
(B.
18)ThepolynomialsHen(x)obey(B.
5)with:g2(x)=1,g1(x)=x,andan=n,(B.
19)andtherecursionrelation:Hen+1(x)=xHen(x)nHen1(x).
(B.
20)Withthestandardizationabove,thesquarenormsoftheHermitepolynomialsHenaregivenby:hn=Hen,Hen=1√2π∞∞[Hen(x)]2expx2/2dx=n!
.
(B.
21)TherstsevenpolynomialsHen(x)aregivenby:He0(x)=1,(B.
22)B.
1ClassicalFamiliesofContinuousOrthogonalPolynomials503He1(x)=x,(B.
23)He2(x)=x21,(B.
24)He3(x)=x33x,(B.
25)He4(x)=x46x2+3,(B.
26)He5(x)=x510x3+15x,(B.
27)He6(x)=x615x4+45x215.
(B.
28)ThepolynomialsHn(x)obey(B.
5)with:g2(x)=1,g1(x)=2x,andan=2n,(B.
29)andtherecursionrelation:Hn+1(x)=2xHn(x)2nHn1(x).
(B.
30)Withthestandardizationabove,thesquarenormsoftheHermitepolynomialsHnaregivenby:hn=Hn,Hn=1√π∞∞[Hn(x)]2expx2dx=2nn!
.
(B.
31)TherstsevenpolynomialsHen(x)aregivenby:H0(x)=1,(B.
32)H1(x)=2x,(B.
33)H2(x)=4x22,(B.
34)H3(x)=8x312x,(B.
35)H4(x)=16x448x2+12,(B.
36)H5(x)=32x5160x3+120x,(B.
37)H6(x)=64x6480x4+720x2120.
(B.
38)B.
1.
3LaguerrePolynomialsTheLaguerrepolynomials,{Ln(x),n=0,1,.
.
.
},areanorthogonalbasisofL2w[0,∞]withrespecttotheweightfunctionw(x)=exp(x).
Theyareconven-tionallynormalizedsothatthefactorsappearinginRodrigues'formula(B.
6)aregivenby[1]:en=n!
.
(B.
39)504BOrthogonalPolynomialsThefunctiongappearingin(B.
6)isg(x)=x.
Accordingly,thepolynomialsLaaredenedby:La(x)=1n!
exp(x)dndxnexp(x)xn,(B.
40)andadmitthefollowingexplicitrepresentation:La(x)=nm=o(1)mn!
(nm)!
(m!
)2xm.
(B.
41)ThepolynomialsLan(x)obey(B.
5)with:g2(x)=x,g1(x)=1x,andan=n,(B.
42)andtherecursionrelation:Lan+1(x)=1n+1(2n+1x)Lan(x)nLan1(x).
(B.
43)Withthestandardizationabove,theLaguerrepolynomialsLanformanorthonormalbasisofL2w[0,∞),i.
e.
Lan,Lam=∞0Lan(x)Lam(x)exp(x)dx=δnm.
(B.
44)Inparticular,thesquarenormhn=1.
TherstsixLaguerrepolynomialsare:La0(x)=1,(B.
45)La1(x)=x+1,(B.
46)La2(x)=12(x24x+2),(B.
47)La3(x)=16(x3+9x218x+6),(B.
48)La4(x)=124(x416x3+72x296x+24),(B.
49)La5(x)=1120(x5+25x4200x3+600x2600x+120).
(B.
50)B.
2GaussQuadratureIntheapplicationofPCmethods,oneneedsinparticulartoevaluatemomentsofPCexpansions.
AsfurtherdiscussedinAppendixC,thistaskcanbegenerallyboileddowntoevaluatinginnerproductsofone-dimensionalpolynomials.
ThissectiondiscussesevaluationoftheseinnerproductsbymeansofGaussquadratureswhich,B.
2GaussQuadrature505duetotheirhighorderaccuracy,provideefcientmeansofcomputingthecorre-spondingintegrals.
WefocusourattentiontotheLegendre,HermiteandLaguerrefamiliesdiscussedintheprevioussection.
Accordingly,weareinterestedinevaluatingintegralsofpolynomialfunctionsdenedovertheintervals[1,1]and[0,∞).
B.
2.
1Gauss-LegendreQuadratureTheGauss-Legendrequadratureisbasedonthefollowingformula:11f(x)dx=nqi=1wif(xi)+Rnq,(B.
51)whereRnq=22nq+1(nq!
)4(2nq+1)[(2nq)!
]3f(2nq)(ξ),1<ξ<1.
(B.
52)Here,nqisthenumberofcollocationpoints,xidenotethecoordinateofthei-thcollocationpoint,andwithecorrespondingweight.
Rndenotestheremainderofthequadrature;itindicatesthattheformulaisexactiftheorderofthepolynomialsNo<2nq.
ThecoordinatesxiarethezerosofLenq(x),andtheweightsaregivenby:wi=2(1x2i)[Lenq(xi)]2.
(B.
53)ExplicitformulasfortheGauss-Legendrequadraturenodesarenotknown,andsotheymustbeevaluatednumerically.
TheGAUSSQroutineavailablefromnetlib(http://netlib.
org)canbeusedtode-terminetheweightsandabscissasappearingin(B.
51).
TableB.
1providesresultsusingthisroutineforselectedvaluesofnq.
B.
2.
2Gauss-HermiteQuadraturesGauss-Hermitequadraturescanbebasedonthefollowingformula(see[1]p.
890):∞∞f(x)exp(x2)dx=nqi=1wif(xi)+Rnq(B.
54)wherenqisthenumberofcollocationpoints,xidenotethequadraturenodesandwiarethecorrespondingweights.
xiisthei-thzeroofthepolynomialHnq,andthe506BOrthogonalPolynomialsTableB.
1NodesandweightsforGauss-Legendrequadraturefordifferentvaluesofnqnq=5nq=10nq=15xiwixiwixiwi0.
9061800.
236927E+000.
9739070.
666713E010.
9879930.
307532E010.
5384690.
478629E+000.
8650630.
149451E+000.
9372730.
703660E010.
0000000.
568889E+000.
6794100.
219086E+000.
8482070.
107159E+000.
5384690.
478629E+000.
4333950.
269267E+000.
7244180.
139571E+000.
9061800.
236927E+000.
1488740.
295524E+000.
5709720.
166269E+00––0.
1488740.
295524E+000.
3941510.
186161E+00––0.
4333950.
269267E+000.
2011940.
198431E+00––0.
6794100.
219086E+000.
0000000.
202578E+00––0.
8650630.
149451E+000.
2011940.
198431E+00––0.
9739070.
666713E010.
3941510.
186161E+00––––0.
5709720.
166269E+00––––0.
7244180.
139571E+00––––0.
8482070.
107159E+00––––0.
9372730.
703660E01––––0.
9879930.
307532E01weightwiisgivenby:wi=2nq1nq!
√πnq2[Hnq1(xi)]2.
(B.
55)Theremainderinthequadrature(B.
54)isgivenby:Rnq=nq!
√π2nq(2nq)!
f(2nq)(ζ)(B.
56)forsomeniteζ.
Thus,theformula(B.
54)isexactifthepolynomialf(x)hasdegreesmallerthan2nq.
TheGAUSSQroutineavailablefromnetlib(http://netlib.
org)canbeusedtode-terminetheweightsandnodesappearingin(B.
54).
TableB.
2providesresultsusingthisroutineforselectedvaluesofnq.
Note,however,theapplicationofPCmethodstypicallyrelyontheHermitefam-ilyHen,andsothequadraturein(B.
54)doesnotcoincidewithtothemeasureinthecorrespondingHilbertspace,whichcanbeexpressedas:f=1√2π∞∞f(x)exp(x2/2)dx.
(B.
57)B.
2GaussQuadrature507TableB.
2Gaussnodesandweightsforthequadraturein(B.
54)fordifferentvaluesofnqnq=5nq=10nq=15xiwixiwixiwi2.
0201830.
199532E013.
4361590.
764043E054.
4999910.
152248E080.
9585720.
393619E+002.
5327320.
134365E023.
6699500.
105912E050.
0000000.
945309E+001.
7566840.
338744E012.
9671670.
100004E030.
9585720.
393619E+001.
0366110.
240139E+002.
3257320.
277807E022.
0201830.
199532E010.
3429010.
610863E+001.
7199930.
307800E01––0.
3429010.
610863E+001.
1361160.
158489E+00––1.
0366110.
240139E+000.
5650700.
412029E+00––1.
7566840.
338744E010.
0000000.
564100E+00––2.
5327320.
134365E020.
5650700.
412029E+00––3.
4361590.
764043E051.
1361160.
158489E+00––––1.
7199930.
307800E01––––2.
3257320.
277807E02––––2.
9671670.
100004E03––––3.
6699500.
105912E05––––4.
4999910.
152248E08Inordertomakeuseoftheresultsabove,onecanapplythefollowingchangeofvariables:f=1√π∞∞g(x)exp(x2)dx(B.
58)whereg(x)=f(√2x).
Consequently,applicationofthepreviousquadraturetotheaboveintegralimmediatelyresultsin:f1√πnqi=1wig(xi)(B.
59)oralternatively,fnqi=1wif(xi)(B.
60)wherewi≡wi/√πandxi≡√2xi.
Equation(B.
60)cannowbereadilyusedtoevaluatemomentsandproductsofHermitepolynomials,Hen.
TableB.
3providesvaluesofxiandwiforselectedvaluesofnq.
Notethatourearlierobservationthatthequadraturein(B.
54)isexactforpolyno-mialsofdegreelessthan2nqstillappliestothequadraturein(B.
60).
Inparticular,508BOrthogonalPolynomialsTableB.
3Gaussnodesandweightsforthequadraturein(B.
60)fordifferentvaluesofnqnq=5nq=10nq=15xiwixiwixiwi2.
8569700.
112574E014.
8594630.
431065E056.
3639480.
858965E091.
3556260.
222076E+003.
5818230.
758071E035.
1900940.
597542E060.
0000000.
533333E+002.
4843260.
191116E014.
1962080.
564215E041.
3556260.
222076E+001.
4659890.
135484E+003.
2890820.
156736E022.
8569700.
112574E010.
4849360.
344642E+002.
4324370.
173658E01––0.
4849360.
344642E+001.
6067100.
894178E01––1.
4659890.
135484E+000.
7991290.
232462E+00––2.
4843260.
191116E010.
0000000.
318260E+00––3.
5818230.
758071E030.
7991290.
232462E+00––4.
8594630.
431065E051.
6067100.
894178E01––––2.
4324370.
173658E01––––3.
2890820.
156736E02––––4.
1962080.
564215E04––––5.
1900940.
597542E06––––6.
3639480.
858965E09whensufcientquadraturepointsareprovided,(B.
60)reproducesthecorrectnorm:He2m=m!
.
(B.
61)B.
2.
3Gauss-LaguerreQuadratureTheGauss-Laguerrequadratureisbasedonthefollowingformula:∞0f(x)exp(x)dx=nqi=1wif(xi)+Rnq,(B.
62)whereRnq=(nq!
)2(2nq)!
f(2nq)(ξ),0<ξ<∞.
(B.
63)Asbefore,nqisthenumberofcollocationpoints,whilexiandwirespectivelyde-notethenodesandweights.
From(B.
63),itfollowsimmediatelythatthequadrature(B.
62)isexactwhentheorderofthepolynomial,f,issmallerthan2nq.
B.
3AskeyScheme509TableB.
4NodesandweightsforGauss-Laguerrequadraturefordifferentvaluesofnqnq=5nq=10nq=15xiwixiwixiwi0.
2635600.
521756E+000.
1377930.
308441E+000.
0933080.
218235E+001.
4134030.
398667E+000.
7294550.
401120E+000.
4926920.
342210E+003.
5964260.
759424E011.
8083430.
218068E+001.
2155950.
263028E+007.
0858100.
361176E023.
4014340.
620875E012.
2699500.
126426E+0012.
6408010.
233700E045.
5524960.
950152E023.
6676230.
402069E01––8.
3301530.
753008E035.
4253370.
856388E02––11.
8437860.
282592E047.
5659160.
121244E02––16.
2792580.
424931E0610.
1202290.
111674E03––21.
9965860.
183956E0813.
1302820.
645993E05––29.
9206970.
991183E1216.
6544080.
222632E06––––20.
7764790.
422743E08––––25.
6238940.
392190E10––––31.
4075190.
145652E12––––38.
5306830.
148303E15––––48.
0260860.
160059E19ThecoordinatesxiarethezerosofLanq(x),andtheweightsaregivenby:wi=(nq!
)2xi(nq+1)2[Lanq+1(xi)]2.
(B.
64)TheGAUSSQroutineavailablefromnetlib(http://netlib.
org)canbeusedtodeter-minetheweightsandabscissasappearingin(B.
62).
TableB.
4providesresultsusingthisroutineforselectedvaluesofnq.
B.
3AskeySchemeAsdiscussedbyXiuandKarniadakis[249],theabovefamiliesoforthogonalpolynomialsaremembersoftheso-calledAskeyschemeofpolynomials[6].
Theschemesclassiescontinuousanddiscretehypergeometricorthogonalpolynomialsthatobeycertaindifferenceanddifferenceequations,andidentieslimitrelation-shipbetweenthem[117,203].
TheinterestingaspectoftheAskeyfamilyisthattheweightingfunctionsassociatedwithsomeitsmemberscorrespondstofrequentlyusedprobabilitydistributions.
Specically,inadditiontotheorthogonalpolynomialsystemsdiscussedabove,theAskeyfamilyincludestheJacobipolynomials,whichinturnincludetheLegen-drefamily,aswellastheCharlier,Meixner,KrawtchoukandHahnfamiliesofdis-creteorthogonalpolynomials.
TheweightingfunctionsassociatedwiththeJacobi510BOrthogonalPolynomialspolynomialscorrespondtothebetadistribution,whereastheweightingfunctionsoftheCharlier,Meixner,KrawtchoukandHahnfamiliesrespectivelycorrespondtothePoisson,negativebinormal,binormal,andhypergeometricdistributions.
Forbrevity,andsincetheseadditionalfamiliesarenotusedintheexamplespro-videdinthepresentvolume,werestrictourselvestoprovidingdenitionsofcorre-spondingpolynomials,andoutliningtheorthogonalityrelationshipsthattheyobey.
ImplementationofthesefamiliesintothecorrespondingWienerchaoscanberead-ilyperformedoncetheorthogonalityrelationshipsareexploitedtodeneasuitableinnerproduct,specicallyinasimilarfashiontothedevelopmentabove.
B.
3.
1JacobiPolynomialsTheJacobipolynomialsP(α,β)n(x),n=0,1,.
.
.
,areanorthogonalfamilyofpoly-nomialsover[1,1]withrespecttotheweightfunctionw(x)=(1x)α(1+x)β.
ThearetypicallynormalizedsothatP(α,β)n(1)=n+αninwhichcasetheyaregivenby[1]:P(α,β)n(x)=12nnm=0n+αmn+βnm(x1)nm(x+1)m.
(B.
65)TheJacobipolynomialssatisfy(B.
5)with[1]:g2(x)=1x2,g1(x)=βα(α+β+2)x,andan=n(n+α+β+1),(B.
66)theRodriguesformula(B.
6)with:g(x)=1x2,anden=(1)n2nn!
,(B.
67)theorthogonalitycondition:11P(α,β)n(x)P(α,β)m(x)w(x)dx=2α+β+12n+α+β+1(n+α+1)(n+β+1)n!
(n+α+β+1)δnm,(B.
68)andtherecurrencerelation:2(n+1)(n+α+β+1)(2n+α+β)P(α,β)n+1(x)B.
3AskeyScheme511=(2n+α+β+1)(α2β2)+(2n+α+β)3xP(α,β)n(x)2(n+α)(n+β)(2n+α+beta+n)P(α,β)n1(x)(B.
69)where(m)n=m(m+1)···(m+n1)denotesthePochhammersymbol.
Notethattheorthogonalitycondition(B.
68)maybereadilyusedtorescalew(x)sothattheresultingspaceL2w[1,1]hasmeasure1,andthattheGAUSSQroutinealsoprovidesnodesandweightsforGauss-Jacobiquadratures.
B.
3.
2DiscretePolynomialsTheCharlier,Meixner,Krawtchouk,andHahnpolynomialsarerespectivelydenotedbyCn(x,a),Mn(x;β,c),Kn(x;p,N),andQn(x;α,β,N).
Inallcases,xisdis-cretevariable.
FortheHahnandKrawtchoukpolynomials,xisdenedovertheset{0,1,.
.
.
,N},whereasfortheCharlierandMeixnerpolynomialsxisdenedovertheintegers0,1,.
.
.
.
Asdiscussedin[117],theCharlier,Meixner,Krawtchouk,andHahnpolynomialsmaybedenedintermsofthehypergeometricseries:τFsa1,.
.
.
,aτb1,.
.
.
,bsz≡∞k=0(a1)k.
.
.
(aτ)k(b1)k.
.
.
(bs)kzkk!
.
(B.
70)RelevantdenitionsaresummarizedTableB.
5Thersttwopolynomialsare:C0(x,a)=1,(B.
71)C1(x,a)=1xa,(B.
72)M0(x;β,c)=1,(B.
73)TableB.
5DenitionofCharlier,Meixner,Krawtchouk,andHahnpolynomialsintermsofhypergeometricseriesPolynomialDenitionCharlier2F0n,x1aMeixner2F1n,xβ11cKrawtchouk2F1n,xN1pHahn3F2n,n+α+β+1,xα+1,N1512BOrthogonalPolynomialsTableB.
6WeightsandsquarenormsoftheCharlier,Meixner,Krawtchouk,andHahnpolynomi-alsPolynomialw(x)hnCharlieraxx!
anexp(a)n!
Meixner(β)xcxx!
cnn!
(β)n(1c)βKrawtchoukNxpx(1p)Nx(1)nn!
(N)n1ppnHahnα+xxβ+NxNx(1)n(n+α+β+1)N+1(β+1)nn!
(2n+α+β+1)(α+1)n(N)nN!
M1(x;β,c)=1+xβ11c,(B.
74)K0(x;p,N)=1,(B.
75)K1(x;p,N)=1xNp,(B.
76)Q0(x;α,β,N)=1,(B.
77)Q1(x;α,β,N)=1(α+β+2)x(α+1)N.
(B.
78)Withthedenitionsabove,thesefamiliesofdiscretepolynomialssatisfythedis-creteorthogonalityrelation:ifn(xi)fm(xi)w(xi)=hnδnm(B.
79)wheretheindexirangesovertheentiredomainofthediscretevariablex.
Relevantformulasfortheweights,w,andsquarenorms,hn,aregiveninTableB.
6.
ThediscretepolynomialssatisfytheRodrigues-typeformula:w(x)fn(x)=rnng(x)(B.
80)wherew(x)istheweightfunction,rnisanormalizingfunctionindependentofx,gisapolynomialinx,anddenotesthebackwarddifferenceoperatorf(x)=f(x)f(x1).
RelevantformulasforthefactorsrandgaregiveninTableB.
7.
Forcomputationalpurposes,polynomialevaluationsmaybeconvenientlyper-formedonthebasisofrecursionrelations.
RelevantformulasfortheCharlier,Meixner,KrawtchoukandHahnpolynomialsaregivenby:aCn+1=(n+ax)CnnCn1,(B.
81)c(n+β)Mn+1=n+(n+β)c(c1)xMnnMn1,(B.
82)B.
3AskeyScheme513TableB.
7Factorsrandgin(B.
80)fortheCharlier,Meixner,Krawtchouk,andHahnpolynomialsPolynomialrng(x)Charlier1axx!
Meixner1(β+n)xcxx!
Krawtchouk(1p)NNnxp1pxHahn(1)n(β+1)n(N)nα+n+xxβ+NxNnxp(Nn)Kn+1=p(Nn)+n(1p)xKnn(1p)Kn1,(B.
83)AnQn+1(x)=(An+Dnx)Qn(x)DnQn1(x)(B.
84)whereCn≡Cn(x;a),Mn≡Mn(x;β,c),Kn≡Kn(x;p,N),Qn(x)≡Qn(x:α,β,N),andAn=(n+α+β+1)(n+α+1)(Nn)(2n+α+β+1)(2n+α+β+2),Dn=n(n+α+β+N+1)(n+β)(2n+α+β)(2n+α+β+1).
Wenallynotethat,withthestandardizationsabove,theweightingfunctionsofCharlier,Meixner,Krawtchouk,andHahnpolynomialscorrespond,uptoascalingfactorindependentofx,totheprobabilitydensityfunctionsofthePoisson,negativebinormal,binormal,andhypergeometricdistributions.
Thescalingfactorscanbereadilyobtainedfromtheorthogonalityexpressions,andcanbeusedtorescalethestandardformsofwtodeneasuitableinnerproduct,namelyoneforwhichthemeasureofthecorresponding2spaceisunity.
AppendixCImplementationofProductandMomentFormulasInthedevelopmentofcomputercodesimplementingPCexpansions,oneisfacedwiththeproblemofevaluatingmomentsofclassicalorthogonalpolynomials,ormoreoftenofmultidimensionalgeneralizationsofthesepolynomials.
ThisAp-pendixoutlinesthemainingredientofageneralapproachthathasproventobequiteusefulintheconstructionofutilitylibrariesthatarecapableofperformingfunda-mentaloperationsandnonlineartransformationsofPCrepresentationsofrandomvariables.
C.
1One-DimensionalPolynomialsFormostsituations,itisquiteadvantageousforuserstocomputeandstorepoly-nomialsappearinginPCexpansions.
AsmentionedinChapter2,theseexpansionsinvolveorthogonalpolynomialsthataretypicallydenedintermsofseriesexpan-sions.
Forcomputationalpurposes,however,itismoreconvenienttoevaluatethepolynomialsrecursionrelations.
AsdiscussedinAppendixB,intheHermitecase,onehasψ0(x)=1,ψ1(x)=x,(C.
1)ψn(x)=xψn1(x)(n1)ψn2(x),n=2,3,.
.
.
andsimilarrecursionrelationsexistfororthogonalpolynomialsthatarememberoftheAskeyfamily.
Theexploitationoftheserelationshipsisrecommendedwheneverpossible.
O.
P.
LeMatre,O.
M.
Knio,SpectralMethodsforUncertaintyQuantication,ScienticComputation,DOI10.
1007/978-90-481-3520-2,SpringerScience+BusinessMediaB.
V.
2010515516CImplementationofProductandMomentFormulasC.
1.
1MomentsofOne-DimensionalPolynomialsWeareinterestedinevaluatingsecond-ordermomentsoftheformfwheref(x)≡ψi(x)ψj(x),(C.
2)andmoregenerallyinhigher-ordermomentswhichmaybeexpressedas:fwheref(x)≡Mm=1ψim(x).
(C.
3)Suchmomentsmaybeevaluatedonthebasisofanalyticalexpressions.
AlternativelyasdiscussedinAppendixB,Gaussquadraturesmaybereadilyapplied,andtheseyieldexactvaluessolongasasufcientnumberofcollocationpointsisprovided,namelytoensureavanishingremainder.
C.
2MultidimensionalPCBasisAsdiscussedinChap.
2,inthemultidimensionalcasethePCbasisfunctionsarede-nedfromanincompletetensorizationofthecorresponding1Dbasisfunctions.
Asaresult,eachmemberofthemultidimensionalPCbasiscanwrittenasaproductof1Dpolynomialsintheappropriaterandomvariable.
Inordertospecify(andhenceimmediatelyevaluate)theelementsofthemultidimensionalbasis,itisadvanta-geoustoassociatewitheachbasisfunction,iamulti-indexαi=(αi1,αi2,.
.
.
,αiN),whereαikdenotestheorderofthe1Dpolynomialinξk.
Oncethesemulti-indicesareavailable,evaluationofthemultidimensionalbasisfunctionisstraightforward.
WeillustratethisforthecaseoftheN-dimensionalHermitepolynomials.
Wedenotethelatterbyk,inordertodistinguishthemfrom,ψj,their1Dcounterparts.
Bydenitionofthemulti-indices,theN-dimensionalpolynomialchaosescanbeobtainedfrom:k(ξ1,ξ2,.
.
.
,ξN)=Ni=1ψαki(ξi).
(C.
4)Thus,knowledgeofthemulti-indices,togetherwiththe1Dbasisfunctions,providesaveryefcientandconvenientmeansofevaluatingtheN-dimensionalPCs.
C.
2.
1Multi-IndexConstructionTocompletethisconstruction,oneneedstodenethemulti-indicesαi,i=0,.
.
.
,P.
Whilemultipledenitionsarepossible,wehaveadoptedthesameindexingschemeusedin[90].
Usingthisindexingscheme,themulti-indicesaredenedrecursivelyforarbitraryorder,No,anddimension,N,asdescribedinthefollowingpseudo-code:C.
2MultidimensionalPCBasis5171.
Forthe0-thorderpolynomial,set:α0i=0,i=1,.
.
.
,N.
NotethatifNo=0,thenwesetP=0andtheprocessisinterrupted.
2.
FortheNrst-orderpolynomials,set:αij=δijfor1≤i,j≤N.
NotethatifNo=1,thenwesetP=Nandtheprocessisinterrupted.
3.
SetP=N4.
Setpi(1)=1,i=1,.
.
.
,N5.
Fork=2,.
.
.
,No:SetL=PSetpi(k)=Nm=ipm(k1),i=1,.
.
.
,NForj=1,.
.
.
,N:+Form=Lpj(k)+1,.
.
.
,L:·SetP=P+1·SetαPi=αmi,i=1,.
.
.
,N·SetαPj=αPj+1Inthisalgorithm,foraspecick,Ldenotesthenumberoftermsthathavebeenconstructedsofarwithorderlessthank.
Thetermsoforderkarethenconstructedfromthetermsoforder(k1)byincreasingtheorderofthefactorsinthosetermsbelongingtoaspecicdimension,oneatthetime.
Forexample,forthethree-dimensionalstochasticcase,thereare6termsoforder2.
Therst6termsoforder3areconstructedbytakingall6termsoforder2,andincreasingthemulti-indexcorrespondingtotheorderoftherststochasticdimensionby1.
Thenext3termsoforder3areconstructedbytakingthelast3termsoforder2,andincreasingtheorderofthecontributionofthesecondstochasticdimension.
Thelasttermisob-tainedbyincreasingthemulti-indexcorrespondingtothethirddimensionwith1inthelasttermoforder2.
Investigationoftheprocessshowsthatthetermsatthecur-rentorder,inwhichtheorderofthecontributionofdimensionjwillbeincreasedtogettermsatthenextorder,arethetermsthatweregeneratedatthecurrentorderfromthepreviousorderbyincreasingtheorderofdimensionj.
Thisbookkeepingisdonebythepi(k).
Notethattheprocedureabovealsoprovidesthenumber,P,ofN-dimensionalPolynomialChaosesoforder≤No.
C.
2.
2MomentsofMultidimensionalPolynomialsWearenowinterestedingeneralizingtheresultsoftheprevioussectiontotheN-dimensionalcase.
Tothisend,weexploitthetensorproductconstructionof518CImplementationofProductandMomentFormulasthemultidimensionalinnerproduct,andagainrelyonthemulti-indicesαitoex-pressthemomentsofN-dimensionalPolynomialChaosesintermsofproductsof1-dimensionalmoments.
Forinstance,thesecond-ordermomentijcanbeex-pressedas:ijN=Nk=1ψαikψαjk1(C.
5)wherewehaveusedsubscriptstodistinguishbetweentheinnerproductsin1DandN-dimensionalspace.
Similarly,foramomentoforderMwehaveMm=1imN=Nk=1Mm=1ψαimk1.
(C.
6)Theadvantageofthisproductfactorizationisthatinpracticeonlyafewone-dimensionalmomentsneedtobeactuallycomputed,andthisresultsinorder-of-magnitudereductionintheeffortrequiredtoformtherequiredN-dimensionalmo-ments.
Italsogreatlyfacilitatesthecomputationofmomentsinvolvingmixedbases,namelythoseinvolvingtensorproductsofdifferentfamiliesoforthogonalpolyno-mials.
C.
2.
3ImplementationDetailsWeconcludewithabriefoutlineofanattractiveapproachfortheimplementationofPCcomputations,especiallyinthemultidimensionalcase.
Thisapproachisbasedontheestablishmentofacodeinfrastructureinwhichthemulti-indicesdenedinSect.
C.
2.
1arecomputedinapre-processingstep,togetherwiththenecessarymul-tidimensionalmoments,namelythoseappearinginthedenitionofmultiplicationtensorsusedtoevaluateproductsandothernonlineartransformations(seeSect.
4.
5).
Ascanbeanticipated,itisgenerallyadvantageoustocomputeandstoretheseten-sorsinapre-processingstep.
Oneshouldparticularlyseektotakeadvantageofthesparsenatureofthesetensors,namelybystoring(andusing)onlythenon-vanishingentriesofthesetensors.
Onceproperlydenedinthiscompactformat,these"trans-formation"tensorscanbeusedasabasisfortheconstructionofutilitylibrariesthatimplementvarioustransformationsofPCrepresentations.
TheapproachoutlinedaboveinfactformsthebasisoftheUQtoolkit,1whichconstitutesoneincarnationofaPC-utility-librarysoftwareinfrastructure.
1B.
J.
Debusschereetal.
,http://www.
sandia.
gov/UQToolkit/.
References1.
Abramowitz,M.
,Stegun,I.
:HandbookofMathematicalFunctions.
Dover,NewYork(1970)2.
Agarwal,R.
,Wong,P.
:ErrorInequalitiesinPolynomialInterpolationandTheirApplicationsvol.
262.
Kluwer,Dordrecht(1993)3.
Alpert,B.
:AclassofbasesinL2forthesparserepresentationofintegraloperators.
J.
Math.
Anal.
24,246–262(1993)4.
Alpert,B.
,Beylkin,G.
,Gines,D.
,Vozovoi,L.
:Adaptivesolutionofpartialdifferentialequa-tionsinmultiwaveletbases.
J.
Comput.
Phys.
182,149–190(2002)5.
Andreev,V.
,Pliss,N.
,Righetti,P.
:Computersimulationofafnitycapillaryelectrophoresis.
Electrophoresis23(6),889–895(2002)6.
Askey,R.
,Wilson,J.
:SomebasichypergeometricpolynomialsthatgeneralizeJacobipoly-nomials.
Mem.
Am.
Math.
Soc.
54,1–55(1985)7.
Babuka,I.
,Nobile,F.
,Tempone,R.
:Astochasticcollocationmethodforellipticpartialdif-ferentialequationswithrandominputdata.
SIAMJ.
Numer.
Anal.
45(3),1005–1034(2007)8.
Babuka,I.
,Tempone,R.
,Zouraris,G.
E.
:Solvingellipticboundaryvalueproblemswithun-certaincoefcientsbytheniteelementmethod:thestochasticformulation.
Comput.
Meth-odsAppl.
Mech.
Eng.
194,1251–1294(2005)9.
Babuska,I.
,Chatzipantelidis,P.
:Onsolvingellipticstochasticpartialdifferentialequations.
Comput.
MethodsAppl.
Mech.
Eng.
191,4093–4122(2002)10.
Babuska,I.
,Liu,K.
M.
:Onsolvingstochasticinitial-valuedifferentialequations.
Technicalreport,TICAMReport02–17,TheUniversityofTexasatAustin(2002)11.
Babuska,I.
,Miller,A.
:Afeedbackniteelementmethodwithaposteriorierrorestimation.
Comput.
MethodsAppl.
Mech.
Eng.
61,1–40(1987)12.
Babuska,I.
,Rheinboldt,W.
:Aposteriorierrorestimatesfortheniteelementmethod.
Int.
J.
Numer.
MethodsEng.
12,1597–1615(1978)13.
Bachman,G.
,Narici,L.
:FunctionalAnalysis.
Dover,NewYork(2000)14.
Batchelor,G.
:AnIntroductiontoFluidDynamics.
CambridgeUniversityPress,Cambridge(1985)15.
Beale,J.
:Ontheaccuracyofvortexmethodsatlargetimes.
In:ComputationalFluidDynam-icsandReactingGasFlows.
Springer,Berlin(1988)16.
Beaudouin,A.
,Huberson,S.
,Rivoalen,E.
:Simulationofanisotropicdiffusionbymeansofadiffusionvelocitymethod.
J.
Comput.
Phys.
186,122–135(2003)17.
Becker,R.
,Rannacher,R.
:Anoptimalcontrolapproachtoaposteriorierrorestimationinniteelementmethods.
ActaNumer.
10,1–102(2001)18.
Benth,F.
E.
,Gjerde,J.
:Convergenceratesforniteelementapproximationsofstochasticpartialdifferentialequations.
Stoch.
Stoch.
Rep.
63(3–4),313–326(1998)19.
Beran,P.
,Pettit,C.
,Millman,D.
:Uncertaintyquanticationoflimit-cycleoscillations.
J.
Comput.
Phys.
217,217–247(2006)O.
P.
LeMatre,O.
M.
Knio,SpectralMethodsforUncertaintyQuantication,ScienticComputation,DOI10.
1007/978-90-481-3520-2,SpringerScience+BusinessMediaB.
V.
2010519520References20.
Berveiller,M.
,Sudret,B.
,Lemaire,M.
:Stochasticniteelement:anonintrusiveapproachbyregression.
Eur.
J.
Comput.
Mech.
15,81–92(2006)21.
Bielewicz,E.
,Górski,J.
:Shellswithrandomgeometricimperfectionssimulation-basedap-proach.
Int.
J.
Non-LinearMech.
37,777–784(2002)22.
Bier,M.
,Palusinski,O.
,Mosher,R.
,Saville,D.
:Electrophoresis:Mathematicalmodelingandcomputersimulation.
Science219(4590),1281–1287(1983)23.
Blatman,G.
,Sudret,B.
:Sparsepolynomialchaosexpansionsandadaptivestochasticniteelementsusingaregressionapproach.
C.
R.
Méc.
336,518–523(2008)24.
Boyd,J.
:ChebishevandFourierSpectralMethods.
Dover,NewYork(2001)25.
Cameron,R.
,Martin,W.
:TheorthogonaldevelopmentofnonlinearfunctionalsinseriesofFourier-Hermitefunctionals.
Ann.
Math.
48,385–392(1947)26.
Canuto,C.
,Hussaini,M.
,Quarteroni,A.
,Zang,T.
:SpectralMethodsinFluidsDynamics.
SpringerSeriesinComputationalPhysics.
Springer,Berlin(1990)27.
Chenoweth,D.
,Paolucci,S.
:Naturalconvectioninanenclosedverticallayerwithlargehorizontaltemperaturedifferences.
J.
FluidMech.
169,173–210(1986)28.
Choquin,J.
,Lucquin-Desreux,B.
:AccuracyofadeterministicparticlemethodforNavier-Stokesequations.
J.
Numer.
MethodsFluids8,1439–1458(1988)29.
Chorin,A.
:Anumericalmethodforsolvingincompressibleviscousowproblems.
J.
Com-put.
Phys.
2,12–26(1967)30.
Chorin,A.
:Numericalstudyofslightlyviscousow.
J.
FluidMech.
57,785–796(1973)31.
Chorin,A.
:Gaussianeldsandrandomow.
J.
FluidMech.
63,21–32(1974)32.
Christiansen,I.
:Numericalsimulationofhydrodynamicsbythemethodofpointvortex.
J.
Comput.
Phys.
13(3),363–379(1973)33.
Christon,M.
,Gresho,P.
,Sutton,S.
:Computationalpredictabilityofnaturalconvectionowsinenclosure.
In:Bathe,K.
(ed.
)ComputationalFluidandSolidMechanics.
ProceedingsofFirstMITConferenceonComputationalFluidandSolidMechanics,pp.
1465–1468.
Else-vier,Amsterdam(2001)34.
Christopher,D.
:Numericalpredictionofnaturalconvectionowsinatallenclosure.
In:Bathe,K.
(ed.
)ComputationalFluidandSolidMechanics,pp.
1469–1471.
Elsevier,Amster-dam(2001)35.
Clément,P.
:Approximationbyniteelementfunctionsusinglocalregularization.
Rev.
Fr.
Autom.
Inf.
Rech.
Opér.
,Sér.
RougeAnal.
Numér.
9(R-2),77–84(1975)36.
Clenshaw,C.
,Curtis,A.
:Amethodfornumericalintegrationonanautomaticcomputer.
Numer.
Math.
2,197–205(1960)37.
Comini,G.
,Manzan,M.
,Nonino,C.
,Saro,O.
:Finiteelementsolutionsfornaturalconvec-tioninatallrectangularcavity.
In:Bathe,K.
(ed.
)ComputationalFluidandSolidMechanics,pp.
1472–1476.
Elsevier,Amsterdam(2001)38.
Cools,R.
,Maerten,B.
:Ahybridsubdivisionstrategyforadaptiveintegrationroutines.
J.
Univ.
Comput.
Sci.
4(5),486–500(1998)39.
Cormick,S.
M.
:MultigridsMethods.
SIAMBooks,Philadelphia(1987)40.
Cottet,G.
:LargetimebehaviorofdeterministicparticleapproximationstotheNavier-Stokesequations.
Math.
Comput.
56,45–59(1991)41.
Cottet,G.
,Koumoutsakos,P.
:VortexMethodsTheoryandPractice.
CambridgeUniversityPress,Cambridge(2000)42.
Cottet,G.
,Mas-Gallic,S.
:AparticlemethodtosolvetheNavier-Stokessystem.
Numer.
Math.
57,805–827(1990)43.
Courant,D.
,Hilbert,C.
:MethodofMathematicalPhysics.
Interscience,NewYork(1943)44.
Crow,S.
,Canavan,G.
:RelationshipbetweenaWiener-Hermiteexpansionandanenergycascade.
J.
FluidMech.
41,387–403(1970)45.
Das,S.
,Ghanem,R.
,Spall,J.
:AsymptoticsamplingdistributionforPolynomialChaosrep-resentation,fromdata:amaximumentropyandFisherinformationapproach.
SIAMJ.
Sci.
Comput.
30(5),2207–2234(2007)46.
Daube,O.
:Aninuencematrixtechniquefortheresolutionof2dNavier-Stokesequationinvelocity-vorticityformulation.
In:Int.
Conf.
Num.
Meth.
inLaminarandTurbulentFlows(1986)References52147.
Daubechies,I.
:TenLecturesonWavelets.
SIAMBooks,Philadelphia(1992)48.
DeVahlDavis,G.
,Jones,I.
:Naturalconvectioninasquarecavity:acomparisonexercise.
Int.
J.
Numer.
MethodsFluids3,227–248(1983)49.
DeValhDavis,G.
,Jones,I.
:Naturalconvectioninasquarecavity:acomparisonexercise.
Int.
J.
Numer.
MethodsFluids3,227–248(1983)50.
Deb,M.
,Babuska,I.
,Oden,J.
:SolutionofstochasticpartialdifferentialequationsusingGalerkinniteelementtechniques.
Comput.
MethodsAppl.
Mech.
Eng.
190,6359–6372(2001)51.
Debusschere,B.
,Najm,H.
,Matta,A.
,Knio,O.
,Ghanem,R.
,LeMatre,O.
:Proteinlabel-ingreactionsinelectrochemicalmicrochannelow:numericalpredictionanduncertaintypropagation.
Phys.
Fluids15(8),2238–2250(2003)52.
Debusschere,B.
,Najm,H.
,Pébray,P.
,Knio,O.
,Ghanem,R.
,LeMatre,O.
:Numericalchallengesintheuseofpolynomialchaosrepresentationsforstochasticprocesses.
J.
Sci.
Comput.
26(2),698–719(2004)53.
Degond,P.
,Mas-Gallic,S.
:Theweightedparticlemethodforconvectiondiffusionequations,part1:thecaseofanisotropicviscosity.
Math.
Comput.
33(53),485–507(1989)54.
Degond,P.
,Mustieles,F.
J.
:Adeterministicapproximationofdiffusionusingparticles.
J.
Sci.
Stat.
Comput.
1(2),293–310(1990)55.
Desceliers,C.
,Ghanem,R.
,Soize,C.
:Maximumlikelihoodestimationofstochasticchaosrepresentationsfromexperimentaldata.
Int.
J.
Numer.
MethodsEng.
66(6),978–1001(2006)56.
Desceliers,C.
,Soize,C.
,Ghanem,R.
:Identicationofchaosrepresentationsofelasticprop-ertiesofrandommediausingexperimentalvibrationtests.
Comput.
Mech.
39(6),831–838(2007)57.
Ditlevsen,O.
,Tarp-Johansen,N.
:Choiceofinputeldsinstochasticnite-elements.
Probab.
Eng.
Mech.
14,63–72(1999)58.
Doostan,A.
,Ghanem,R.
,Red-Horse,J.
:Stochasticmodelreductionforchaosrepresenta-tions.
Comput.
MethodsAppl.
Mech.
Eng.
196,3951–3966(2007)59.
Drazin,P.
,Reid,W.
:HydrodynamicStability.
CambridgeUniversityPress,Cambridge(1982)60.
Edwards,W.
,Tuckerman,L.
,Friesner,R.
,Sorensen,D.
:Krylovmethodsfortheincompress-ibleNavier-Stokesequations.
J.
Comput.
Phys.
110(1),82–102(1994)61.
Efron,B.
,Tibshirani,R.
:AnIntroductiontotheBootstrap.
MonographsonStatisticsandAppliedProbability,vol.
57.
Chapman&Hall/CRCPress,London(1994)62.
Eldred,M.
:Optimizationstrategiesforcomplexengineeringapplications.
SandiaTechnicalReportSAND98-0340,SandiaNationalLaboratories(1998)63.
Eldred,M.
,Giunta,A.
,Wojkiewicz,S.
,vanBloemenWaanders,B.
,Hart,W.
,Alleva,M.
:Dakota,amultilevelparallelobject-orientedframeworkfordesignoptimization,parameterestimation,sensitivityanalysis,anduncertaintyquantication.
version3.
0referencemanual.
SandiaTechnicalReportSAND02-XXXX(inpreparation),SandiaNationalLaboratories(2002).
http://endo.
sandia.
gov/DAKOTA/papers/Dakota_hardcopy.
pdf64.
Eldredge,J.
:Numericalsimulationoftheuiddynamicsof2Drigidbodymotionwiththevortexparticlemethod.
J.
Comput.
Phys.
221,626–648(2007)65.
Eldredge,J.
,Leonard,A.
:Ageneraldeterministictreatmentofderivativesinparticlesmeth-ods.
J.
Comput.
Phys.
180,686–709(2002)66.
Ermakov,S.
,Jacobson,S.
,Ramsey,J.
:Computersimulationsofelectrokinetictransportinmicrofabricatedchannelstructures.
Anal.
Chem.
70,4494–4504(1998)67.
Ermakov,S.
,Jacobson,S.
,Ramsey,J.
:Computersimulationsofelectrokineticinjectiontech-niquesinmicrouidicdevices.
Anal.
Chem.
72,3512–3517(2000)68.
Ermakov,S.
,Righetti,P.
:Computersimulationforcapillaryzoneelectrophoresis,aquanti-tativeapproach.
J.
Chromatogr.
A667,257–270(1994)69.
Ermakov,S.
,Zhukov,M.
,Capelli,L.
,Righetti,P.
:Artifactualpeaksplittingincapillaryelec-trophoresis,2:defocussingphenomenaforampholytes.
Anal.
Chem.
67,2957–2965(1995)70.
Ern,A.
,Guermond,J.
L.
:TheoryandPracticeofFiniteElementsvol.
159.
Springer,Berlin(2004)522References71.
Ernst,O.
,Powell,C.
,Silvester,D.
,Ullmann,E.
:EfcientsolversforalinearstochasticGalerkinmixedformulationofdiffusionproblemswithrandomdata.
SIAMJ.
Sci.
Comput.
31(2),1424–1447(2009)72.
Nobile,F.
R.
T.
,Webster,C.
:Ananisotropicsparsegridstochasticcollocationmethodforpartialdifferentialequationswithrandominputdata.
SIAMJ.
Numer.
Anal.
46(5),2411–2442(2008)73.
Nobile,F.
R.
T.
,Webster,C.
:Asparsegridstochasticcollocationmethodforpartialdifferen-tialequationswithrandominputdata.
SIAMJ.
Numer.
Anal.
46(5),2309–2345(2008)74.
Fletcher,C.
:ComputationalTechniquesforFluidDynamics,vol.
1.
Springer,Berlin(1988)75.
Frauenfelder,P.
,Schwab,C.
,Todor,R.
A.
:Finiteelementsforellipticproblemswithstochas-ticcoefcients.
Comput.
MethodsAppl.
Mech.
Eng.
194(2–5),205–228(2005)76.
Furnival,D.
,Elman,H.
,Powell,C.
:H(div)preconditioningforamixteniteelementformu-lationofthestochasticdiffusionproblem.
TechnicalReport2008-66,MIMSEPrint(2008)77.
Ganapathysubramanian,B.
,Zabaras,N.
:Sparsegridcollocationschemesforstochasticnat-uralconvectionproblems.
J.
Comput.
Phys.
225,652–685(2007)78.
Gardiner,C.
:HandbookofStochasticMethods:forPhysics,ChemistryandtheNaturalSci-ences.
SpringerseriesinSynergetics.
Springer,Berlin(2004)79.
Ge,L.
,Cheung,K.
,Kobayashi,M.
:Stochasticsolutionforuncertaintypropagationinnon-linearshallow-waterequations.
J.
Hydraul.
Eng.
134,1732–1743(2008)80.
Gebhart,B.
,Jaluria,Y.
,Mahajan,R.
,Sammakia,B.
:Buoyancy-inducedFlowsandTransport.
Hemisphere,Washington(1988)81.
Gentleman,W.
:ImplementingtheClenshaw-Curtisquadrature,I:methodologyandexperi-ence.
Commun.
ACM15(5),337–342(1972)82.
Gentleman,W.
:ImplementingtheClenshaw-Curtisquadrature,II:computingthecosinetransform.
Commun.
ACM15(5),343–346(1972)83.
Genz,A.
,Keister,B.
:FullysymmetricinterpolatoryrulesformultipleintegralsoverinniteregionswithGaussianweight.
J.
Comput.
Appl.
Math.
71,299–309(1996)84.
Gerstner,T.
,Griebel,M.
:Numericalintegrationusingsparsegrids.
Numer.
Algorithms18,209–232(1998)85.
Ghanem,R.
:Probabilisticcharacterizationoftransportinheterogeneousmedia.
Comput.
MethodsAppl.
Mech.
Eng.
158,199–220(1998)86.
Ghanem,R.
:ThenonlinearGaussianspectrumoflognormalstochasticprocessesandvari-ables.
ASMEJ.
Appl.
Mech.
66,964–973(1999)87.
Ghanem,R.
,Dham,S.
:Stochasticniteelementanalysisformultiphaseowinheteroge-neousporousmedia.
Transp.
PorousMedia32,239–262(1998)88.
Ghanem,R.
,Kruger,R.
:Numericalsolutionofspectralstochasticniteelementsystems.
Comput.
MethodsAppl.
Mech.
Eng.
129,289–303(1996)89.
Ghanem,R.
,Spanos,P.
:Aspectralstochasticniteelementformulationforreliabilityanal-ysis.
J.
Eng.
Mech.
ASCE117,2351–2372(1991)90.
Ghanem,R.
,Spanos,P.
:StochasticFiniteElements:ASpectralApproach,2ndedn.
Dover,NewYork(2002)91.
Ghosh,D.
,Ghanem,R.
:Stochasticconvergenceaccelerationthroughbasisenrichmentofpolynomialchaosexpansions.
Int.
J.
Numer.
MethodsEng.
73,162–184(2008)92.
Goluh,G.
,VanLoan,C.
:MatrixComputations,3rdedn.
JohnsHopkinsUniversityPress,Baltimore(1996)93.
Gresho,P.
,Sutton,S.
:8:1thermalcavityproblem.
In:Bathe,K.
(ed.
)ComputationalFluidandSolidMechanics,pp.
1482–1485.
Elsevier,Amsterdam(2001)94.
Grigoriu,M.
:StochasticCalculus,ApplicationsinSciencesandEngineering.
Birkhuser,Basel(2002)95.
Groce,G.
,Favero,M.
:Simulationofnaturalconvectionowinenclosuresbyanunstaggeredgridnitevolumealgorithm.
In:Bathe,K.
(ed.
)ComputationalFluidandSolidMechanics,pp.
1477–1481.
Elsevier,Amsterdam(2001)96.
Grudmann,H.
,Waubke,H.
:Non-linearstochasticdynamicsofsystemswithrandomprop-erties,aspectralapproachcombinedwithstatisticallinearization.
Int.
J.
Non-LinearMech.
31(5),619–630(1997)References52397.
Hairer,E.
,Nrsett,S.
,Wanner,G.
:SolvingOrdinaryDifferentialEquationsI,NonstiffProb-lems,2ndedn.
Springer,Berlin(1992)98.
Halton,J.
:Ontheefciencyofcertainquasi-randomsequencesofpointsinevaluatingmul-tidimensionalintegrals.
Numer.
Math.
2(1),84–90(1960)99.
Hardin,R.
,Sloane,N.
:Anewapproachtotheconstructionofoptimaldesigns.
J.
Stat.
Plann.
Inference37,339–369(1993)100.
Heuveline,V.
,Rannacher,R.
:Duality-baseadaptivityinthehp-niteelementmethod.
J.
Nu-mer.
Math.
12(2),95–113(2003)101.
Hien,T.
,Kleiber,M.
:Stochasticniteelementmodelinginlineartransientheattransfer.
Comput.
MethodsAppl.
Mech.
Eng.
144,111–124(1997)102.
Holden,H.
,Oksendal,B.
,Uboe,J.
,Zhang,T.
:StochasticPartialDifferentialEquations.
Birkhuser,Basel(1996)103.
Hortmann,M.
,Peric,M.
,Scheuerer,G.
:Finitevolumemultigridpredictionoflaminarnatu-ralconvection:Benchmarksolutions.
Int.
J.
Numer.
MethodsFluids11,189–207(1990)104.
Jacod,J.
,Protter,P.
:ProbabilityEssentials.
Universitext/Springer,Berlin(1991)105.
Johnston,H.
,Krasny,R.
:Computationalpredictabilityofnaturalconvectionowsinenclo-sures:Abenchmarkproblem.
In:Bathe,K.
(ed.
)ComputationalFluidandSolidMechanics,pp.
1486–1489.
Elsevier,Amsterdam(2001)106.
Kac,M.
,Siegert,A.
:AnexplicitrepresentationofastationaryGaussianprocess.
Ann.
Math.
Stat.
18,438–442(1947)107.
Kaminski,M.
,Hien,T.
:Stochasticniteelementmodelingoftransientheattransferinlay-eredcomposites.
Int.
Commun.
HeatMassTransfer26(6),801–810(1999)108.
Karhunen,K.
:UberlineareMethodeninderWahrscheinlichkeitsrechnung.
Am.
Acad.
Sci.
,Fennicade,Ser.
A,I37,3–79(1947)109.
Keese,A.
:Numericalsolutionofsystemswithstochasticuncertainties:ageneralpurposeframeworkforstochasticniteelements.
PhDthesis,Tech.
Univ.
Braunschweigh(2004)110.
Keese,A.
,Matthies,H.
:NumericalmethodsandSmolyakquadraturefornonlinearstochasticpartialdifferentialequations.
Technicalreport,InstituteofScienticComputingTUBraun-schweigBrunswick(2003)111.
Keryszig,E.
:AdvancedEngineeringMathematics.
Wiley,NewYork(1979)112.
Kim,J.
,Moin,P.
:Applicationofafractional-stepmethodtoincompressibleNavier-Stokesequations.
J.
Comput.
Phys.
59,308–323(1985)113.
Kim,S.
E.
,Choudhury,D.
:NumericalinvestigationoflaminarnaturalconvectionowinsideatallcavityusinganitevolumebasedNavier-Stokessolver.
In:Bathe,K.
(ed.
)Computa-tionalFluidandSolidMechanics,pp.
1490–1492.
Elsevier,Amsterdam(2001)114.
Knio,O.
,Ghanem,R.
:Polynomialchaosproductandmomentformulas:Auserutility.
Tech-nicalreport,JohnsHopkinsUniversity(2001)115.
Knio,O.
,LeMatre,O.
:UncertaintypropagationinCFDusingpolynomialchaosdecompo-sition.
FluidDyn.
Res.
38,616–640(2006)116.
Knio,O.
,Najm,H.
,Wyckoff,P.
:Asemi-implicitnumericalschemeforreactingow,II:stiff,operator-splitformulation.
J.
Comput.
Phys.
154,428–467(1999)117.
Koekoek,R.
,Swarttouw,R.
:TheAskey-schemeofhypergeometricorthogonalpolynomi-alsanditsq-analogue.
Technicalreport98-17,DepartmentofTechnicalMathematicsandInformatics,DelftUniversityofTechnology(1998).
http://fa.
its.
tudelft.
nl/~koekoek/askey118.
Koumoutsakos,P.
,Leonard,A.
,Pépin,P.
:Boundaryconditionsforviscousvortexmethods.
J.
Comput.
Phys.
113,52–61(1994)119.
LeMatre,O.
:ANewtonmethodfortheresolutionofsteadystochasticNavier-Stokesequa-tions.
Comput.
Fluids(2007,submitted)120.
LeMatre,O.
:PolynomialchaosexpansionofaLagrangianmodelfortheowaroundanairfoil.
C.
R.
Acad.
Sci.
334(11),693–698(2006)121.
LeMatre,O.
,Knio,O.
:Astochasticparticle-meshschemeforuncertaintypropagationinvorticalows.
J.
Comput.
Phys.
226,645–671(2007)122.
LeMatre,O.
,Knio,O.
,Debusschere,B.
,Najm,H.
,Ghanem,R.
:Amultigridsolverfortwo-dimensionalstochasticdiffusionequations.
Comput.
MethodsAppl.
Mech.
Eng.
192(41–42),4723–4744(2003)524References123.
LeMatre,O.
,Knio,O.
,Najm,H.
,Ghanem,R.
:Astochasticprojectionmethodforuidow.
I.
Basicformulation.
J.
Comput.
Phys.
173,481–511(2001)124.
LeMatre,O.
,Knio,O.
,Najm,H.
,Ghanem,R.
:UncertaintypropagationusingWiener-Haarexpansions.
J.
Comput.
Phys.
197(1),28–57(2004)125.
LeMatre,O.
,Najm,H.
,Ghanem,R.
,Knio,O.
:Multi-resolutionanalysisofWiener-typeuncertaintypropagationschemes.
J.
Comput.
Phys.
197(2),502–531(2004)126.
LeMatre,O.
,Najm,H.
,Pébay,P.
,Ghanem,R.
,Knio,O.
:Multi-resolution-analysisschemeforuncertaintyquanticationinchemicalsystems.
SIAMJ.
Sci.
Comput.
29(2),864–889(2007)127.
LeMatre,O.
,Reagan,M.
,Debusschere,B.
,Najm,H.
,Ghanem,R.
,Knio,O.
:Naturalcon-vectioninaclosedcavityunderstochastic,non-Boussinesqconditions.
SIAMJ.
Sci.
Com-put.
26(2),375–394(2004)128.
LeMatre,O.
,Reagan,M.
,Najm,H.
,Ghanem,R.
,Knio,O.
:Astochasticprojectionmethodforuidow.
II.
Randomprocess.
J.
Comput.
Phys.
181,9–44(2002)129.
LeMatre,O.
,Scanlan,R.
,Knio,O.
:NumericalestimationoftheutterderivativesofaNACAairfoilbymeansofNavier-Stokessimulation.
J.
FluidsStruct.
17,1–28(2003)130.
LeQuéré,P.
:AccuratesolutiontothesquarethermallydrivencavityathighRayleighnum-ber.
Comput.
Fluids20(1),29–41(1991)131.
LeQuéré,P.
,deRoquefort,T.
:Computationofnaturalconvectionintwo-dimensionalcavi-tieswithChebyshevpolynomials.
J.
Comput.
Phys.
57,210–228(1985)132.
LeQuéré,P.
,Masson,R.
,Perrot,P.
:AChebyshevcollocationalgorithmfor2dnon-Boussinesqconvection.
J.
Comput.
Phys.
103,320–334(1992)133.
Leonard,A.
:Vortexmethodsforowsimulations.
J.
Comput.
Phys.
37,289–335(1980)134.
Liu,J.
:MonteCarloStrategiesinScienticComputing.
Springer,Berlin(2001)135.
Liu,N.
,Hu,B.
,Yu,Z.
W.
:Stochasticniteelementmethodforrandomtemperatureincon-cretestructures.
Int.
J.
SolidsStruct.
38,6965–6983(2001)136.
Loève,M.
:Fonctionsaléatoiresdusecondordre.
In:Lévy,P.
(ed.
)ProcessusStochastiqueetMouvementBrownien.
GauthierVillars,Paris(1948)137.
Loève,M.
:ProbabilityTheory.
Springer,Berlin(1977)138.
Lumley,J.
:StochasticToolsinTurbulence.
AcademicPress,NewYork(1970)139.
Ma,X.
,Zabaras,N.
:Anadaptivehierarchicalsparsegridcollocationalgorithmforthesolu-tionofstochasticdifferentialequations.
J.
Comput.
Phys.
(2009,inpress)140.
Majda,A.
,Sethian,J.
:ThederivationandnumericalsolutionoftheequationsforzeroMachnumbercombustion.
Comb.
Sci.
Technol.
42,185–205(1985)141.
Makeev,A.
,Maroudas,D.
,Kevrekidis,I.
:Coarsestabilityandbifurcationanalysisusingstochasticsimulators:kineticMonteCarloexamples.
J.
Chem.
Phys.
116,10083–10091(2002)142.
Mao,Q.
,Pawliszyn,J.
,Thormann,W.
:Dynamicsofcapillaryisoelectricfocusingintheabsenceofuidow:high-resolutioncomputersimulationandexperimentalvalidationwithwholecolumnopticalimaging.
Anal.
Chem.
72,5493–5502(2000)143.
Marzouk,Y.
,Najm,H.
:DimensionalityreductionandpolynomialchaosaccelerationofBayesianinferenceininverseproblems.
J.
Comput.
Phys.
228,1862–1902(2009)144.
Marzouk,Y.
,Najm,H.
,Rahn,L.
:StochasticspectralmethodsforefcientBayesiansolutionofinverseproblems.
J.
Comput.
Phys.
224,560–586(2007)145.
Mathelin,L.
,Hussaini,M.
:Astochasticcollocationalgorithmforuncertaintyanalysis.
Tech-nicalReportNASA/CR-2003-212153,NASALangleyResearchCenter(2003)146.
Mathelin,L.
,Hussaini,M.
,Zang,T.
:StochasticapproachestouncertaintyquanticationinCFDsimulations.
Numer.
Algorithms38(1),209–236(2005)147.
Mathelin,L.
,LeMatre,O.
:Dualbasedaposteriorierrorestimationforstochasticniteelementmethods.
Commun.
Appl.
Math.
Comput.
Sci.
2(1),83–115(2007)148.
Mathelin,L.
,Zang,T.
,Bataille,F.
:Uncertaintypropagationforturbulent,compressiblenoz-zleowusingstochasticmethods.
AIAAJ.
42(8),1669–1676(2004)149.
Matthies,H.
,Brenner,C.
,Bucher,C.
,Soares,C.
:Uncertaintiesinprobabilisticnumericalanalysisofstructuresandsolids—stochasticnite-elements.
Struct.
Saf.
19(3),283–336(1997)References525150.
Matthies,H.
,Bucher,C.
:Finiteelementforstochasticmediaproblems.
Comput.
MethodsAppl.
Mech.
Eng.
168,3–17(1999)151.
Matthies,H.
G.
,Keese,A.
:Galerkinmethodsforlinearandnonlinearellipticstochasticpartialdifferentialequations.
Comput.
MethodsAppl.
Mech.
Eng.
194(12–16),1295–1331(2005)152.
McGufn,V.
,Tavares,M.
:Computerassistedoptimizationofseparationsincapillaryzoneelectrophoresis.
Anal.
Chem.
69,152–164(1997)153.
McKay,M.
,Conover,W.
,Beckman,R.
:Acomparisonofthreemethodsforselectingvaluesofinputvariablesintheanalysisofoutputfromacomputercode.
Technometrics21,239–245(1979)154.
Meecham,W.
,Jeng,D.
:UseoftheWiener-Hermiteexpansionfornearlynormalturbulence.
J.
FluidMech.
32,225(1968)155.
Micheletti,S.
,Perotto,S.
,Picasso,M.
:Stabilizedniteelementsonanisotropicmeshes:apriorierrorestimatesfortheadvection-diffusionandthestokesproblems.
SIAMJ.
Numer.
Anal.
41(3),1131–1162(2003)156.
deMontigny,P.
,Stobaugh,J.
,Givens,R.
,Carlson,R.
,Srinivasachar,K.
,Sternson,L.
,Higuchi,T.
:Naphtalene-2,3-dicarboxaldehyde/cyanideion:arationallydesigneduorogenicreagentforprimaryamines.
Anal.
Chem.
59,1096–1101(1987)157.
Moré,J.
J.
,Garbow,B.
S.
,Hillstrom,K.
E.
:UserguideforMINPACK-1.
TechnicalReportANL-80-74,ArgoneNationalLab(1980)158.
Morokoff,W.
,Caisch,R.
:Quasi-MonteCarlointegration.
J.
Comput.
Phys.
122(2),218–230(1995)159.
Mosher,R.
,Dewey,D.
,Thormann,W.
,Saville,D.
,Bier,M.
:Computersimulationandex-perimentalvalidationoftheelectrophoreticbehaviorofproteins.
Anal.
Chem.
61,362–366(1989)160.
Mosher,R.
,Gebauer,P.
,Thormann,W.
:Computersimulationandexperimentalvalidationoftheelectrophoreticbehaviorofproteins.
III.
useoftitrationdatapredictedbytheprotein'saminoacidcomposition.
J.
Chromatogr.
638,155–164(1993)161.
Nair,P.
,Keane,A.
:Stochasticreducedbasismethods.
AIAAJ.
40,1653–1664(2002)162.
Najm,H.
:AconservativelowMachnumberprojectionmethodforreactingowmodeling.
In:Chan,S.
(ed.
)TransportPhenomenainCombustion,vol.
2,pp.
921–932.
TaylorandFrancis,Washington(1996)163.
Najm,H.
:Uncertaintyquanticationandpolynomialchaostechniquesincomputationaluiddynamics.
Annu.
Rev.
FluidMech.
41,35–52(2009)164.
Najm,H.
,Wyckoff,P.
,Knio,O.
:Asemi-implicitnumericalschemeforreactingow,I:stiffchemistry.
J.
Comput.
Phys.
143(2),381–402(1998)165.
Nicola,B.
,Verlinden,B.
,Beuselinck,A.
,Jancsok,P.
,Quenon,V.
,Scheerlinck,N.
,Verbosen,P.
,deBaerdemaeker,J.
:Propagationofstochastictemperatureuctuationsinrefrigeratedfruits.
Int.
J.
Refrig.
222,81–90(1999)166.
Nobile,F.
,Tempone,R.
:Analysisandimplementationissuesforthenumericalapproxi-mationofparabolicequationswithrandomcoefcients.
TechnicalReport22/2008,MOX(2008)167.
Nouy,A.
:Generalizedspectraldecompositionmethodforsolvingstochasticniteelementequations:invariantsubspaceproblemanddedicatedalgorithms.
Comput.
MethodsAppl.
Mech.
Eng.
(2007,submitted)168.
Nouy,A.
:Ageneralizedspectraldecompositiontechniquetosolveaclassoflinearstochasticpartialdifferentialequations.
Comput.
MethodsAppl.
MethodsEng.
196(45–48),4521–4537(2007)169.
Nouy,A.
:Méthodedeconstructiondebasesspectralesgénéraliséespourl'approximationdeproblèmesstochastiques.
Méc.
Ind.
8(3),283–288(2007)170.
Nouy,A.
,Clément,A.
,Schoefs,F.
,Mos,N.
:Anextendedstochasticniteelementmethodforsolvingstochasticpartialdifferentialequationsonrandomdomains.
Comput.
MethodsAppl.
Mech.
Eng.
(2007,submitted)171.
Nouy,A.
,LeMatre,O.
:Generalizedspectraldecompositionforstochasticnonlinearprob-lems.
J.
Comput.
Phys.
228,202–235(2009)526References172.
Nouy,A.
,Schoefs,F.
,Mos,N.
:X-SFEM,acomputationaltechniquebasedonX-FEMtodealwithrandomshapes.
Eur.
J.
Comput.
Mech.
16,277–293(2007)173.
Novak,E.
,Ritter,K.
:Simplecubatureformulaswithhighpolynomialexactness.
Constr.
Approx.
15,499–522(1999)174.
Ogden,R.
T.
:EssentialsWaveletsforStatisticalApplicationsandDataAnalysis.
Birkhuser,Basel(1997)175.
Oksendal,B.
:StochasticDifferentialEquations.
AnIntroductionwithApplications,5thedn.
Springer,Berlin(1998)176.
Paillere,H.
,LeQuéré,P.
:Modellingandsimulationofnaturalconvectionowswithlargetemperaturedifferences:abenchmarkproblemforlowMachnumbersolvers.
In:12thSem-inar"ComputationalFluidDynamics"CEA/NuclearReactorDivision(2000)177.
Paillere,H.
,Viozat,C.
,Kumbaro,A.
,Toumi,I.
:ComparisonoflowMachnumbermodelsfornaturalconvectionproblems.
HeatMassTransfer36,567–573(2000)178.
Palusinski,O.
,Graham,A.
,Mosher,R.
,Bier,M.
,Saville,D.
:Theoryofelectrophoreticsep-arations,partII:constructionofanumericalsimulationschemeanditsapplications.
AIChEJ.
32(2),215–223(1986)179.
Pan,T.
W.
,Glowinski,R.
:Aprojection/wave-likeequationmethodfornaturalconvectionowsinenclosures.
In:Bathe,K.
(ed.
)ComputationalFluidandSolidMechanics,pp.
1493–1496.
Elsevier,Amsterdam(2001)180.
Panton,R.
:IncompressibleFlow.
Wiley,NewYork(1984)181.
Papoulis,A.
,Pillai,S.
:Probability,RandomVariablesandStochasticProcesses.
McGraw-Hill,NewYork(2002)182.
Patankar,N.
,Hu,H.
:Numericalsimulationofelectroosmoticow.
Anal.
Chem.
70,1870–1881(1998)183.
Pellissetti,M.
,Ghanem,R.
:Iterativesolutionofsystemsoflinearequationsarisinginthecontextofstochasticniteelements.
Adv.
Eng.
Softw.
31,607–616(2000)184.
Peraire,J.
,Peiró,J.
,Morgan,K.
:Adaptiveremeshingforthree-dimensionalcompressibleowcomputations.
J.
Comput.
Phys.
103(2),269–285(1992)185.
Pernice,M.
,Walker,H.
:NITSOL:ANewtoniterativesolverfornonlinearsystems.
SIAMJ.
Sci.
Comput.
19(1),302–318(1998)186.
Petras,K.
:OntheSmolyakcubatureerrorforanalyticfunctions.
Adv.
Comput.
Math.
12,71–93(2000)187.
Petras,K.
:FastcalculationintheSmolyakalgorithm.
Numer.
Algorithms26,93–109(2001)188.
Peyret,R.
:SpectralMethodsforIncompressibleViscousFlow.
Springer,Berlin(2002)189.
Potte,G.
,Després,B.
,Lucor,D.
:Uncertaintyquanticationforsystemsofconservationlaws.
J.
Comput.
Phys.
228,2443–2467(2009)190.
Powell,C.
:Robustpreconditioningforsecond-orderellipticPDEswithrandomeldcoef-cients.
TechnicalReport2006.
419,MIMSEprint(2006)191.
Powell,C.
,Elman,H.
:Blockdiagonalpreconditioningforspectralstochasticniteelementsystems.
IMAJ.
Numer.
Anal.
29(2),350–375(2009)192.
Probstein,R.
:PhysicochemicalHydrodynamics:AnIntroduction,2ndedn.
Wiley,NewYork(1994)193.
Pulkelsheim,F.
:OptimalDesignofExperiments.
ClassicsinAppliedMathematics,vol.
50.
SIAM,Philadelphia(2006)194.
Raviart,P.
:AnAnalysisofParticlesMethods.
LectureNotesinMath.
,vol.
1127,pp.
243–324.
Springer,Berlin(1985)195.
Reagan,M.
,Najm,H.
,Debusschere,B.
,Matre,O.
L.
,Knio,O.
,Ghanem,R.
:Spectralstochasticuncertaintyquanticationinchemicalsystems.
Combust.
TheoryModel.
8,607–632(2004)196.
Rivoalen,E.
,Huberson,S.
:Numericalsimulationofaxisymmetricowsbymeansofapar-ticlemethod.
J.
Comput.
Phys.
152,1–31(1999)197.
Roberts,G.
,Rhodes,P.
,Snyder,R.
:Dispersioneffectsincapillaryzoneelectrophoresis.
J.
Chromatogr.
480,35–67(1989)198.
Rosenhead,L.
:Theformationofvorticesfromasurfaceofdiscontinuity.
Proc.
R.
Soc.
134(1931)References527199.
Saad,Y.
:IterativeMethodsforSparseLinearSystems,2ndedn.
SIAM,Philadelphia(2003)200.
Saad,Y.
,Schultz,M.
:GMRES:ageneralizedminimalresidualalgorithmforsolvingnon-symmetriclinearsystem.
J.
Sci.
Stat.
Comput.
7,856–869(1986)201.
Salinger,A.
,Lehoucq,R.
,Pawlowski,R.
,Shadid,J.
:Understandingthe8:1cavityproblemviascalablestabilityanalysisalgorithms.
In:Bathe,K.
(ed.
)ComputationalFluidandSolidMechanics,pp.
1497–1500.
Elsevier,Amsterdam(2001)202.
Scales,P.
,Grieser,F.
,Healy,T.
,White,L.
,Chan,D.
:Electrokineticsofthesilica-solutioninterface:aatplatestreamingpotentialstudy.
Langmuir8,965–974(1992)203.
Schoutens,W.
:StochasticProcessesandOrthogonalPolynomials.
Springer,Berlin(2000)204.
Schuller,G.
:Computationalstochasticmechanics—recentadvances.
Comput.
Struct.
79,2225–2234(2001)205.
Shafer-Nielsen,C.
:Acomputermodelfortime-basedsimulationofelectrophoresissystemswithfreelydenedinitialandboundaryconditions.
Electrophoresis16,1369–1376(1995)206.
Shen,W.
Z.
,Loc,T.
P.
:Simulationof2dexternalviscousowbymeansofadomaindecom-positionmethodusinganinuencematrixtechnique.
Int.
J.
Numer.
MethodsFluids220,1111(1995)207.
Sleijpen,G.
,Fokkema,D.
:BICGSTAB(L)forlinearequationsinvolvingunsymmetricma-triceswithcomplexspectrum.
Electon.
Trans.
Numer.
Anal.
1,11–32(1993)208.
Slepian,D.
,Pollak,H.
:Prolatespheroidalwavefunctions;Fourieranalysisanduncertainty,I.
BellSyst.
Tech.
J.
,3–63(1961)209.
Slepian,D.
,Pollak,H.
:Prolatespheroidalwavefunctions;Fourieranalysisanduncertainty,IV:extensiontomanydimensions;generalizesprolatespheroidalfunctions.
BellSyst.
Tech.
J.
,009–3057(1964)210.
Sluzalec,A.
:Randomheatowwithphasechange.
Int.
J.
HeatMassTransfer43,2303–2312(2000)211.
Sluzalec,A.
:Stochasticniteelementanalysisoftwo-dimensionaleddycurrentproblems.
Appl.
Math.
Model.
24,401–406(2000)212.
Smolyak,S.
:Quadratureandinterpolationformulasfortensorproductsofcertainclassesoffunctions.
Dokl.
Akad.
NaukSSSR4,240–243(1963)213.
Sobol,I.
:Uniformlydistributedsequenceswithanadditionaluniformproperty.
USSRCom-put.
Math.
Math.
Phys.
16,236–242(1976)214.
Soize,C.
:TheFokker-PlanckEquationforStochasticDynamicalSystemsansItsExplicitSteadySolutions.
WorldScientic,Singapore(1994)215.
Soize,C.
:Acomprehensiveoverviewofanon-parametricprobabilisticapproachofmodeluncertaintyforpredictivemodelsinstructuraldynamics.
J.
SoundVib.
288,623–652(2005)216.
Soize,C.
:Randommatrixtheoryformodelinguncertaintyincomputationalmechanics.
Comput.
MethodsAppl.
Mech.
Eng.
194(12–16),1333–1366(2005)217.
Soize,C.
:Constructionofprobabilitydistributionsinhighdimensionusingthemaximumentropyprinciple:Applicationstostochasticprocesses,randomeldsandrandommatrices.
Int.
J.
Numer.
MethodsEng.
76(10),1583–1611(2008)218.
Soize,C.
,Ghanem,R.
:Physicalsystemswithrandomuncertainties:chaosrepresentationswitharbitraryprobabilitymeasure.
J.
Sci.
Comput.
26(2),395–410(2004)219.
Soize,C.
,Ghanem,R.
:Reducedchaosdecompositionwithrandomcoefcientsofvector-valuedrandomvariablesandrandomelds.
Comput.
MethodsAppl.
Mech.
Eng.
198(21–26),1926–1934(2009)220.
Stoer,J.
,Bulirsch,R.
:IntroductiontoNumericalAnalysis,2ndedn.
TextsinAppliedMath-ematics,vol.
12.
Springer,Berlin(1993)221.
Strang,G.
:IntroductiontoAppliedMathematics.
Wellesley-CambridgePress,Wellesley(1986)222.
Stroud,A.
,Secrest,D.
:GaussianQuadratureFormula.
Prentice-Hall,NewYork(1966)223.
Suslov,S.
,Paolucci,S.
:APetrov-Galerkinmethodforowsincavities.
In:Bathe,K.
(ed.
)ComputationalFluidandSolidMechanics,pp.
1501–1504.
Elsevier,Amsterdam(2001)224.
Theting,T.
:SolvingWick-stochasticboundaryvalueproblemsusinganiteelementmethod.
Stoch.
Stoch.
Rep.
70,241–270(2000)528References225.
Thormann,W.
,Mosher,R.
:Theoryofelectrophoretictransportandseparations:thestudyofelectrophoreticboundariesandfundamentalseparationprinciplesbycomputersimulation.
In:Chrambach,A.
,Dunn,M.
,Radola,B.
(eds.
)AdvancesinElectrophoresis,vol.
2,pp.
45–108.
VCH,Weinheim(1988)226.
Thormann,W.
,Zhang,C.
X.
,Caslavska,J.
,Gebauer,P.
,Mosher,R.
:Modelingoftheimpactofionicstrengthontheelectroosmoticowincapillaryelectrophoresiswithuniformanddiscontinuousbuffersystems.
Anal.
Chem.
70,549–562(1998)227.
Trefethen,L.
:IsGaussquadraturebetterthanClenshaw-CurtisSIAMRev.
50(1),67–87(2008)228.
Trottenberg,U.
,Oosterlee,C.
,Schüller,A.
:Multigrid.
AcademicPress,NewYork(2001)229.
Tryoen,J.
:Méthodeintrusivedemodélisationd'incertitudesdanslessystèmeshyperboliquesdeloisdeconservation.
MSthesis,UniversitéParisVI(2008)230.
Tuckerman,L.
:Steady-statesolvingviastokespreconditioning;recursionrelationsofellipticoperators.
In:11thInt.
Conf.
onNumer.
Meth.
inFluidDynamics.
Springer,Berlin(1989)231.
Tuckerman,L.
,Huepe,C.
,Brachet,M.
:Numericalmethodsforbifurcationproblems.
In:Descalzi,O.
,Martinez,J.
,Rica,S.
(eds.
)InstabilitiesandNon-equilibriumStructuresIX,pp.
75–86.
KluwerAcademic,Dordrecht(2004)232.
Vanel,J.
,Peyret,R.
,Bontoux,P.
:Apseudo-spectralsolutionofvorticity-stream-functionequationsusingtheinuencematrixtechnique.
Numer.
MethodsFluidDyn.
II,463–475(1986)233.
Waldvogel,J.
:FastconstructionoftheFejérandClenshaw-Curtisquadraturerules.
BITNu-mer.
Math.
43(1),001–018(2003)234.
Walnut,D.
F.
:AnIntroductiontoWaveletsAnalysis.
AppliedandNumericalHarmonicAnal-ysis.
Birkhuser,Basel(2002)235.
Walter,G.
:WaveletsandOtherOrthogonalSystemswithApplications.
CRCPress,BocaRaton(1994)236.
Walters,R.
,Huyse,L.
:Uncertaintyanalysisforuidmechanicswithapplications.
Technicalreport,NASACR-2002-211449(2002)237.
Wan,Q.
H.
:Effectofelectroosmoticowontheelectricalconductivityofpackedcapillarycolumns.
J.
Phys.
Chem.
B101(24),4860–4862(1997)238.
Wan,X.
,Karniadakis,G.
:Anadaptivemulti-elementgeneralizedpolynomialchaosmethodforstochasticdifferentialequations.
J.
Comput.
Phys.
209,617–642(2005)239.
Webster,C.
:Sparsegridcollocationtechniquesforthenumericalsolutionofpartialdiffer-entialequationswithrandominputdata.
PhDthesis,FloridaStateUniversity(2007)240.
Westerberg,K.
:ThermallydrivenowinacavityusingtheGalerkinniteelementmethod.
In:Bathe,K.
(ed.
)ComputationalFluidandSolidMechanics,pp.
1505–1508.
Elsevier,Am-sterdam(2001)241.
Wiener,S.
:Thehomogeneouschaos.
Am.
J.
Math.
60,897–936(1938)242.
Wojtkiewicz,S.
,Eldred,M.
,Field,R.
,Urbina,A.
,Red-Horse,J.
:Atoolkitforuncer-taintyquanticationinlargecomputationalengineeringmodels.
In:Proceedingsofthe42ndAIAA/ASME/ASCE/AHS/ASCStructures,StructuralDynamics,AIAAPaper2001-1455(2001)243.
Xin,S.
,LeQuéré,P.
:AnextendedChebyshevpseudo-spectralcontributiontocpncfebench-mark.
In:Bathe,K.
(ed.
)ComputationalFluidandSolidMechanics,pp.
1509–1513.
Elsevier,Amsterdam(2001)244.
Xiu,D.
,Hesthaven,J.
:High-ordercollocationmethodsfordifferentialequationswithran-dominputs.
SIAMJ.
Sci.
Comput.
27(3),1118–1139(2005)245.
Xiu,D.
,Karniadakis,G.
:Modelinguncertaintyinsteadystatediffusionproblemsviagener-alizedpolynomialchaos.
Comput.
MethodsAppl.
Mech.
Eng.
191,4927–4948(2002)246.
Xiu,D.
,Karniadakis,G.
:TheWiener-Askeypolynomialchaosforstochasticdifferentialequations.
SIAMJ.
Sci.
Comput.
24,619–644(2002)247.
Xiu,D.
,Karniadakis,G.
:Modelinguncertaintyinowsimulationsviageneralizedpolyno-mialchaos.
J.
Comput.
Phys.
187,137–167(2003)248.
Xiu,D.
,Karniadakis,G.
:Supersensitivityduetouncertainboundaryconditions.
Int.
J.
Nu-mer.
MethodsEng.
61(12),2114–2138(2004)References529249.
Xiu,D.
,Lucor,D.
,Su,C.
,Karniadakis,G.
:Stochasticmodelingofowstructureinteractionsusinggeneralizedpolynomialchaos.
J.
FluidsEng.
124,51–59(2002)250.
Xiu,D.
,Shen,J.
:EfcientstochasticGalerkinmethodsforrandomdiffusionequations.
J.
Comput.
Phys.
228,266–281(2009)251.
Youla,D.
:ThesolutionofthehomogeneousWiener-Hopfintegralequationoccurringintheexpansionofsecond-orderstationaryrandomfunctions.
IRETans.
Inf.
Theory,18–193(1957)IndexAaposteriorierror,413aposteriorierrorestimation,392,406dual-based,408Adaptivehξrenement,423Adaptivemethod,13Adaptivepartitioning,391,396Adaptivescheme,391Adaptivesparsegrid,59Adjointproblem,414Adjointvariable,413Afneoperator,295Afnespace,441Analysisofvariance,40Anisotropicerrorestimator,418Anisotropich/qrenement,429Arnoldimethod,290ARPACK,26Askeyscheme,509Autocorrelationfunction,19BBayesformula,485Bayesianinference,43Betadistribution,509Bi-conjugategradientmethod,291BiCGStabalgorithm,323,329Binormaldistribution,509Biot-Savartintegral,232Bootstrapmethod,152Borelset,484Boussinesqequations,181linearized,325rotationform,231vorticity-streamfunctionformulation,325Burgersequation,107,141Monte-Carlomethod,150NISP,148percentileanalysis,154randomviscosity,144stochastic,141,392variationalform,142CCauchy-Schwarzinequality,488Centralmoment,490orderr,490Chaosexpansion,37Charlierpolynomials,509recurrencerelation,513Rodriguesformula,512Choleskydecomposition,293Clémentinterpolant,418,424Clenshaw-Curtisformula,54Coercivebilinearform,438Collocationmethod,46,68Compressionratio,395Conditionnumber,291Conditionaldensity,492Conditionalprobability,485Conformalmapping,336Conjugategradientmethod,110,291Convection-dominatedow,250Convergencealmostsure,490indistribution,490inLp,490O.
P.
LeMatre,O.
M.
Knio,SpectralMethodsforUncertaintyQuantication,ScienticComputation,DOI10.
1007/978-90-481-3520-2,SpringerScience+BusinessMediaB.
V.
2010531532IndexConvergence(cont.
)inprobability,490mean-square,490Correlationkernel,19Correlationlength,22Correlationtime,22Cubature,51Cubatureformula,11,46multidimensional,46Cumulativedistributionfunction,53,154,489Curseofdimensionality,46,48,56DDAKOTAtoolkit,208Dataerror,2Dataparametrization,78,79Detailfunction,346continuouscascade,396Diffusionvelocity,234Dimension-adaptivesparsegrid,60Distribution,485Distributionfunction,374,489Drivencavity,181Dualproblem,414EElectrokinetically-drivenow,158Electrolytebuffer,263Electroneutralitycondition,268Electroosmoticow,263Electrophoreticmobility,265Electrostaticeldpotential,264Empiricalcorrelation,49Empiricalorthogonalitycondition,66Enrichedapproximationspace,416Errorindicator,399Expectationoperator,77,78,486,487FFastmultipolemethod,236Fejèrformula,54Finitedimensionaldensity,497Finitedimensionaldistribution,497Finitemeasure,484Finite-rate/partialequilibriumformulation,263First-orderMarkovprocess,22Fredholmintegralequation,19GGalerkindivision,92Galerkinformulation,11Galerkininversion,92Galerkinmethod,73,81Galerkinproduct,90repeated,92Galerkinprojection,80absolutevalue,96integrationapproach,99linearproblem,82maxoperator,97minoperator,97non-intrusive,103polynomialnonlinearity,90squareroot,95Taylorexpansion,103tripleproduct,91Galerkinsystem,289blockstructure,289Gaussquadrature,11,51,504Gauss-Hermitequadrature,52,183,203,505Gauss-Jacobiquadrature,511Gauss-Laguerrequadrature,508Gauss-Legendrequadrature,52,377,505Gauss-Seideliteration,303Gaussianpdf,30Gaussianprocess,27Gaussianvariable,11GAUSSQpackage,505,506,509,511Generalizedminimalresidual,290Generalizedpolynomialchaos,35,47Generalizedspectraldecomposition,392,433Germ,6Gibbsphenomenon,343,353Globalapproximationerror,417GMResalgorithm,323,329Gram-Schmidtorthogonalization,377Hh–pspectralapproximation,408Haarscalingfunction,345Haarwavelets,344Hahnpolynomials,509recurrencerelation,513Rodriguesformula,512Haltonsequence,49Heatequation,107,108Galerkinproblem,114Index533Heatequation(cont.
)log-normalconductivity,119semi-discretespectralproblem,113spectralproblem,114stochasticvariationalformulation,111variationalformulation,109Hermitepolynomials,11,501recurrencerelation,503Rodriguesformula,502seriesrepresentation,502Hessianmatrix,418Hilbertspace,75,488Hoeffdingdecomposition,138Homogeneouschaos,11,18orderp,28Hypergeometricdistribution,509Hypergeometricorthogonalpolynomials,509IImageprobabilityspace,78Importancesampling,9IncompleteLUdecomposition,293Independentrandomvariables,6Inducedprobability,485Innerproduct,488Integralapproximation,232,235Integralrepresentation,234Integrationoperator,486Intrinsicvariability,4Intrusivemethod,12Iso-probabilistictransformation,53Isotropichξ,xrenement,424Isotropichξrenement,421Iterativemethod,12,89JJacobipolynomials,509,510recurrencerelation,510Rodriguesformula,510seriesrepresentation,510Jointdensityfunction,492Jointdistributionfunction,491KKarhunen-Loèvedecomposition,6,18Krawtchoukpolynomials,509recurrencerelation,513Rodriguesformula,512Krylovmethod,13,287,288Krylovspace,290Krylovsubspace,87,324LL2norm,488L2space,488Lagrangemultiplier,413Laguerrepolynomials,503recurrencerelation,504Rodriguesformula,503seriesrepresentation,504Lanczosmethod,290LAPACK,26Latinhypercubesampling,9,49,183,207,388Lawoflargenumbers,49Lawoftotalprobability,485Leastsquarest,63Leastsquaresminimization,46,64Legendrepolynomialrescaled,376Legendrepolynomials,500recurrencerelation,501Rodriguesformula,501seriesrepresentation,500Localerror,417Localrenement,391Localvariance,429Lorentzsystem,344,382Low-Mach-numberow,157LUdecomposition,293MMarginaldensity,492Marginaldistribution,492,497Marginalization,492Massdivergenceconstraint,228Massmatrix,26Matrix-freemethod,115,323Maximumentropyprinciple,43Meansquareconvergence,343,490Meansquareerror,20,489Meansquareestimator,489Measurablefunction,485Measurablespace,484Measurabletransformation,486Measure,484Measurespace,484Measurementuncertainty,5534IndexMeixnerpolynomials,509recurrencerelation,513Rodriguesformula,512Metric,488Microchannelow,263Modecoupling,83Modelerror,1MonteCarlomethod,8,48MonteCarlosampling,387Motherwavelet,346Multi-resolutionanalysis,13Multidimensionalindex,397Multigridcycle,307Multigridmethod,13,287,297Multiplicationtensor,85Multipoleexpansion,26Multiresolutionanalysis,343,373Multiwaveletresolutionlevel,380Multiwaveletbasis,373,375Multiwaveletexpansion,344adaptive,392h–pconvergence,382localrenement,391NNaturalconvection,181,253Boussinesqequations,181Navier-Stokesequations,157,160Boussinesqlimit,157linearizedstochasticform,321zero-Mach-numberlimit,212Negativebinormaldistribution,509Nernst-Einsteinequation,265Nestedquadrature,53Nestedsolvers,296Neumannexpansion,433Newtoniteration,321Newtonmethod,287,316stochasticincrement,317,321,322Newton-Cotesformula,54Newton-Raphsoniteration,95Non-intrusivemethod,11,45Non-intrusivespectralprojection,46,47cubaturemethod,48simulationapproach,48Non-parametricanalysis,2Non-rationalspectrum,24Nonlinearfunctional,89Numericalerror,2Nusseltnumber,187OOrthogonalpolynomials,499Orthogonalprojection,489PP1niteelements,109Parametershock,344Parametricbifurcation,343Parametrizationuncertainty,4Partialtensorization,11Particlemethod,229,231Particlestrengthexchange,231Particle-meshscheme,158,236Partition,485Passivescalar,248Percentileanalysis,108Poiseuilleow,164Poissondistribution,509Polynomialchaoscoefcients,30expansion,6,18,28,29mean-squareconvergence,29,35momentformula,515multidimensionalbasis,31onedimensionalbasis,31orderp,11,28productformula,515system,30Polynomialinterpolation,69Poweralgorithm,392,439Preconditioner,89,287,291blockdiagonal,295,296blockJacobi,294diagonal,292incompleteLU,293Jacobi,292operatorexpectation,295Preconditioning,13Principalcomponentanalysis,18Probabilitymeasure,484Probabilityspace,74,484Productprobabilityspace,484Projectionoperator,306Prolongationoperator,306Properorthogonaldecomposition,18Proteinlabelingreaction,263Index535Pseudo-randomnumbergenerator,49Pseudo-randomsampling,8,48,121Pseudo-transientmethod,317QQuadratureformula,46,51,504QuasiMonteCarlo,49RRandomvariables,485independent,20,493mean,39orthogonal,494second-order,17uncorrelated,494variance,39Randomvector,486Randomwalk,234Rationalspectrum,21Rayleigh-Bénardow,344,350,394Rayleigh-Bénardinstability,360Realization,17,495Reducedbasis,13Renementξ,418h,418p,418x,418Renementthreshold,394Regressionmethod,11Remeshingscheme,237Resolutionlevel,347Responsesurfacemethod,45Riskanalysis,6SSamplepath,495Samplesolutionset,8Samplespace,483Samplingerror,152Second-orderprocess,11Sensitivityindex,139total,139σalgebra,74σeld,483Simulationapproach,48Smolyakformula,57Smoothingkernel,241Soboldecomposition,138Sobolsequence,49Sobolevspace,109Solutionexpansion,79Solvabilitycondition,116Solvabilityconstraint,214Sparsecollocation,71Sparsecubature,56Sparsegridmethod,46adaptive,46Sparsestoragemethod,115Sparsesystem,287Sparsetensorization,56Spectraldecomposition,19Spectralexpansion,17Spectralmethod,6,9Spectralproblem,74,80preconditioned,89Standarderror,152Stationaryprocess,21Stiffnessmatrix,26Stochasticdiscretization,77Stochasticelement,411Stochasticeld,496StochasticGalerkinproblem,12,287StochasticGalerkinprojection,288Stochasticmatrix,288Stochasticmode,79Stochasticmultigrid,297Stochasticprocess,495almostsurelycontinuous,496centered,18continuousinprobability,496continuousinthep-thmean,496meansquarecontinuous,18,496secondorder,17,497stationaryinthestrictsense,497weaklystationary,497Stochasticprojectionmethod,157,394Boussinesqow,183incompressibleow,158Stochasticreducedbasis,341Stochasticresidual,80Stokesproblem,318Subspacemethod,289Successiveoverrelaxation,304Symmetricpositivedenitesystem,291Symmetricsemi-positiveoperator,19536IndexTTensorproductformula,55Tensorproductspace,75Testrandomvariable,81Trapezoidalrule,54Triangularkernel,24Triangulation,109UUncertaintymanagement,6Uncertaintypropagation,5Uncertaintyquantication,5,6UQtoolkit,272,518VV-cycle,307Validation,5Variance,22Varianceanalysis,6,137Variancereduction,9Vorticity,231WW-cycle,307Waveletexpansion,343Weakformulation,77Whitenoise,21band-limited,24Wiener-Haarexpansion,344,345Wiener-Legendreexpansion,344ZZetapotential,263Zufallpackage,121