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AppendixABasicTheoreticalToolsThischaptercollectsbasicnotionsoffunctionalanalysisandnumericalanalysis,togetherwithtoolsthatareextensivelyusedthroughoutthebook.
Foramorein-depthreading,werefertoe.
g.
[265,36,166,66,1,236].
A.
1LinearMaps,FunctionalsandBilinearFormsLetVandWbetwonormedvectorspacesoverR.
Amap(oroperator)L:V→Wissaidtobelinearifforanytwovectorsx,y∈Vandanyscalarα∈RL(x+y)=L(x)+L(y),L(αx)=αL(x),thatis,ifitpreservesadditionandmultiplicationbyascalar.
Alineartransformationalwaysmapslinearsubspacesontolinearsubspaces–theselatterpossiblyoflowerdimension.
AlinearoperatorisboundedifthereexistsaconstantM>0suchthatLvW≤MvVv∈V.
Alinearboundedoperatorisalsocontinuous.
ThevectorspaceL(V,W)formedbyalllinearcontinuousoperatorsfromVintoWcanbeendowedwiththenormLL(V,W)=supv∈V\{0}LvWvV.
(A.
1)Letusintroducearelevantclassofmaps,theso-calledafnemaps,whichareextensivelyexploitedthroughoutthebook.
Anafnemap(orafnetransformation)L:V→Wisafunctionoftheformv→Mv+b,whereM:V→WisalinearmaponVandbisanelementofW.
Unlikea(purely)linearmap,anafnemapdoesnotmapthezeropointinitself.
Everylineartrans-formationisafne,butnoteveryafnetransformationislinear.
SpringerInternationalPublishingSwitzerland2016A.
Quarteroni,A.
Manzoni,F.
Negri,ReducedBasisMethodsforPartialDifferentialEqua-tions,UNITEXT–LaMatematicaperil3+292,DOI10.
1007/978-3-319-15431-2_A266AppendixA:BasicTheoreticalToolsAnafnemappreservespreservescollinearity(thatis,allpointslyingonalineini-tiallystilllieonalineaftertransformation)andratiosofdistances.
Moreover,setsofparallellinesremainparallelunderanafnetransformation.
IntheEuclideanspaceV=Rd,examplesofafnetransformationsincludetranslations,rotations,scal-ings,homotheties,similaritytransformations,reections,andtheircompositions.
Wepointoutthatalthoughanafnetransformationpreservesproportionsonlines,itdoesnotpreserveanglesorlengths.
Alltrianglesareafne,thatis,anytrianglecanbetransformedintoanyotherbyanafnetransformation.
Letusdenetwoimportant(linear)subspacesofVandW,respectively.
GivenalinearmapL:V→W,wedenethekernelandtheimageorrangeofLbyKer(L)={x∈V:L(x)=0},Range(L)={w∈W:w=L(x),x∈V}.
ThedimensionsofthesetwosubspacesaretherankandthenullityofL,respectively,dim(Range(L))=rank(L),dim(Ker(L))=null(L).
Thesetwosubspacesarerelatedthroughtheso-calledrank-nullitytheoremdim(Ker(L))+dim(Range(L))=null(L)+rank(L)=dim(V).
RemarkA.
1.
Matricesyieldexamplesoflinearmapsovernite-dimensionalvectorspaces:ifA∈Rm*nthenL(x)=Ax,x∈Rn,describesalinearmapbetweenRnandRm.
Inthiscase,therankandnullityofLcoincidewiththewell-knownnotionsofrankandnullityofthematrixA,respectively.
Aremarkablepropertyexploitedthroughoutthebookisthefollowingone.
Letusdenoteby{j}nj=1abasisforthespaceV,wheren=dim(V).
EachelementofVisuniquelydeterminedbyalinearcombinationofthebasisfunctions,undertheformc11+.
.
.
+cnn,withc1,.
.
.
,cn∈R.
Thus,ifL:V→Wisalinearmap,L(c11+.
.
.
+cnn)=c1L(1)+.
.
.
+cnL(n),thatis,themapLisentirelydeterminedbythevectorsL(1),.
.
.
,L(n).
Bydenoting{ζi}mi=1abasisofW,wecanrepresenteachelementL(j)∈WasL(j)=a1jζ1+.
.
.
+amjζm.
Inthisway,themapisentirelydeterminedbythematrixA∈Rm*n,whereAij=aij,i=1,.
.
.
,m,j=1,.
.
.
,n,sothatforanyv∈VwecanevaluateitsimagethroughLasL(v)=∑nj=1aijvj.
InthespecialcaseW=R,linearmapsarecalledfunctionals.
DenitionA.
1.
GivenavectorspaceV,wecallfunctionalonVanoperatorF:V→RassociatingarealnumbertoeachelementofV.
AfunctionalFisoftendenotedbymeansofthedualityF(v)=F,v.
Afunc-tionalissaidtobelinearifitislinearwithrespecttoitsargument,thatisifF(λv+μw)=λF(v)+μF(w)λ,μ∈R,v,w∈V.
A.
1LinearMaps,FunctionalsandBilinearForms267AlinearfunctionalisboundedifthereisaconstantC>0suchthat|F(v)|≤CvVv∈V.
(A.
2)AlinearandboundedfunctionalonaBanachspace(i.
e.
anormedandcompletespace)isalsocontinuous.
WecandenethespaceV,calleddualofV,asthesetoflinearandboundedfunctionalsonV,thatisV={F:V→RsuchthatFislinearandbounded}andweequipitwiththenorm·VdenedasFV=supv∈V\{0}|F(v)|vV.
(A.
3)TheconstantCappearingin(A.
2)isgreaterorequaltoFV.
Alinearoperatorbetweentwospacesforwhichtheinversealsoexistsiscalledisomorphism,accordingtothefollowingDenitionA.
2.
Alinearandbounded(hencecontinuous)operatorTbetweentwofunctionalspacesVandWisanisomorphismifitmapsbijectivelytheelementsofthespacesVandWanditsinverseT1exists.
IfalsoVWholds,suchisomor-phismiscalledcanonical.
Wefurtherintroducethedenitionofanotherfundamentalingredientofabstractvariationalproblems,bilinearforms.
DenitionA.
3.
GivenanormedfunctionalspaceVwecallformanapplicationwhichassociatestoeachpairofelementsofVarealnumbera:V*V→R.
Aformiscalled:1.
bilinearifitislinearwithrespecttobothitsarguments,i.
e.
ifa(λu+μw,v)=λa(u,v)+μa(w,v)λ,μ∈R,u,v,w∈V,a(u,λw+μv)=λa(u,v)+μa(u,w)λ,μ∈R,u,v,w∈V;2.
continuousifthereexistsaconstantM>0suchthat|a(u,v)|≤MuVvVu,v∈V;(A.
4)3.
symmetricifa(u,v)=a(v,u)u,v∈V;(A.
5)4.
positive(orpositivedenite)ifa(v,v)>0v∈V;(A.
6)5.
coerciveifthereexistsaconstantα>0suchthata(v,v)≥αv2Vv∈V.
(A.
7)268AppendixA:BasicTheoreticalToolsA.
2HilbertSpacesHilbertspacesrepresenttheidealsettingtoformulatethemostcommonboundaryvalueproblems.
Tothisaim,letusgivethefollowingDenitionA.
4.
Amap(·,·):V*V→RisaninnerorscalarproductinVifitsatisesthefollowingproperties:forallx,y,z∈V,α,β∈R,1.
positivity:(x,x)≥0and(x,x)=0ifandonlyifx=0;2.
symmetry:(x,y)=(y,x);3.
bilinearity:(αx+βz,y)=α(x,y)+β(z,y).
AnyscalarproductyieldsaninducednormoverV,denedasx=(x,x)x∈V.
(A.
8)Aninnerproductspace–thatis,avectorspacewithaninnerproduct–isalsoanormedvectorspace.
Moreover,thefollowingCauchy-Schwarzinequalityholds:|(x,y)|≤xyx,y∈V;(A.
9)equalityholdsin(A.
9)ifandonlyify=αx,forsomeα∈R.
IfVisa(linear)spaceendowedwithaninnerproduct–thatis,aninnerproductspace–wesaythatVisaHilbertspaceifitiscompletewithrespecttotheinducednorm(A.
8).
Wewilloftendenotethecorrespondingscalarproductby(·,·)V.
Twovectorsx,y∈VaresaidtobeorthogonalwithrespecttotheV-scalarproductandwritex⊥yif(x,y)V=0.
AvectorxissaidtobeorthogonaltoasetW(writtenx⊥W)ifx⊥wforeachw∈W.
Wenowconsiderthefollowingproblem:givenavectorx∈VandaclosedsubspaceWinV,nd(ifitexists)thevectorw∈Wclosesttox,inthesensethatitminimizesxwV.
Thefollowingtheoremgivesananswertothisproblem,aswellasausefulchar-acterizationofitssolutionintermsoforthogonalityproperties.
TheoremA.
1(Orthogonalprojections).
LetVbeaHilbertspaceandWaclosedsubspaceofV.
Correspondingtoanyvectorx∈V,thereisauniquevectorw∈W(calledtheprojectionofxontoW)suchthatxwV=infw∈WxwV.
Furthermore,anecessaryandsufcientconditionforw∈Wtobetheuniquemin-imizingvectoristhatxwbeorthogonaltoW,i.
e.
(xw,w)V=0w∈W.
TheorthogonalcomplementofasubspaceWofaHilbertvectorspaceV,denotedbyW⊥,isthesetofofallvectorsinVthatareorthogonaltoeveryvectorinWW⊥={v∈V|(v,w)V=0w∈W}.
(A.
10)A.
3AdjointOperators269TheorthogonalcomplementofasubspaceWofHisasubspaceofV,too,anditisalwaysclosed(inthetopologyinducedbythemetricdenedoverV).
IfWisaclosed(linear)subspaceofV,thenWW⊥=V,thatis,eachelementv∈Vcanbeuniquelyexpressedasasumofanelementw∈Wandanelementw⊥∈W⊥.
Finally,dimW+dimW⊥=dimV.
Thefollowingtheorem,calledidenticationorrepresentationtheorem(seee.
g.
[236]or[265]fortheproof),holds.
TheoremA.
2(Rieszrepresentationtheorem).
LetVbeaHilbertspace.
ForeachlinearandboundedfunctionalfonVthereexistsauniqueelementxf∈Vsuchthatf(y)=(y,xf)Vy∈V,andfV=xfV.
(A.
11)Conversely,eachelementx∈VidentiesalinearandboundedfunctionalfxonVsuchthatfx(y)=(y,x)Vy∈VandfxV=xV.
(A.
12)IfVisaHilbertspace,itsdualspaceVoflinearandboundedfunctionalsonVisaHilbertspacetoo.
Moreover,thankstoTheoremA.
2,thereexistsabijectiveandisometric(i.
e.
norm-preserving)transformationfxfbetweenVandVthankstowhichVandVcanbeidentied.
Wecandenotethistransformation–calledRiesz(isometric)isomorphism–asfollows:RV:V→V,x→fx=RVx,R1V:V→V,f→xf=R1Vx.
(A.
13)ThankstotheRieszisomorphism,wecandeneaninnerproductoverVasfollows:(F,G)V=(R1VF,R1VG)VF,G∈V.
Asaconsequence,FV=R1VFVF∈V.
(A.
14)DenitionA.
5.
LetXandYbetwoHilbertspaces.
WesaythatXisembeddedinYwithcontinuousembeddingifthereexistsaconstantCsuchthatwY≤CwXw∈X.
MoreoverXisdenseinYifeachelementbelongingtoYcanbeobtainedasthelimit,inthe·Ynorm,ofasequenceofelementsofX.
A.
3AdjointOperatorsRiesz'sTheoremenablesthedenitionofadjointoperatorofL,whichextendsthenotionoftransposeATofamatrixA∈Rm*n,thatis(Ax,y)=(x,ATy)x∈Rn,y∈Rm.
Actually,thenotionofadjointoperatorcanbedenedforanygivenL∈L(V,W),beingVandWtwoBanachspaces.
Tothisend,letusconsiderthereal-valuedmapTy:x→Wy,LxW.
Foranyprescribedy∈W,TydenesanelementofV.
270AppendixA:BasicTheoreticalToolsInfact,|Tyx|=|Wy,LxW|≤LxWyW≤LL(V,W)xVyW,sothatTy≤LL(V,W)yW.
TheoperatorL:W→VdenedbythemapWy→Ty∈ViscalledtheadjointofL.
Moreprecisely:DenitionA.
6.
TheoperatorL:W→VdenedbytheidentityVLy,xV=Wy,LxWx∈V,y∈W,(A.
15)iscalledtheadjointofL.
LisalinearandboundedoperatorbetweenWandV,thatisL∈L(W,V),moreoverLL(W,V)=LL(V,W).
InthecasewhereVandWaretwoHilbertspaces,anadditionaladjointoperator,L:W→V,calledtransposeorHilbertspaceadjointorsimplyadjointofL,canbeintroduced.
Itisdenedby(Ly,x)V=(y,Lx)Wx∈V,y∈W.
(A.
16)Here,(·,·)VdenotesthescalarproductofV,while(·,·)Wdenotesthescalarprod-uctofW.
Theabovedenitioncanbeexplainedasfollows:foranygivenelementy∈W,thereal-valuedfunctionx→(y,Lx)Wislinearandcontinuous,henceitde-nesanelementofV.
ByRiesz'stheorem(TheoremA.
2)thereexistsanelementxofV,whichwenameLy,thatsatises(A.
16).
SuchoperatorbelongstoL(W,V)(thatis,itislinearandboundedfromYtoX),moreoverLL(W,V)=LL(V,W).
(A.
17)WesaythatLisselfadjoint(orHermitian)ifV=WandL=L.
Then,(A.
16)inthiscasereducesto(Lx,y)V=(x,Ly)V.
Inparticular,symmetricmatricesarespecialcasesofselfadjointoperators.
InthecasewhereVandWaretwoHilbertspaces,wethushavetwonotionsofadjointoperator,LandL,whicharelinkedthroughthefollowingrelationship:RVL=LRW,(A.
18)RVandRWbeingRiesz'scanonicalisomorphismsfromVtoVandfromWtoW,respectively(see(A.
13)).
Indeed,x∈V,y∈W,VRVLy,xV=(Ly,x)V=(y,Lx)W=WRWy,LxW=VLRWy,xV.
A.
4CompactOperatorsInthissection,webrieyrecallthedenitionandsomeremarkablepropertiesofcompactoperators.
Forafurtheranalysisofthistopic,see,e.
g.
,[226,72,66].
InthefollowingVandWwilldenotetwoHilbertspaces.
A.
4CompactOperators271DenitionA.
7.
AnoperatorL∈L(V,W)iscompactifLmapsboundedsetsintoprecompactsets,i.
e.
L(E)iscompactinWforeveryboundedEV.
DenitionA.
8.
L∈L(V,W)issaidtohaveniterankifRange(L)Wisnitedimensional.
IfL∈L(V,W)isaniterankoperator,thenLiscompact.
Inparticular,ifeitherdim(V)0,δ>0suchthatF(x+h)F(x)LxhW≤εhVh∈Vwith||h||Vm+n/2.
Inparticular,inonespatialdimension(n=1),thefunctionsofH1(Ω)arecontinuous(theyareindeedabsolutelycontinuous,see[236]and[36],whileintwoorthreedimensionstheyarenotnecessarilyso.
Instead,thefunctionsofH2(Ω)arealwayscontinuousforn=1,2,3.
276AppendixA:BasicTheoreticalToolsAnotherSobolevspace,ubiquitousinthevariationalformulationofPDEs,when-everweimposeDirichletboundaryconditions,isH10(Ω),thatis,thespaceoffunc-tionsv∈H1(Ω)vanishingontheboundaryΩ.
WecanthusdeneH10(Ω)={v∈H1(Ω):v|Ω=0}ifwerequirethatv=0overtheentireboundaryΩ,orH1ΓD(Ω)={v∈H1(Ω):v|ΓD=0,ΓDΩ}ifwerequirethatv=0overtheportionΓDΩ.
Togiveaprecisemeaningtothevaluev|Ωofv∈H1(Ω)onΩ–theso-calledtraceofvonΩ–weneedtointroduceatraceoperator,whichassociatestoeachfunctionv∈L2(Ω),withgradientinL2(Ω),afunctionv|ΩrepresentingitsvaluesonΩ:TheoremA.
5.
LetΩbeadomain3ofRdprovidedwithasufcientlyregularbound-aryΩ,andletk≥1.
Thereexistsoneandonlyonelinearandcontinuousappli-cationγ0:Hk(Ω)→L2(Ω)suchthatγ0v=v|Ωforanyv∈Hk∩C0(Ω);γ0viscalledtraceofvonΩ.
Inparticular,thereexistsaconstantC>0suchthatγ0vL2(Γ)≤CvHk(Ω).
TheresultstillholdsifweconsiderthetraceoperatorγΓD:Hk(Ω)→L2(ΓD)whereΓDisasufcientlyregularportionoftheboundaryofΩwithpositivemeasure.
Owingtothisresult,DirichletboundaryconditionsmakesensewhenseekingsolutionsvinHk(Ω),withk≥1,providedweinterprettheboundaryvalueinthesenseofthetrace.
ThetraceoperatorsallowforaninterestingcharacterizationofthepreviouslydenedspaceH10(Ω).
Indeed,wehavethefollowingproperty:PropositionA.
1.
LetΩbeadomainofRdprovidedwithasufcientlyregularboundaryΩandletγ0bethetraceoperatorfromH1(Ω)inL2(Ω).
ThenH10(Ω)=Ker(γ0)={v∈H1(Ω):γ0v=0}.
Inotherwords,H10(Ω)isformedbythefunctionsofH1(Ω)havingnulltraceontheboundary.
ThefunctionsofH10(Ω)and,moreingeneral,thoseofH1ΓD(Ω),foreveryΓDΩ,meas(ΓD)>0,enjoythefollowingrelevantproperties:PropositionA.
2(Poincareinequality).
LetΩbeaboundedsetinRd;thenthereexistsaconstantCΩsuchthatvL2(Ω)≤CΩ|v|H1(Ω)v∈H10(Ω).
(A.
22)PropositionA.
3.
Theseminorm|v|H1(Ω)isanormonthespaceH10(Ω)thatturnsouttobeequivalenttothenormvH1(Ω).
3AdomainΩofRdisaboundedconnectedopensubsetofRdwithaLipschitzcontinuousboundaryΩ,see,e.
g.
,[66,Chap.
1].
A.
8PolynomialInterpolationandOrthogonalPolynomials277A.
7BochnerSpacesWhenconsideringparametrizedfunctionsv(x,μμμ),μμμ∈PRP,P>0,itisnaturaltointroducethedenitionofLp-spacesofWk,q(Ω)-valuedfunctions.
LetusdenotebySacompactsubsetPofRP.
For1≤p,q2n2n2andλn(T)2n+1enlogn,n→∞sothatforlargenLagrangeinterpolationshallbecomeunstable.
ThisfallsunderthenameofRungephenomenon:theinterpolantInfmightshowoscillationsclosetotheextremaoftheintervalnearly2ntimeslargerthanf,eveniffisanalytic.
See,e.
g.
,[221,Sect.
8.
1].
InterpolationbasedonChebyshev(orLegendre)polynomialsallowstoovercometheRungephenomenon.
ChebyshevandLegendrepolynomialsaretwoexamplesoffamiliesoforthogonalpolynomials,whichprovideageneraltoolinapproximationtheory(see,e.
g.
,[51,221,249]).
Letw=w(x)beapositive,integrablefunctionon(1,1)anddenoteby{pk∈Pk,k=0,1,.
.
.
}asystemofalgebraicpolynomialswhicharemutuallyorthogonalon(1,1)withrespecttow,thatis,11pk(x)pm(x)w(x)dx=0ifk=m.
A.
8PolynomialInterpolationandOrthogonalPolynomials279Letusdenoteby(·,·)wtheinnerproductdenedby(f,g)w=11f(x)g(x)w(x)dxandfw=(f,f)1/2w;(·,·)wand·warerespectivelythescalarproductandthenormfortheweightedL2w(1,1)space.
LegendreandChebyshevpolynomialsover[1,1]correspondtothefollowingtwocases,forwhichwecanalsoprovidetheexpressionofGausspointsandco-efcients–respectively,Gauss-Lobattopointsandcoefcients,inthecasewealsoincludetheextremaoftheintervalamongthesetofpoints:Chebyshevweightw(x)=(1x2)1/2,resultingintheGausspointsandweightsxj=cos(2j+1)π2(n+1),wj=πn+1,0≤j≤n;thecorrespondingGauss-Lobattopointsandweightsarexj=cosπjn,wj=πdjn,0≤j≤n,n≥1(A.
27)beingd0=dn=2,Dj=1forj=1,.
.
.
,n1.
TheGausspointsare,foraxedn≥0,thezerosoftheChebyshevpolynomialTn+1∈Pn+1,beingTk(x)=coskθ,θ=arccosx,k=0,1,.
.
.
whereas,forn≥1,theinternalGauss-LobattopointsarethezerosofTn;Legendreweightw(x)=1,resultingintheGausspointsandweightsxjzerosofLn+1(x),wj=2(1x2j)(Ln+1(xj))2,0≤j≤n;thecorrespondingGauss-Lobattopointsandweightsarex0=1,xn=1,xjzerosofLn(x),1≤j≤n1(A.
28)wj=2n(n+1)1(Ln(xj))2,0≤j≤nwhereLk(x)=12k[k/2]∑l=0(1)lkl2(kl)kxk2lk=0,1,.
.
.
isthek-thLegendrepolynomial.
280AppendixA:BasicTheoreticalToolsTheChebyshevinterpolantoffisthepolynomialIGLn,wfofdegreenthatinterpo-latesfattheGauss-Lobattopoints(A.
27),andcanbeexpressedasIGLn,wf(x)=n∑i=0f(xi)li(x)(A.
29)whereli∈Pnisthei-thcharacteristicLagrangepolynomialdenedby(A.
25),suchthatli(xj)=δijforanyi,j=0,.
.
.
,n.
InthesamewaywecanobtaintheLegendreinterpolantoff,byreplacingthepoints(A.
27)withthosedenedin(A.
28).
Amoreefcient(andstable)interpolationisobtainedbyrelyingontheso-calledbarycentricformulae,seee.
g.
[249]forfurtherdetails.
TheclusteringofChebyshevpointsclosetotheextremaoftheintervalisin-deedakeyfeature–Legendrepointshaveasimilardistributionandsharethesamegoodbehavior.
Fromaquantitativestandpoint,theLebesgueconstantgrowsonlylogarithmicallyifChebyshevpointsareused,sinceΛn(T)≤2πlog(n+1)+1andλn(T)2πlogn,n→∞.
(A.
30)ThisresultmakesChebyshevpointsabetterchoiceforpolynomialinterpolation.
Thankstothisresult,wecanalsostatethatChebyshevinterpolantsarenear-best:puttingtogether(A.
26)and(A.
30),wehavethatthemaximumnormaccuracydifferencebetweenChebyshevinterpolantsandthebestapproximantcanneverbelarge.
Infact,ifIGLn,wfistheChebyshevinterpolantoffatthen+1Gauss-Lobattopoints,thenfIGLn,wf∞≤2+2πlog(n+1)fpn∞wheretheconstantappearingattheright-handsideisoforder102forn>1066.
Finally,anotherrelevantfeatureofChebyshev(orLegendre)interpolationisthatthesmootherthefunctionbeinginterpolated,thefasterthedecayoftheerrorwithrespectton.
Inparticular,theinterpolationerrorcanbeboundedasfIGLn,wfw≤Cnsfs,w,s≥1inthecaseofbothChebyshevandLegendreinterpolation,providedforsomes≥1f(k)∈L2w(1,1)foranyk=0,.
.
.
,sandfs,w=(∑sk=0f(k)2w)1/2.
Weunderlinethattheconvergencedependsonthedegreeofregularityoffinadditiontothenumberofinterpolationpoints.
Wealsoobtainanexponentialconvergenceresultinthecaseofanalyticfunctions,s→∞.
Moreover,foranycontinuousfunctionf,fIGLn,wf∞≤Cn1/2sfs,w,s≥1.
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Springer-Verlag,NewYork(1990)IndexAafneexpansion163parametricdependence7,8,52transformation157,162,175approximationfull-order1Galerkin24,31high-delity1,41low-rank117Petrov-Galerkin27Bbestapproximation96,118branchofnonsingularsolutions216Ccompliance47computeraideddesign179conditiondiscreteinf-sup27,32,187inf-sup16,18uniformellipticity20constantcoercivity14continuity14,228discretecoercivity26discretecontinuity31discreteinf-sup30Lebesgue278Sobolevembedding217,232,242controlvariable3convergence25DderivativeFrechet215,217,219,272Gateaux272inthesenseofdistributions273deviation96directsum79discrepancy96distribution273domain276original40,155reference40,155,161domaindecomposition179Eequationadvection-diffusion-reaction12,20,159,181equationslinearelasticity172linearelasticity12,22,184Navier-Stokes13,23,173,217,232Oseen243Stokes13,22,173,186erroraprioriestimate34approximation24errorestimateoptimal24,27,32errorestimatoraposteriori56effectivity59Ffactorcoercivity41continuity41stability41,46niteelementsLagrangianbasis34SpringerInternationalPublishingSwitzerland2016A.
Quarteroni,A.
Manzoni,F.
Negri,ReducedBasisMethodsforPartialDifferentialEqua-tions,UNITEXT–LaMatematicaperil3+292,DOI10.
1007/978-3-319-15431-2294Indexform267afne52bilinear14,267coercive14,41,267continuous14,40,267inf-supstable16,41nonafne167,205parametric52positive267stronglycoercive14symmetric15,267trilinear23,217weaklycoercive16free-formdeformation179full-ordermodelseehigh-delitymodelfunctioncompactsupport273functional266bounded267continuous14,265cost246linear14,266norm267objective246quadratic15,246GGalerkinorthogonality24,46Gram-Schmidtorthonormalization109,143greedyalgorithm8,142,143Hhigh-delitymodel1,41hpreducedbasis109hyper-reduction227Iimage266inequalityG¨arding17,21Korn35Poincare21,276interpolationChebyshev279Lagrange102,194,277Legendre279isogeometricanalysis179isomorphism267KKarhunen-Lo`evedecomposition123kernel266Kolmogorovn-width96,97,149,199LLagrangemultiplier250,257Lagrangiancharacteristicpolynomials277lemmaCea25Lax-Milgram15Strang206liftingfunction20,22,165linearprogram64Mmagicpoints195mapafne265linear265matrixcorrelation124inversegeneralized117Jacobian221,224,233,234Moore-Penrosegeneralizedinverse117,140stiffness25,42transformation6,54,75methodcollocation5,103,194descent249discreteempiricalinterpolation193,203,227empiricalinterpolation193,195,227niteelement33Galerkin24,76convergence25Gauss-Newton225generalizedGalerkin206gradient249least-squares77Newton220,221,225,232Petrov-Galerkin78projection-based6reducedbasis43EIM-G-RB205element179G-RB76,224LS-RB77,224,225nonlinearLS-RB225PG-RB78reducedbasis(RB)EIM-G-RB206stochasticcollocation194stronglyconsistent24successiveconstraint62Index295modelorderreduction1Nnodes34nonafneparametrization167problems167normenergy15,24Frobenius116,117Hilbert-Schmidt271normalequations77nullity266numberPeclet68Reynolds174Oofine/onlinestrategy7operator265adjoint270bounded265compact271continuous265discretesupremizer30niterank271Hilbertspaceadjoint270Hilbert-Schmidt129,271linear265projection79selfadjoint270semilinearelliptic218supremizer28,49,188,233trace276transpose270optimaltestspace28optimalitysystem257,258orthogonalcomplement268Pparameterset39parameterscontrol247geometric39physical39scenario247parametriccomplexity89,104parametrizedPDE1Piolatransformation171pointsequispaced278Gauss279Gauss-Lobatto279polynomialsChebyshev102,278Legendre102,278orthogonal104,278PrincipalComponentAnalysis123principalcomponentanalysis121problemreducedbasis(RB)6abstractvariational14adjoint250,251,259constrainedminimization19dataassimilation245forward2heattransfer3high-delity1,41identication3,245inverse3many-query4minimization15mixedvariational17optimalcontrol3,245optimaldesign3,245parametricoptimization247,248parametrized1,40parametrizedoptimalcontrol247,256PDE-contrainedoptimization245saddlepoint17stronglycoercive14variational15weaklycoercive16projection6projection-basedmethod5projector79oblique81orthogonal81properorthogonaldecomposition8,123basis124gappy204Rradialbasisfunction66,179range266rank266reducedbasiserrorestimator59functions44Galerkin45,76Hermite110Lagrange109least-squares48,77method2Petrov-Galerkin45,78296Indexsolution6,44space44Taylor110reduced-ordermodel2,45reduced-ordermodeling1residual6,58,76Rieszrepresentative30,48,269Rungephenomenon278Ssaddlepoint19sampletest69training69,143samplingfullfactorial135latinhypercube135random135sparsegrid135tensorial135scalarproductdiscrete73seminorm20,276sensitivityequations94,250sequentialquadraticprogramming251shapeoptimization3,246singularvaluedecomposition115singularvalues115singularvectors115snapshots2,5,43,124,143solutionmanifold87map87regular219,228set87spaceBanach267dual267Hilbert268ofdistributions273Sobolev274sparsegrid195stabilityestimate24,41stabilityfactor26,62,65,188,228interpolant65lowerbound62supportcompact273ofafunction273systemapproximation227TtheoremBabuˇska27Banachxed-point230,243Brezzi18,32ImplicitFunction93,216Kantorovich220,231Leray-Schauder217,218Neˇcas16orthogonalprojection268Riesz269Schmidt130Schmidt-Eckart-Young118spectral271trace276transformationPiola158,178Vvertices34