Multigrid Methods for Constrained Minimization Problems and Application to Saddle Point Problems

0.65 MB·

by  Long Chen

MULTIGRID METHODS FOR CONSTRAINED MINIMIZATION PROBLEMS
AND APPLICATION TO SADDLE POINT PROBLEMS
LONGCHEN
6
1
0
ABSTRACT. Thefirstorderconditionoftheconstrainedminimizationproblemleadsto
2
a saddle point problem. A multigrid method using a multiplicative Schwarz smoother
n forsaddlepointproblemscanthusbeinterpretedasasuccessivesubspaceoptimization
a methodbasedonamultileveldecompositionoftheconstraintspace.Convergencetheoryis
J developedforsuccessivesubspaceoptimizationmethodsbasedontwoassumptionsonthe
spacedecomposition:stabledecompositionandstrengthenedCauchy-Schwarzinequality,
5
and successfully applied to the saddle point systems arising from mixed finite element
1
methodsforPoissonandStokesequations. Uniformconvergenceisobtainedwithoutthe

fullregularityassumptionoftheunderlyingpartialdifferentialequations.Asabyproduct,
]
A aV-cyclemultigridmethodfornon-conformingfiniteelementsisdevelopedandprovedto
beuniformconvergentwithevenonesmoothingstep.
N
.
h
t
a
m
1. INTRODUCTION
[
Given a quadratic energy E(v) defined on a Hilbert space , we consider the con-

1 V
strainedminimizationproblem:
v
1
(1) minE(v),
9 v
0 ∈K
4 where is the null space of a linear and bounded operator B defined on . By
K ⊂ V V
0 introducingtheLagrangemultiplierfortheconstraint,wecanfindtheminimizerof(1)by
. solvingasaddlepointsystem.Inthispaper,weshalldesignandanalyzemultigridmethods
1
0 fortheconstrainedminimizationproblem(1)andapplythemtothesaddlepointsystems
6 arisingsfrommixedfiniteelementdiscretizationofPoisson,Darcy,andStokesequations.
1 We shall adapt the constraint decomposition methods developed by Tai for nonlinear
v: variationalinequalities[55]totheconstrainedminimizationproblem. Let = N
K i=1Ki
i beaspacedecomposition.Ourmethodconsistsofsolvingalocalconstrainedminimization
X
(cid:80)
problemineachsubspace whichisequivalenttosolvingasmallsaddlepointproblem.
r Ki
a Thus our relaxation can be interpreted as a multiplicative overlapping Schwarz method
which is known as Vanka smoother [58] in the context of computational fluid dynamics.
Withapropermultileveldecomposition,ourmethodbecomestheclassicalV-cyclemulti-
gridmethod.
Assuming that the decomposition = N satisfies two assumptions: energy
K i=1Ki
stable decomposition (SD) and strengthened Cauchy-Schwarz inequality (SCS), we are
(cid:80)
Date:January19,2016.
2010MathematicsSubjectClassification. 65N55;65F10;65N22;65N30;
Key words and phrases. Constrained optimization, saddle point system, mixed finite elements, multigrid
methods.
LChasbeensupportedbyNSFGrantDMS-1418934.
1
2 L.CHEN
abletoprovetheconvergenceofourmethod
1
E(uk+1) E(u) 1 E(uk) E(u) ,
− ≤ − 1+C C −
(cid:18) A S(cid:19)
(cid:2) (cid:3)
where uk is the k-th iteration, and C and C are positive constants in (SD) and (SCS).
A S
Wealsoextendtheanalysistothecasewherethelocalconstrainedminimizationproblem
isnotsolvedexactlybutonegradientiterationisapplied.
ItisknownthatnumericallymultiplicativeSchwarzsmootherleadstoanefficientmulti-
gridmethodsforsaddlepointproblems[52,53],however,theoreticalanalysisforthecon-
vergenceisonlyavailableforlessefficientadditiveversions[52,53]. Ournewframework
canfillthisgap. Furthermore, theoptimalchoiceoftherelaxationparameterusedinthe
inexactsolversoflocalproblemscanbederivedfromtheminimizationpointofview.
We then apply our method to the saddle point systems arising from mixed finite ele-
ment methods of Poisson, Darcy, and Stokes equations. By verifying assumptions (SD)
and(SCS)formultileveldecompositionsofH(div)elementspaces,wewillprovetheuni-
formconvergenceofaV-cyclemultigridmethodformixedfiniteelementmethodsforthe
PoissonandDarcyequations. OursmootherisrelatedtotheoverlappingSchwarzmethod
developed for H(div) problems in [32, 59, 42, 41, 3]. But our analysis from the energy
minimization point of view is more transparent. We note that a similar stable multilevel
decompositionfortheRaviart-Thomasspacehasbeenproposedin[59]intwodimensions
andin[34,3]inthreedimensions. Ourdecompositionforthreedimensionalcaseisnew
and does not require the duality argument and thus relax the full regularity assumption
neededin[34,3].
WeusetheequivalencebetweenCrouzeix-Raviart(CR)non-conformingmethodsand
mixed methods to develop a V-cycle multigrid method for non-conforming methods of
Poissonequationandproveitsuniformconvergence.Existingconvergenceproofsofmulti-
gridmethodsfornon-conformingmethods[11,48,15,16,49]cannotcoverV-cycleswith
fewsmoothingstepswhileournewframeworkcan. Thetwoingredientsofournewmulti-
grid method for non-conforming methods are: the overlapping Schwarz smoothers, and
inter-gridtransferoperatorsthroughthenestedfluxspaces.
FordiscreteStokesequations,weapplyourtheorytodivergencefreeandnestedfinite
element spaces, e.g., Scott-Voligious elements [54]. Again traditional multigrid conver-
genceproofsforStokesequationsrequiresthefullregularityassumption[60,12,14,8,68,
47]. Using the framework developed in this paper, we can obtain multigrid convergence
withoutthefullregularityassumption. Veryrecently,Brenner,Li,andSung[17]havede-
velopednewmultigridmethodsforStokesequationsandhaveprovedtheuniformconver-
gencewithoutthefullregularityassumption. Theconvergenceresultof[17]is,however,
restricted to W-cycle multigrid methods with sufficient many smoothing steps. Here we
consider V-cycle multigrid with only one smoothing. Furthermore, smoothers developed
in[17]arelessefficientthanVanka-typesmoothersconsideredhere;seenumericalexam-
plesin[52,17]. Ontheotherhand,theframeworkdevelopedin[17]canbeappliedtoany
stablemixedfiniteelementdiscretizationofStokesequationandin[18]suchconvergence
theoryisalsoextendedtotheDarcysystems,whilethecurrenttheorycanbeonlyapplied
to the case when the constrained subspaces are nested. For non-nested constrained sub-
spaces,anadditionalprojectorisneededandananalysisforW-cyclemultigridwithoutthe
fullregularityassumptioncanbefoundin[23]. ForpopularfiniteelementpairsofStokes
equations, a fast multigrid method using least square distributive Gauss-Sedel smoother
hasbeendevelopedin[62]forStokesequationsandgeneralizetoOseenproblemin[25].
MULTIGRIDMETHODSFORSADDLEPOINTPROBLEMS 3
Althoughmostoftheabstracttheory,eitherbasedontheXu-Zikatanovidentity[67]or
followingthe Tai-Xuapproach [56], hasbeen developedin certainformin theliterature,
theapplicationtomultigridmethodsforsolvingsaddlepointsystemsarenewandleadto
severalnewcontributionofthemultigridtheoryforsaddlepointsystems: aconvergence
proofofV-cyclewithevenonesmoothingstep,aconvergenceproofwithoutfullregularity
assumption,andaconvergenceproofforthemultiplicativeSchwarzsmoother. Stablede-
compositionofseveralfiniteelementspacesestablishedinthispaperalsohavetheirown
interest.
Therestofthispaperisstructuredasfollows. InSection2,weintroducethealgorithm.
In Section 3, we give a convergence proof using the X-Z identity and in Section 4, we
present an alternative proof based on the constraint subspace optimization method. We
extend the convergence proof to the inexact local solver in Section 5. In Section 6, 7,
and 8, we apply our method to mixed finite element methods for the Poisson and Darcy
equations, non-conforming finite element methods for the Poisson equation, and mixed
finiteelementmethodsfortheStokesequations,respectively. Inthelastsection,wegive
conclusionandoutlookforfuturework.
2. ALGORITHM
Let be a Hilbert space equipped with inner product (, ) and be a closed
H · · V ⊂ H
subspace and thus is also a Hilbert space. Suppose A : is a symmetric and
V V → V
positivedefinite(SPD)operatorwithrespectto(, ),whichintroducesanewinnerproduct
· ·
(u,v) :=(Au,v)=(u,Av)on .Thenormassociatedto(, )or(, ) willbedenoted
A A
V · · · ·
by or , respectively. Let be another Hilbert space and let B : be
A
(cid:107)·(cid:107) (cid:107)·(cid:107) P V → P
a linear operator. With a slight abuse of notation, we still denote the inner product of
P
by (, ). In most problems of consideration, the inner product (, ) for is the vector
· · · · H
L2-innerproductwhilefor itisthescalarL2-innerproduct. ThetransposeBT :
P P →V
istheadjointofBinthe(, )innerproduct,i.e.,(Bv,q)=(v,BTq)forallv ,q .
· · ∈V ∈P
Foranf ,wedefinetheDirichlet-typeenergy:
∈H
1
(2) E(v)= v 2 (f,v), forv .
2(cid:107) (cid:107)A− ∈V
In this paper we always identify a functional in the dual space as an element in
(cid:48)
H H
throughtheRieszmapinducedby(, ).Denoteby =ker(B)thesubspacesatisfyingthe
· · K
constraintBv =0,i.e.,thenullspaceofB. Weareinterestedinthefollowingconstrained
minimizationproblem:
(3) minE(v).
v
∈K
Sincetheenergyisquadraticandconvex, thereexistsauniquesolutionto(3)andthe
minimizeruof(3)ischaracterizedasthesolutionoftheequation: Findu suchthat
∈K
(4) (Au,v)=(f,v) forallv .
∈K
WeintroducetheoperatorA : as(A u,v) = (Au,v)forallu,v andthe
K K → K K ∈ K
operatorQ : asthe(, )-projection,i.e.,foragivenfunctionf ,Q f
K H → K · · ∈ H K ∈ K
satisfies(Q f,v) = (f,v)forallv . Thentheoperatorformof(4)is: Findu
K ∈ K ∈ K
suchthat
(5) A u=Q f in .
K K K
Asitmightbedifficulttofindbasesforthesubspace ,insteadofsolvingthesymmetric
K
positivedefiniteformulation(5),weshallconsideranequivalentsaddlepointformulation.
4 L.CHEN
Let us introduce the Lagrange multiplier p , equation (4) can be rewritten as the
∈ P
followin4gsaddlepointsystem: Findu ,pL.CHEsNuchthat
∈V ∈P
Let us(iAntruo,dvu)ce+th(ep,LBagvr)an=ge(fm,uvl)tiplier p , equfaotiroanll(4v) can,be rewritten as the
2 P ∈V
following(sBadud,leq)pointsystem:=Fi0ndu ,p suchthfoatrallq ,
2V 2P ∈P
whichwillbewritteni(nAtuh,evo)p+er(apto,Brfvo)rm=(f,v) forallv ,
2V
(Bu,q) =0 forallq ,
A BT u f 2P
(6) = .
whichwillbewrittenintheoBperatOorformp 0
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
Let and be two approApriatBeTnormus for spface and , respectively. It is
wellkn(cid:107)(o6w·)(cid:107)nVthat(6(cid:107))·is(cid:107)wPellposedifanBdonOlyifthpefo=llow0ing.Vso-callPedBrezziconditions
✓ ◆✓ ◆ ✓ ◆
[19]hold:Let and betwoappropriatenormsforspace and ,respectively. Itis
V P
k·k k·k V P
(1)wCelolnktninouwintythoafto(p6)eriastworesllApoasneddBif:anthdeorenleyxiifstthceofnosltlaonwtisncgs,oc-ca>lle0dsBurcehzztihcaotnditions
a b
[19]hold:
(Au,v) c u v , (Bv,q) c v q , forallu,v ,q .
(1)≤Coan(cid:107)tinu(cid:107)iVty(cid:107)of(cid:107)oVperatorsAand≤B:b(cid:107)the(cid:107)reVe(cid:107)xi(cid:107)stPconstantsca,cb >∈0Vsuch∈thPat
(2) CoercivityofAinthekernelspace. Thereexistsaconstantα>0suchthat
(Au,v) c u v , (Bv,q) c v q , forallu,v ,q .
a V V b V P
 k k k k  k k k k 2V 2P
(Au,u) α u 2 forallu ker(B).
(2) CoercivityofAinth≥eke(cid:107)rne(cid:107)lVspace.There∈existsaconstant↵>0suchthat
(3) Inf-supconditionofB(.ATuh,eur)eex↵istusa2confsotraanltlβu >k0ers(uBch).that
� k kV 2
(Bv,p)
(3) Inf-supconditioninoffB.Tshuepreexistsaconstant�β>. 0suchthat
p∈P,p(cid:54)=0v∈V,τ(cid:54)=0(cid:107)v(cid:107)V((cid:107)Bp(cid:107)vP,p)≥
inf sup �.
aChgooiocdeschoofincoersminsc(cid:107)e·B(cid:107)Vmaanydn(cid:107)o·t(cid:107)bPepa2crPoen,npt6=oin0tuuvo2nuViqs,⌧ui6=ne0[k6v9k]Vankndpokr(cid:107)Pm·(cid:107),�Vc.f=. th(cid:107)e·(cid:107)mAixmeadyfonromtbuelaatilownayosf
PoissonCehqouicaetisoonfninorSmesctki·oknV6a.nTdhkr·okuPgahroeuntotthuisnipq(cid:107)au·pe(cid:107)e[Ar6,9w]aenwdikll·kaVss=umke·kthAemwaeylnl-optobseeadlnweasyss
agoodchoicesinceBmaynotbecontinuousin norm,c.f.themixedformulationof
A
of(6)andfocusonitsefficientsolvers. k·k
PoissonequationinSection6. Throughoutthispaper,wewillassumethewell-posedness
Problems (5) and (6) are equivalent theoretically but will lead to different algorithms.
of(6)andfocusonitsefficientsolvers.
In practice, the saddle point formulation will be easier to solve when bases of are not
Problems(5)and(6)areequivalenttheoreticallybutwillleadtodifferentalgorithms.
K
availablIenoprraecxtpiceen,stihveestaoddfolermpo.intformulationwillbeeasiertosolvewhenbasesof arenot
K
Wesahvaalilladbelveeolroepxapnednsaivnealtyozfeormmu.ltigridmethodsforsolvingthesaddlepointsystem(6)
basedonsWubespshaacleldceovrerelocptioanndmaentahlyozdesm[6u5lt]igarniddmitsetahdoadpstafotirosnoltvoinogpttihmeiszaadtdiolenpporionbtlseymstsem[5(66,)
55]. Lebtasedonsubspacecorrectionmethods[65]anditsadaptationtooptimizationproblems[56,
55].Let = + + + , ,i=1,...,N,
1 2 N i
V V V ··· V V ⊂V
= + + + , ,i=1,...,N,
beaspacedecompositionVof Vs1atisfVy2ing·t·h·ecoVnNditVioin⇢V
beaspacedecompositioVnof satisfyingthecondition
= + + V+ , = ker(B),i=1,...,N.
1 2 N i i
K K = K + ···+ K+ K, V=∩ ker(B),i=1,...,N.
1 2 N i i
Fork 0andagiveKnappKroximKated·s·o·lutiKonukK V,o\nestepoftheSuccessiveSubspace
≥Fork 0andagivenapproximatedsolution∈ukK ,onestepoftheSuccessiveSubspace
Optimization�(SSO)method[56]isasfollows: 2K
Optimization(SSO)method[56]isasfollows:
Algorithm:uk+1 =SSO(uk)
v =uk;
0
fori=1:N do
e =argmin E(v +w );
vi =v +wei;2Ki i�1 i
i i 1 i
�
end
uk+1 =v ;
N
Algorithm1:SuccessiveSubspaceOptimizationMethod.
MULTIGRIDMETHODSFORSADDLEPOINTPROBLEMS 5
IfwewritetheEulerequationofthelocalminimizationproblem,itreadsas
(7) (Ae ,φ )=(f Av ,φ ) forallφ .
i i i 1 i i i
− − ∈K
Namely e is the solution of the residual equation restrict to . We can thus treat SSO
i i
K
as the subspace correction method for solving (4) using the space decomposition =
K
N
. Wecananalyzetheconvergencefromthispointofview.
i=1Ki
UsingthefactAu = f in andv = v +e ,equation(7)isalsoequivalenttothe
(cid:48) i i 1 i
(cid:80) K −
A-orthogonality
(8) (u v ,φ ) =0 forallφ ,
i i A i i
− ∈K
whichcanbealsowrittenas
(9) (E(cid:48)(vi),φi)=0 forallφi i.
∈K
Let = B( ). Define A : as for u , Au such that
i i i i i i i i i
P P ∩ V V → V ∈ V ∈ V
(A u ,v ) = (Au ,v )forallv ,andB : asforu ,Bu such
i i i i i i i i i i i i i i
∈ V V → P ∈ V ∈ P
that (B u ,q ) = (Bu ,q ) for all q . Let Q : be the projection in (, )
i i i i i i i i i
∈ P H → V · ·
inner product. The constrained minimization problem in the constraint subspace will
i
K
besolvedbysolvingasmallsaddlepointsystemin :
i
V
A BT e Q (f Av )
(10) i i i = i − i−1 .
B O p 0
i i
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
A typical multilevel decomposition is given as follows. First we construct a macro-
decomposition = J withnestedsubspaces ... = . Usually
V k=1Vk V1 ⊂ V2 ⊂ ⊂ VJ V
theyarebasedonasequenceofsuccessivelyrefinedmeshes. Foreachsubspace ,k =
k
1,...,J, we introdu(cid:80)ce a micro-decomposition = Nk and set =V
Vk i=1Vk,i Kk,i Vk,i ∩
ker(B). Notethattheassumption = J Nk requiresacarefulchoiceofthe
K k=1 i=1K(cid:80)k,i
micro-decompositionof . Roughlyspeaking,eachsubspace shouldbebigenough
k k,i
V (cid:80) (cid:80) V
to contain a basis function of and each basis function of should be contained in at
K K
least one . Similar decomposition is required to design robust multigrid methods for
k,i
V
nearlysingularsystem[38].
Remark 2.1. Solving local saddle problems in k,i sequentially in the k-th level can
V
be interpret as a multiplicative Schwarz smoother which is better known as the Vanka
smoother[58]forNavier-Stokesequations. (cid:3)
Duetothenestednessofthemacro-decomposition,restrictionandprolongationopera-
torsareneededonlyfortwoconsecutivelevels. Insummary,SSObasedonthismultilevel
decompositionleadstoaV-cyclemultigridmethodforthesaddlepointproblem(6)witha
multiplicativeSchwarzsmoother.
Thanks to the assumption , if uk , then uk+1 = SSO(uk) is still in
k,i
K ⊂ K ∈ K
. Namely the iteration remains in the constrained subspace. Uzawa method [57], an-
K
otherpopulariterativemethodforsolvingthesaddlepointproblem,willnotpreservethe
constraintandthusisnotconsideredhere.
WeshalluseeithertheunconstrainedSPDformulation(4)and(7)orconstrainedsaddle
pointformulation(6)and(10). Theyareequivalentformsfortheconvergenceanalysisbut
differentalgorithmically.
Weendthissectionwithadiscussionofthenon-homogenousconstraint,i.e.,thesaddle
pointproblem
A BT u f
(11) = .
B O p g
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
6 L.CHEN
To change to the form (6), we can first find a u satisfying Bu = g and let u =
∗ ∈ V ∗
u +δu. Thentheequationforδuisintheform(6).
∗
Thereareseveralwaystofindsuchu . Onewayistosolve
∗
I BT u 0
(12) ∗ = ,
B O p g
(cid:18) (cid:19)(cid:18) ∗(cid:19) (cid:18) (cid:19)
whichissupposedtobeeasierthansolving(11). ForStokesequations,solving(12)essen-
tiallyrequiresaPoissonsolverforpressureforwhichfastsolversareavailable. ForDarcy
equations,Aisaweightedmassmatrixwithpossiblyhighlyoscillatorycoefficients,while
(12)isagainjustaPoissonoperator.
Whenthespace consistsofdiscontinuouselements,whichisthecaseofmostappli-
P
cationsconsideredinthispaper,wecanfindsuchu byoneV-cyclewithpost-smoothing
∗
only;seeSection6fordetails.
3. CONVERGENCEANALYSISBASEDONTHEXZIDENTITY
Inthissection,weprovideaconvergenceanalysisusingtheSPDformulation(4)and(7).
TheanalysisisbasedontheXZidentity[67]forthemultiplicativeiterativemethodsand
canbefoundin[66].
DenotedbyP theA-orthogonalprojectiononto fori = 1,...,N. Thentheerror
i i
K
operator of SSO can be written as (I P )(I P ) (I P ), i.e., u uk+1 =
N N 1 1
N (I P )(u uk),whereuk+1 =−SSO(uk−). Th−efo·ll·o·win−gXZidentity−wasestab-
i=1 − i −
lishedin[67]
(cid:81)
N
2 1
(13) (I P ) =1 ,
i
− A − 1+c0
(cid:13)i(cid:89)=1 (cid:13)
(cid:13) (cid:13)
where (cid:13) (cid:13)
N J
2
c = sup inf P v .
0 i j
(cid:107)v(cid:107)A=1(cid:80)Ji=1vi=v,vi∈Ki(cid:88)i=1(cid:13) j=(cid:88)i+1 (cid:13)A
Foranelementaryproofof(13),werefertoChen[22(cid:13)]. (cid:13)
(cid:13) (cid:13)
Inordertoestimatetheconstantc ,weproposetwoimportantpropertiesofthespace
0
decomposition.
Stabledecomposition(SD):foreveryv ,thereexistsvi i,i=1,...,N suchthat
∈K ∈K
N N
v = v , and v 2 C v 2.
i (cid:107) i(cid:107)A ≤ A(cid:107) (cid:107)A
i=1 i=1
(cid:88) (cid:88)
StrengthenedCauchySchwarzinequality(SCS):foranyui iandvj j
∈K ∈K
1/2 1/2
N N N N
(u ,v ) C1/2 u 2 v 2 .
i j A ≤ S (cid:32) (cid:107) i(cid:107)A(cid:33)  (cid:107) j(cid:107)A
i=1j=i+1 i=1 j=1
(cid:88) (cid:88) (cid:88) (cid:88)
Withassumptions(SD)and(SCS),weshallprovideaupperboundofcandthusobtaina
0
convergenceproofofSSOmethodforsolvingthesaddlepointproblem(6).
Theorem 3.1. Assume that the space decomposition K = Ni=1Ki satisfy assumptions
(SD)and(SCS).ForSSOmethod,wehave
(cid:80)
N
2 1
(I P ) 1 .
i
− A ≤ − 1+CACS
(cid:13)i(cid:89)=1 (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
MULTIGRIDMETHODSFORSADDLEPOINTPROBLEMS 7
Proof. Weapply(SCS)withu =P N v toobtain
i i j=i+1 j
N N (cid:80) N N N
u 2 = (u ,P v ) = (u ,v )
(cid:107) i(cid:107)A i i j A i j A
i=1 i=1 j=i+1 i=1j=i+1
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
1/2 1/2
N N
C1/2 u 2 v 2 ,
≤ S (cid:32) (cid:107) i(cid:107)A(cid:33) (cid:32) (cid:107) i(cid:107)A(cid:33)
i=1 i=1
(cid:88) (cid:88)
whichleadstotheinequality
N N
(14) u 2 C v 2.
(cid:107) i(cid:107)A ≤ S (cid:107) i(cid:107)A
i=1 i=1
(cid:88) (cid:88)
Consequently,wechoosev = N v asastabledecompositionsatisfying(SD)toget
i=1 i
N N N N
2 (cid:80)
P v = u 2 C v 2 C C v 2,
i j A (cid:107) i(cid:107)A ≤ S (cid:107) i(cid:107)A ≤ S A(cid:107) (cid:107)A
(cid:88)i=1(cid:13) j=(cid:88)i+1 (cid:13) (cid:88)i=1 (cid:88)i=1
whichimpliesc0(cid:13)(cid:13) CSCA.T(cid:13)(cid:13)hedesiredresultthenfollowsfromtheX-Zidentity(13). (cid:3)
≤
The assumption (SCS) is relatively easy to verify. The key is to construct a stable
decompositionoftheconstraintspace .
K
4. CONVERGENCEANALYSISBASEDONCONSTRAINEDOPTIMIZATION
In this section we provide an alternative proof using the constraint optimization ap-
proachestablishedbyTai[55].Italsoprovidesabetterapproachtoextendtheconvergence
prooftoinexactand/ornonlinearlocalsolvers.
Wewillalwaysdenotebyutheglobalminimizerof(3). Givenaninitialguessu0 ,
∈K
let uk be the kth iteration in SSO algorithm for k = 1,2, . We aim to prove a linear
···
reductionoftheenergydifference
(15) E(uk+1) E(u) ρ E(uk) E(u) ,
− ≤ −
with a contraction factor ρ (0,1). Ideally ρ(cid:2)is independent o(cid:3)f the size of the problem.
∈
Theproofisdevelopedin[56,55]for anonlinearandconvexenergybutsimplifiedhere
forthequadraticenergy.
WefirstexploretherelationbetweentheenergyandtheA-normoftheerror.
Lemma4.1. Foranyw,v ,wehave
∈V
1
(16) E(w)−E(v)= 2(cid:107)w−v(cid:107)2A+(E(cid:48)(v),w−v).
Consequentlyfortheminimizeru andanyw ,
∈K ∈K
1
(17) E(w) E(u)= w u 2.
− 2(cid:107) − (cid:107)A
Proof. Verificationof(16)and(17)isstraightforward. (cid:3)
Basedontheidentity(17),thetargetinequality(15)becomesamorefamiliarone
(18) uk+1 u ρ1/2 uk u .
A A
(cid:107) − (cid:107) ≤ (cid:107) − (cid:107)
Letd = E(uk) E(u)andδ = E(uk) E(uk+1).Thequantityd isthedistance
k k k
− −
of the current energy to the lowest one, δ is the amount of the energy decreased in one
k
iteration, and they are connected by the identity δ = d d . By Lemma 4.1, we
k k k+1
−
8 L.CHEN
haved = 1 uk u 2 butingeneralδ = 1 uk uk+1 2 sinceuk+1 maynotbethe
k 2(cid:107) − (cid:107)A k (cid:54) 2(cid:107) − (cid:107)A
minimizer. Foreachv ,i=1,...,N,inSSO,wedohave
i
1
E(v ) E(v )= v v 2,
i−1 − i 2(cid:107) i−1− i(cid:107)A
since v is the local minimizer and v v = e ; see also the orthogonal-
i i 1 i i i
− − − ∈ K
ity (9). Borrowing the terminology of the convergence theory of adaptive finite element
methods[46],weshallpresentourproofbasedonthefollowingtwoinequalities.
DiscreteLowerBound. ThereexistsapositiveconstantCLsuchthatfork =0,1,2,...
N
δ C e 2.
k ≥ L (cid:107) i(cid:107)A
i=1
(cid:88)
UpperBound. ThereexistsapositiveconstantCU suchthatfork =0,1,2,...
N
d C e 2.
k+1 ≤ U (cid:107) i(cid:107)A
i=1
(cid:88)
Theorem4.2. Assumethatthediscretelowerboundandupperboundholdwithconstants
C andC respectively. Wethenhave
L U
c
0
d d ,
k+1 k
≤ 1+c
0
wherec =C /C .
0 U L
Proof. The proof is straightforward by assumptions and rearrangement of the following
inequality
N
d C e 2 C /C δ =c (d d ).
k+1 ≤ U (cid:107) i(cid:107)A ≤ U L k 0 k− k+1
i=1
(cid:88)
(cid:3)
Verifying the lower bound is relatively easy since E is convex. Indeed we have the
followingidentitywhichcharacterizesexactlytheamountofenergydecreasedinonestep
of SSO. Again in the sequel, uk+1 = SSO(uk) and e is the ith correction in , for
i i
K
i=1,...,N.
Theorem4.3.
N
1
E(uk) E(uk+1)= e 2.
− 2 (cid:107) i(cid:107)A
i=1
(cid:88)
Proof. Bytheidentity(16)andtheorthogonality(9),wehave,fori=1,...,N,
1 1
E(v ) E(v )= v v 2 = e 2,
i−1 − i 2(cid:107) i−1− i(cid:107)A 2(cid:107) i(cid:107)A
andconsequently
N N
1
E(uk) E(uk+1)= [E(v ) E(v )]= e 2.
− i−1 − i 2 (cid:107) i(cid:107)A
i=1 i=1
(cid:88) (cid:88)
(cid:3)
Proving the upper bound is more delicate. We first present a lemma which can be
verifieddirectlybydefinitionandLemma4.1.
MULTIGRIDMETHODSFORSADDLEPOINTPROBLEMS 9
Lemma4.4.
(19) (E(cid:48)(uk+1)−E(cid:48)(u),uk+1−u)=(cid:107)uk+1−u(cid:107)2A =2 E(uk+1)−E(u) .
Wethengiveamultileveldecompositionoftheleft-handside(cid:2)of(19). (cid:3)
Lemma4.5. Foranydecompositionuk+1−u= Ni=1wi,wi ∈Ki,i=1,2,...,N,
(cid:80) N N
(E (uk+1) E (u),uk+1 u)= (e ,w ) .
(cid:48) (cid:48) j i A
− −
i=1 j>i
(cid:88)(cid:88)
Proof.
(E (uk+1) E (u),uk+1 u)=(E (uk+1),uk+1 u)
(cid:48) (cid:48) (cid:48)
− − −
N N N N N
= (E (uk+1) E (v ),w )= (E (v ) E (v ),w )= (e ,w ) .
(cid:48) (cid:48) i i (cid:48) j (cid:48) j 1 i j i A
− − −
i=1 i=1 j>i i=1 j>i
(cid:88) (cid:88)(cid:88) (cid:88)(cid:88)
Inthefirststep,weusethefactE (u)=0in sinceuistheminimizeranduk+1 u .
(cid:48) (cid:48)
K − ∈K
InthesecondstepweuseE (v ) = 0in sincev istheminimizerin andw ;
seealso(8). (cid:48) i Ki(cid:48) i Ki i ∈ K(cid:3)i
Lemma4.6. Assumethatthespacedecompositionsatisfiesassumptions(SD)and(SCS).
Thenwehavetheupperbound
N
1
E(uk+1) E(u) C C e 2.
− ≤ 2 S A (cid:107) i(cid:107)A
i=1
(cid:88)
Proof. We shall chose a stable decomposition for uk+1 u = N w ,w ,i =
− i=1 i i ∈ Ki
1,2,...,N. ByLemma4.5and(SCS),wehave
(cid:80)
N N
(E (uk+1) E (u),uk+1 u)= (e ,w )
(cid:48) (cid:48) j i A
− −
i=1 j>i
(cid:88)(cid:88)
1/2 1/2
N N
C1/2 e 2 w 2
≤ S  (cid:107) j(cid:107)A (cid:32) (cid:107) i(cid:107)A(cid:33)
j=1 i=1
(cid:88) (cid:88)
  1/2
N
(C C )1/2 e 2 uk+1 u .
≤ S A (cid:32) (cid:107) i(cid:107)A(cid:33) (cid:107) − (cid:107)A
i=1
(cid:88)
Substitutingtheidentity(seeLemma4.4)
uk+1 u 2 =(E (uk+1) E (u),uk+1 u)
(cid:107) − (cid:107)A (cid:48) − (cid:48) −
intotheaboveinequalityandcancelingone uk+1 u ,wecanobtain
A
(cid:107) − (cid:107)
N
uk+1 u 2 C C e 2.
(cid:107) − (cid:107)A ≤ S A (cid:107) i(cid:107)A
i=1
(cid:88)
UsingtheidentityE(uk+1) E(u)= uk+1 u 2/2,weobtainthedesiredresult. (cid:3)
− (cid:107) − (cid:107)A
Wesummarizeourconvergenceresultintothefollowingtheorem.
10 L.CHEN
Theorem4.7. AssumethatthespacedecompositionK = Ni=1Ki satisfiesassumptions
(SD)and(SCS).Then
(cid:80)
1
E(uk+1) E(u) 1 E(uk) E(u) .
− ≤ − 1+C C −
(cid:18) A S(cid:19)
(cid:2) (cid:3)
Remark 4.8. The estimate is consistent with the one obtained by the XZ identity which
indicatesthatourenergyestimateissharp.
5. CONVERGENCEANALYSISWITHINEXACTLOCALSOLVERS
InthealgorithmSSO,weassumethatthelocalproblemissolvedexactlywhichmaybe
costlywhenthedimensionofthelocalspaceislarge. Inthissection,weconsiderinexact
solvers using one gradient iteration and establish the corresponding convergence proof.
NotethatXZidentitycannotbeappliedtothenonlinearsolversconsideredhere.
Recallthatthelocalconstrainedminimizationproblemis:letr =Q (f Av ),find
i i i 1
− −
e suchthat
∗i ∈Ki
A BT e r
(20) i i ∗i = i .
B O p 0
(cid:18) i (cid:19)(cid:18) ∗i(cid:19) (cid:18) (cid:19)
Hereweusee ,p todenotethesolutionobtainedbytheexactsolver.Intheinexactsolver
∗i ∗i
proposedbelow,theconstraintisstillsatisfiedbutoperatorA isreplacedbyasimplerone
i
D ,e.g.,thediagonalofA . Ingeneral,letD beanSPDoperatoron ,wefirstsolvethe
i i i i
V
localproblem
D BT s r
(21) i i i = i .
B O p 0
i i
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
Thenweapplythelinesearchalongthedirections tofindanoptimalscaling:
i
(22) minE(v αs ),
i 1 i
α R − −
∈
whosesolutionis
(r ,s )
i i
(23) α= .
(As ,s )
i i
Weupdate
v =v αs .
i i 1 i
− −
ThisisonestepofapreconditionedgradientmethodandD isapreconditionerofA .
i i
Inthissection,wewillalwaysdenotebye thesolutionof(20)ande = αs withs
∗i i i i
beingthesolutionof(21)andαgivingby(23). Withsuchchoiceofα, westillhavethe
firstordercondition
(24) (E(cid:48)(vi),ei)=0.
Remark5.1. IntheoriginalVankasmootherforNavier-Stokesequation,Di =ωdiag(Ai)
withasuitableparameterω (0.5,0.8)[58]andnolinesearchisapplied,i.e.,α=1. (cid:3)
∈
Usingthefirstordercondition(24),westillhavethefollowingidentity.
Lemma5.2.
N
1
E(uk) E(uk+1)= e 2.
− 2 (cid:107) i(cid:107)A
i=1
(cid:88)