Multiple shooting and mesh adaptation for PDE constrained optimization problems [Elektronische Ressource] / vorgelegt von Helke Karen Hesse

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Inaugural-DissertationzurErlangung der DoktorwürdederNaturwissenschaftlich-Mathematischen GesamtfakultätderRuprecht-Karls-UniversitätHeidelbergvorgelegt vonDiplom-Mathematikerin Helke Karen Hesseaus OberhausenTag der mündlichen Prüfung: 27. Juni 2008Multiple Shooting and Mesh Adaptationfor PDE Constrained Optimization ProblemsGutachter: Prof. Dr. Rolf RannacherProf. Dr. Dr. h.c. Hans Georg BockAbstractIn this thesis, multiple shooting methods for optimization problems constrained by partial differentialequations are developed, and, furthermore, a posteriori error estimates and local mesh refinementtechniques for these problems are derived. Two different approaches, referred to as the direct and theindirect multiple shooting approach, are developed. While the first approach applies multiple shootingto the constraining equation and sets up the optimality system afterwards, in the latter approachmultiple shooting is applied to the optimality system of the optimization problem. The setup of bothmultiple shooting methods in a function space setting and their discrete analogs are discussed, anddifferent solution and preconditioning techniques are investigated. Furthermore, error representationformulas based on Galerkin orthogonality are derived. They involve sensitivity analysis by means of anadjoint problem and employ standard error representation on subintervals combined with additionalprojection errors at the shooting nodes.

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Inaugural-Dissertation

zur

DoktorwürdederErlangung

der

Naturwissenschaftlich-MathematischenGesamtfakultät

der

ersitätt-Karls-UnivhuprecR

ergHeidelb

vorgelegt

erinDiplom-Mathematik

agT

der

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Obauserhausen

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Prüfung:

Karen

27.

Hesse

Juni

2008

Multiple

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PDE

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Constrained

r:tehGutac

Prof.

Prof.

Dr.

Dr.

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Mesh

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Optimization

Rolf

Dr.

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h.

c.

Hans

Georg

Problems

Bo

ck

Abstract

Inthisequationsthesis,aremdevultipleeloped,shoand,otingfurmethodsthermore,forapoptimizationosteriorierrorproblemsestimatesconstrainedandlobcalypartialmeshdifferenrefinementialt
tecindirecthniquesmforultipletheseshootingproblemsapproacareh,derivareed.devTelwopoed.differenWhiletaptheproacfirsthes,approacreferredhtoappliesasmtheultipledirectshoandotintheg
totheconstrainingequationandsetsuptheoptimalitysystemafterwards,inthelatterapproach
mmultipleultiplesshohootingotingismethoapplieddsintoathefunctionoptimalitspaceysystemsettingofandthetheiroptimizdiscreteationproblem.analogsareThesetupdiscussed,ofbandoth
formdifferenulastbasedsolutiononandGalerkinpreconditionorthogonalitingytechnareiquesderived.areinvTheyinestigatedvolve.Fsensitiviturthermore,yanalyserrorisbyrepresenmeansoftationan
proadjointjectionproblemerrorsatandtheemploshoyotingnostandarddes.Aerrorposterrepioriresenerrortationonestimatessubinandtervalsmeshcomrefinemebinedntwithindicatorsadditionalare
derivedfromthiserrorrepresentation.Severalmeshstructuresoriginatingfromdifferentrestrictions
topresenlocalted.refinemenThistmoaredeldescribdiscussed.esanFinalexplosly,niveumericalsystemresultsthatdoforesthenotsolidallowstatethefuelsolutionignitionbymodelstandarared
domainsolutiontecdecomphniquesositiononthemethodswholeliketimemultipldomaienshoandoting.isatypicalexamplefortheapplicationoftime

Zusammenfassung

IndieserDoktorarbeitwerdenMultipleShootingVerfahrenfürdurchpartielleDifferentialgleichungen
beschränkteOptimierungsproblemeentwickeltundzusätzlichaposterioriFehlerschätzerundMetho-
denzurlokalenGitterverfeinerungfürdieseProblemeausgearbeitet.Eswerdenzweiunterschiedliche
Ansätze,welchealsdirekterundindirekterAnsatzeinesMultipleShootingVerfahrensbezeichnet
werden,betrachtet.WährendderersteAnsatzdasMultipleShootingVerfahrensfürdiebeschrän-
kendeDifferentialgleichungansetztundanschließenddasOptimalitätssystemaufstellt,wendetder
letzteredasMultipleShootingVerfahrenaufdasOptimalitätssysteman.DieDarstellungbeider
AnsätzeimFunktionenraumunddiediskretenEntsprechungenwerdendiskutiert,undverschiede-
neLösungs-undVorkonditionierungstechnikenwerdenuntersucht.Desweiterenwerdenbasierend
aufEigenschaftenderGalerkinorthogonalitätFehlerdarstellungenhergeleitet.Diesebeinhalteneine
SensitivitätsanalyseanhandvonadjungiertenProblemenundverwendenFehlerdarstellungenauf
TeilintervallenzusammenmitzusätzlichenProjektionsfehlernandenZeitknotendesMultipleShooting
Verfahrens.AusgehendvondieserDarstellungwerdenaposterioriFehlerschätzerundIndikatoren
fürdieGitterverfeinerunghergeleitet.VerschiedeneGitterstrukturen,welcheausunterschiedlichen
RestriktionenandielokaleVerfeinerungresultieren,werdendiskutiert.Abschließendwerdennumeri-
scheErgebnissefüreinModell,welchesdieZündungsphaseeinesFestkörperbrennstoffesbeschreibt,
angegeben.DiesesModellbeschreibteinexplosivesSystems,dasdieLösungmitStandardverfahren
aufdemgesamtenZeitgebietnichtzulässt,unddasdahereintypischesBeispielfürdieAnwendung
vonZeitgebietszerlegungsmethoden,wiezumBeispielMultipleShootingVerfahren,darstellt.

Contents

Intro1duction

1

2FormulationandTheoryofPDEConstrainedOptimizationProblems9
2.1Preliminaries....................................9
2.2FormulationofAbstractParabolicOptimizationProblems...........10
2.3ExistenceandUniquenessofSolutions......................14
2.4OptimalityConditions...............................16

3HistoricalBackgroundoftheMultipleShootingApproach19
3.1TheSingleShootingApproachforODEBoundaryValueProblems......19
3.2TheDirectMultipleShootingApproachforODEBoundaryValueProblems.21
3.3CondensingTechniques..............................22
3.4DerivativeGeneration...............................24
3.5TheMultipleShootingApproachforODEConstrainedOptimizationProblems25

4TheMultipleShootingApproachforPDEConstrainedOptimization27
4.1FromODEstoPDEs–DifferencesandChallenges...............27
4.2TheIndirectMultipleShootingApproach....................28
4.3TheDirectMultipleShootingApproach.....................33

5Space-TimeFiniteElementDiscretization45
5.1TimeDiscretization................................45
5.2SpaceDiscretization................................48
5.3DiscretizationofTimeandSpace........................51
5.3.1DiscretizationoftheMultipleShootingVariables............51
5.3.2DynamicallyChangingSpatialMeshes..................51
5.3.3IntervalwiseConstantSpatialMeshes..................54
5.4DiscretizationoftheControls...........................54
5.5TheImplicitEulerTimeSteppingScheme....................57

6SolutionTechniquesfortheMultipleShootingApproach59
6.16.1.1SolutionTSolutionechniquofesthefortheMultipleIndirecShototingMultipleSystemSho.oting.....Approac..h...............6060
6.1.36.1.2TheSolutionGMRESoftheInMethotervdalforProbthelemsSolution–ofNewton’sthemethLinearizedod...System..........6469
6.1.4SolutionoftheLinearProblems–FixedPointIterationandGradient
Method...................................71
6.1.5ApplicabilityofNewton’sMethodfortheIntervalProblems......74

i

tstenCon

6.26.1.6SolutionSoluTectihonniquesoftheforInthetervalDirectProblemsMultiple–TheShootingReducedApproacApproach.h.............8477
6.2.1SolutionoftheMultipleShootingSystem................85
6.2.36.2.2TheCondenGMRESsingTecMethohniquesdforforthetheSolutionSolutionofofthetheLinearizedLinearizedSystemSystem.......8993
6.2.4FromODEstoPDEs–Limitations....................95
6.3NumericalComparisonoftheDirectandIndirectMultipleShootingApproach96

7APosterioriErrorEstimation101
7.1TheClassicalErrorEstimatorfortheCostFunctional.............101
7.2APosterioriErrorEstimationfortheMultipleShootingSystem........107
7.3EvaluationoftheErrorEstimators........................112
7.4NumericalExamples................................114

8MultipleShootingandMeshAdaptation119
8.1MeshAdaptationbytheClassicalDWRErrorEstimator...........119
8.1.1LocalizationoftheErrorEstimator...................119
8.1.2TheProcessofMeshAdaptation.....................121
8.2MeshAdaptationbytheErrorEstimatorfortheMultipleShootingSystem.122
8.2.1LocalizationoftheErrorEstimator...................122
8.2.2TheProcessofMeshAdaptation.....................123
8.3NumericalExamples................................123

9ApplicationtotheSolidFuelIgnitionModel131
9.1TheSolidFuelIgnitionModel...........................131
9.2TheoreticalBackground..............................133
9.3OptimalControloftheSolidFuelIgnitionModel................135

andConclusion10okOutlo

wledgmentscknoA

Bibliography

ii

143

145

147

ductionIntro1

Inthisthesis,wedevelopandinvestigatemultipleshootingmethodsforoptimalcontrol
problemsconstrainedbyparabolicpartialdifferentialequations.Furthermore,wecombine
thesemultipleshootingmethodswithaposteriorierrorestimationtechniquesandadaptive
cedures.proefinementrmeshSystemsofpartialdifferentialequations(PDEs)playanimportantroleasmodelsfordynamic
processes,forexampleinphysics,chemistry,biology,orengineering.Optimizationproblems
occurasparameterestimationproblemsinthecontextofquantitativemodelingorasoptimal
controloroptimaldesignproblemswhereaprocesshastobeconstructedoroperatedto
es.jectivobcertainmeetReactors,builttostudythedetailsofchemicalreactions,mustprovidestableandpredictable
environments(pressure,temperature,mixtureofspecies)inordertoavoidspuriousob-
servations.Therefore,thereactormustbecontrolledtomaintaintheseenvironments.In
dynamicalprocesses,thisshouldbeachievedbyoptimalcontrol,whichcanbeinterpretedas
aconstrainedoptimizationproblem.Inthiscase,theconstraintsconsistofaPDEinitial
boundaryvalueproblemandfurthertechnicalrestrictions.Thus,atypicalexamplefor
optimalcontrolproblemsconstrainedbyPDEswithpathandcontrolconstraintsisthe
cost-minimaloperationofacatalytictubereactorundertemperaturerestrictions[36]orthe
controlofflowconditionsformeasurementsinahigh-temperatureflowreactor[19].Further
examplesofPDEconstrainedoptimizationproblemsareproblemsofcatalyticreactions,
forexamplethecatalyticpartialoxidationofmethaneintubularreactorsorthecatalytic
conversionofexhaustgasinpassengercarsorahigh-temperatureflowreactorwhichhas
extensivelybeenresearchedin[19].
PossibleapproachestothesolutionofPDEconstrainedoptimizationproblemsaregivenby
theclassofshootingmethods.Originallydevelopedforthesolutionofboundaryvalueproblems
(BVPs)inordinarydifferentialequations(ODEs),theseapproachesobtaintheirdenomination
fromthetypicalsolutionprocess:Foraguessedinitialvalue,theapproximationoftheterminal
timevalueisnumericallycalculated,andtheapproximationoftheinitialconditionisimproved
byaniterativeprocedure.Metaphoricallyspeaking,givenanapproximationoftheinitial
value,weshootontotheterminaltimevalueandseektomatchtheprescribedvalueatthis
t.oinptimeIngeneral,wedifferentiatebetweensingleshootingandmultipleshootingmethods,though
singleshootingmerelydisplaysthespecialcaseofmultipleshootingforonetimeintervalas
wewillseelateron.
Multipleshootingmethodshaveproventobethestate-of-the-artforoptimizationproblems
inthecontextofordinarydifferentialordifferentialalgebraicequationsystems.Multiple

1

1ductiontroIn

shootingmethodssimultaneouslysolvetheconstraints(thesimulationorforwardproblem)
andtheoptimizationproblemthroughglobalizedtailoredinfeasibleNewton-likemethods.
Theytypicallyusetime-adaptivestrategiesforthediscretizationofthedifferentialequation
constraintsandtailoreddecompositionmethodsforthesolutionofthestructuredquadratic
problemsineveryiteration([9,27]).
Multipleshootingmethodspossessseveraladvantages.First,multipleshootingmethods
arestableandcanbeappliedforthesolutionofhighlyinstableproblems.Second,thetime
domaindecompositionallowstheintroductionofknowledgeabouttheprocessatalltimepoints
bychoosingadequateinitialguessesforthestates.Furthermore,multipleshootingmethods
allowtheparallelsolutionofthesubproblemsonthedifferenttimesubintervals.
Themultipleshootingmethodasatimedomaindecompositionschemegoesbacktothe
solutionoftwopointboundaryvalueproblemsforordinarydifferentialequationswhichare
ofinterestnotonlyinthecontextofoptimizationproblems,butareoftenencounteredin
physicsandengineering.Startingfromthesingleshootingmethod,earlydevelopmentsinto
thedirectionofmultipleshootingfortwopointboundaryvalueproblemscanbefoundinthe
publicationofMorrison,Riley,andZancanaro[31],thearticleofHolt[23],andthearticleof
Keller[26].AgoodoverviewofthemultipleshootingapproachforODEtwopointboundary
valueproblemsisgiveninthetextbookofStoer[39],wherefurtherextensionstoODE
constrainedoptimizationthroughtheindirectapproachareshortlyintroduced.Thismatter
isalsodiscussedinthereportofBulirsch[13].Theadvantagesofthedirectapproachare
outlinedinthediplomathesisofPlitt[33],inwhichfirstapproachestothesoftwarepackage
MUSCODareimplementedanddiscussed,andfurtherinthearticleofBockandPlitt[11]
andinthethesisofBock[9].Overtheyearsavarietyofdifferenttechniquesforcertain
concreteODEconstrainedproblemshasbeenderivedfromtheoriginalideasofBock,Holt,
Keller,Plitt,andothersandhasbeenappliedforthesolutionofthoseapplicationproblems
mentionedabove.InthiscontextthesolutionofPDEconstrainedoptimizationproblemsby
multipleshootingisbroughtdowntotheODEapproachbyspatialdiscretizationwiththe
methodoflines([36]).Forthisreasontheapproachislimitedtocoarsespatialdiscretizations,
andspatialmeshadaptationisnotpossible.
TheideatoextendthemultipleshootingapproachforODEconstrainedoptimizationto
optimizationproblemswhichareconstrainedbyparabolicPDEsisarathernewtopicof
research.FirstapproacheswerederivedbySerban,Li,andPetzoldwhomadefirstadvances
intothedirectionofadaptivemeshrefinementincombinationwithmultipleshooting.This
approachisassociatedwiththestructuredadaptivemeshrefinementmethod(SAMR)and
wasfirstdiscussedin[37].Adirectmultipleshootingapproachforlinearquadraticoptimal
controlproblemswasdevelopedbyUlbrichin[42]andHeinkenschlossin[20],andfurther
extensionswerepresentedbyComasinherdoctoralthesis[14].Alltheseapproachesare
limitedtolinearquadraticoptimalcontrolproblemsandaremainlymotivatedbythepossible
parallelizationoftheintervalwiseproblemsandbythereducedstoragerequirements.The
efficientparallelizationisdifficultduetoalackofappropriateparallelizablepreconditioners.
Thereducedstoragerequirementsdonotholdforthemultipleshootingmethodincombination
withadaptivemeshrefinementobtainedbythedualweightedresidualmethod(DWRmethod).
Thisrestrictionfollowsfromtherequiredstorageofprimalanddualvariableoverthewhole
timeintervalfortheevaluationoftheaposteriorierrorestimator.

2

Nevertheless,multipleshootingisofcrucialimportanceforthesolutionofhighlyinstable
constrainedoptimizationproblemsinwhichtheconstrainingdifferentialequationcannot
besolvedforslightlydisturbedcontrolparameters.AtypicalexampleofaninstableODE
constrainedoptimizationproblemisgivenbyExample1.1below.Wewanttodetermine
atimedistributedcontrolq:I→Randastatefunctionu:I→Rsuchthatthecost
functional(1.1a)isminimizedwhileufulfillstheordinaryfirstorderdifferentialequation
(1.1b).Inthisoptimizationproblemwesearchthebestpossibleapproximationofu¯:I→R
limitedbyaregularizationtermpenalizingthecontrolcosts.
1.1.ExampleTTqmin,uJ(q,u):=2α|q(t)|2dt+21|u(t)−u¯(t)|2dt(1.1a)
00thathsucu−λeu=qonI=(0,T),
u(0)=u0.(1.1b)
Foranappropriatechoiceofu¯,thesolutionuoftheconstrainingequationexistsandis
boundedfortheoptimalcontrolq.Butforthestandardinitialcontrolq=0thesolutionof
theconstrainingdifferentialequationblowsup:Withtheparameterλ=5andtheinitial
valueu0=0wesearchtocalculatethesolutionforq≡0.Thesolutionhasablowupat
aboutt=0.2asshowninFigure1.1,andthenumericalintegrationoftheequationonthe
wholetimeintervalisthusimpossible.
3530252015105000.050.10.150.20.25
Figure1.1:Behaviorofu(t)fordifferenttimest∈[0,0.25]forExample1.1withλ=5and
u0=0.(x-axis:t,y-axis:u(t))
Multipleshootingasatimedomaindecompositionmethodontheotherhandsplitsthe
timeintervalintosmallsubintervals.Thisenablesustocalculateintervalwisetrajectories

3

ductiontroIn1

)(1.2a

andtherebytoimprovetheintervalwiseapproximationofthecontrolsuccessivelyinthe
cess.prooptimizationAnanalogousexamplecanbestatedforPDEconstrainedoptimalcontrolproblems.We
considerthesolidfuelignitionmodelwhichinthecontextofoptimizationhasbeeninvestigated
in[25]or[24].Thisproblemisapowerfulexamplefordemonstratingthepropertiesofexplosive
systems,andacomprehensivetheoreticalframeworkdiscussingtheexistenceofsolutions
isavailableintheliterature.Wewilldescribethesetheoreticalaspectsandpropertiesof
theproblemlateronwhenconsideringanumericalapplication.Fornow,itissufficient
topresentthepureproblemformulationinExample1.2andtopointoutthemotivation
forapplyingmultipleshootingtechniquestothisproblem.SimilartotheODEexample,
wewanttodetermineacontrolq:I→L2(Ω)asasourcetermontheright-handsideof
theconstrainingequation(1.2b)suchthatthestateu:I→L2(Ω)fulfillstheconstraining
equationandapproximatesthegivenstateu¯:I→L2(Ω)inthecostfunctional(1.2a)best
ossible.pExample1.2.(Thesolidfuelignitionmodel)
αT1T
qmin,uJ(q,u):=20q(t)2dt+20u(t)−u¯(t)2dt(1.2a)
subjecttotheconstrainingequation
∂tu−Δu−λeu=qinI×Ω,
u(0)=u0inΩ,(1.2b)
u(t,∙)=0onI×∂Ω.
ForthetimeintervalI=(0,1),thespatialdomainΩ=(−1,1)×(−1,1),thecontrol
q≡0,theinitialconditionu0=0,andtheparameterλ=7.5,thesolutionblowsupat
approximatelyt=0.14.Thesolutionona6timesgloballyrefinedmeshfordifferenttime
pointsisshowninFigure1.2.Incontrast,thesolutionoftheoptimalcontrolproblem(we
considerthesimplestcaseofu¯≡0inthefollowing)isboundedintime.Byapplication
ofmultipleshooting,weareabletosolvetheintervalwiseproblemsandobtainthecorrect
solutionaftersomestepsofmultipleshooting.Wehavechosenintervalwiseconstantcontrols
intimeandperformedthecalculationfor20intervalswithtimestepsize0.01.Thecontrol
andcorrespondingsolutionobtainedbythecalculationarepresentedinFigure1.3and1.4.
Thesolutionoverthewholetimeintervaldoesnotonlyblowupfortheeasiestcaseof
q≡0,butalsoforseveralothertestedinitialcontrols.Therefore,thebreakdownofstandard
optimizationroutinesislikely,whereasmultipleshootingwithasufficientlylargenumberof
intervalsissuitableforthesolutionoftheproblem.
TheaccurateapproximationofthesolutiontoaPDEusuallyrequireshighcomputational
effortwhichcanbereducedbyusingadaptivegridstrategies.Finiteelementschemeshave
proventobeverysuccessfulinthiscontext.Inparticular,themethodofdualweighted
residuals(DWRmethod)issuitedtospeedupoptimalcontrolproblemsgovernedbypartial
differentialequations,sinceitallowstheefficientapproximationofthegoaloftheoptimization
problem.Therefore,thecombinationofmultipleshootingmethodsforPDEconstrained
optimizationwithmeshadaptationtechniquesisalsodiscussed.

4

(a)t=0.04

(b)t=0.08

(c)t=0.12(d)t=0.1389
Figure1.2:Solutionu(t,x,y)forq≡0,u0=0,λ=7.5atdifferenttimepoints.

(a)t=0.25

(b)t=0.75

(c)t=0.95(d)t=1.00
Figure1.3:Controlq(t,x,y)forλ=7.5,α=10−2.

Wenowgiveashortoverviewofthetopicsrelatedtomeshadaptivemultipleshootingthat
thesis.thisindiscussedare

InChapter2wepresenttheformulationandtheoryofPDEconstrainedoptimization
problemsuniquenessofandgivsolutions.eabriefFovurthermore,erviewonwetheciteandtheoreticalprovebacneededkgroundstandardconcerningresultsexistencefromtheand
literature,forexamplefirstandsecondorderoptimalityconditions.

5

ductiontroIn1

(a)t=0.25

(c)t=0.95

(b)t=0.75

(d)t=1.00

Figure1.4:Primalsolutionu(t,x,y)forλ=7.5,α=10−2.

Weproceedwithanoverviewonthehistoricalbackgroundofthemultipleshooting
approachinChapter3.Here,thehistoricalmotivationanddevelopmentofmultipleshooting
forODEsissummarized,andtheinsufficiencyofthesingleshootingapproachforthesolution
isdiscussed.Weintroducemultipleshootingandexplainbrieflyfurtherdevelopments,such
ascondensingandefficientderivativegeneration.Furthermore,thebasicideaofmultiple
shootingforODEconstrainedoptimizationproblemsisbrieflypresented.
TheideaofmultipleshootingforPDEconstrainedoptimizationisintroducedin
Chapter4.First,wedeveloptheindirectmultipleshootingapproach,whichappliesmultiple
shootingtechniquestotheoptimalitysystemoftheproblem.Afterthat,weintroducethe
directmultipleshootingapproachwhichparameterizestheconstrainingequationandthecost
functionalbymultipleshootingandderivestheoptimalitysystemafterwards.Finally,we
closethechapterwithatheoreticalinvestigationoftherelationbetweendirectandindirect
oting.shoultiplemChapter5isdevotedtotheappropriatediscretizationintimeandspace.Weintroduce
continuousGalerkinfiniteelementmethodsonquadrilateralsasonepossiblemethodfor
thespatialdiscretization.Forthetimediscretization,wepresentdiscontinuousGalerkin
methods,andfinallywediscussdifferentpossibilitiesforthediscretizationofthecontrol
space.Aboveall,thechoiceofthespatialmeshesatthemultipleshootingnodesisofgreat
importance,andavarietyofpossiblechoicesexists.Furthermore,thechapterprovidesus
withtwofundamentallydifferentapproachesforthechoiceofthechangingspatialmeshes
intime.Thefirstapproachallowsdynamicallychangingmeshesineachtimestep,andthe
secondapproachisbasedonconstantmeshesforeachsubinterval.Finally,wepresentthe
implicitEulertimesteppingschemeasthesimplestexampleforadiscontinuousGalerkin

6

heme.scdiscretizationtimeInChapter6wediscusssolutiontechniquesforthemultipleshootingapproach.
Newton’smethodisappliedinbothdirectandindirectmultipleshootingforsolvingthe
optimalitysystem.Theresultinglinearizedproblemissolvedbyapplicationofthegeneralized
minimumresidualmethod.Wereviewdifferentpreconditionersandoutlinethenecessity
andefficiencyofpreconditioningbynumericalexamples.Additionally,inanalogytothe
ODEapproach,wedevelopacondensingtechniquefordirectmultipleshootingwhichreduces
thecomputationaleffort.Thechaptercloseswithanumericalcomparisonofthedifferent
hes.approacChapter7isdevotedtothestudyofaposteriorierrorestimationforthediscretization
errorofthecostfunctional.Thediscretizationofstateandcontrolspaceisnecessaryforthe
computationalsolutionoftheproblemandleadstoinexactapproximationsofbothcontrol
andstatevariables.Thiserrorresultsinanincorrectfunctionalvalue,andtheaimisto
chosethediscretizationsuchthattheerrorisminimalforaprescribednumberofcells.Asa
firstidea,theusualgoalorienteddualweightedresidualerrorestimatorforPDEconstrained
optimizationproblemscanbeusedasanadd-onfunctionalityafterthesolutionoftheproblem
bythemultipleshootingapproach.Inthecontextofmultipleshooting,thisapproachis
limitedtocertaindiscretizations,whereadjacentmeshesonthemultipleshootingnodesare
thesame.Therefore,wedevelopanewerrorestimationapproachfortheconvergedsolution
whichallowstheconsiderationofadditionalprojectionerrorsonthemultipleshootingnodes.
InChapter8,wediscussdifferentstrategiesforthecombinationofmultipleshootingand
meshadaptation.First,wepresentthecommonideaofrefinementduetothecellwise
errorindicators.Thisapproachresultsindynamicallychangingspatialmeshes.Second,
wedeveloparefinementstrategywithintervalwiseconstantmeshes,whichequilibratesthe
projectionerroronthemultipleshootingnodeswiththediscretizationerrorontheintervals.
Finally,numericalexamplesillustratetheefficiencyofbothapproachesincomparisonto
t.refinemenglobalInthecontextofapplications,thesolidfuelignitionmodelisatypicalexamplefor
multipleshooting.WepresentthisexampleindetailinChapter9.Here,thechemicaland
theoreticalbackgroundaresummarized.Wediscussthereasonsfortheunstablebehaviorof
theproblem.Finally,wepresentresultsfromnumericalcomputationsfordifferentsettingsof
theproblemattheendofthechapter.Allcomputationsinthisthesisweredonewiththe
finiteelementsoftwarepackagedeal.II.Theobtainedsolutionswerevisualizedbymeansof
VisuSimple.arewsofttheThefinalChapter10isdevotedtoanoverviewonmultipleshootingforPDEconstrained
optimization,drawingconclusionsandgivinganoutlookonfurtherdevelopments.The
resultsobtainedsofararesummarized,andconclusionsconcerningthepropertiesand
applicabilityofthemethodaredrawn.Possibleextensionsandpromisingfuturewaysfor
developmentsofmultipleshootingmethodsforPDEconstrainedoptimizationarebriefly
outlined.

7

2FormulationandTheoryofPDE
ProblemsOptimizationConstrained

Thisoptimizationchaptercovproblems.ersaWebriefformoutlineulateanofinthetroduformctoryulationexampleandintheorderorytoofdevPDEeloptheconstrainedgeneral
andfundamentalideaofparabolicoptimizationproblemsinSection2.1andcontinuewith
theusualmathematicalsettingandabstractformulationoftheseproblemsinSection2.2.
Wellknownresultsonexistenceanduniquenessofsolutionstoparabolicpartialdifferential
equationsarebrieflyreviewedinSection2.3,andthenecessaryandsufficientoptimality
conditionsforparabolicoptimizationproblemsarerevisedinthefinalSection2.4.

riesPrelimina2.1

Beforeproblems,wewgoeingivtoeandetailintrowithductoryrespectexampletoantotheabstractkindformofulationproblemsofparabconsideredolicinthisoptimizationthesis.
Ourdifferengoaltialofequationoptimizationofparabisolicfindingtype.anTheoptimalsimplestconptrolossibleforacaseissystemalineargovernedproblembyinavpartiolvingal
conthetrolLaplacianparameterwithqisthehomogeneoussourceDirictermhletofbtheoundaryequation.conditionsUtilizingandangivabsentracinitialtcostvalue0.functionalThe
J(q,u)tobediscussedlater,ouroptimalcontrolproblemis:findapair(q,u)insuitable
thathcsuspaces2.1.ExampleJ(q,u)=min
tsconstraintheunder∂tu+Δu=qinI×Ω,
u=0onI×∂Ω,
u(0,.)=0inΩ

inapolygonaldomainΩ⊂RdandonatimeintervalI=(0,T).
Thenextparagraphisdevotedtothedevelopmentofanabstractmathematicalframework
forparabolicoptimizationproblems.Keepingthepreviousexampleinmindwegeneralize
theformulationofthestateequation.Wepresentappropriatespacesforstatesandcontrols
andspecifythecostfunctionalsofinterest.Finally,wegivefurtherexamplesandembed
themintotheabstractformulation.

9

2FormulationandTheoryofPDEConstrainedOptimizationProblems

2.2FormulationofAbstractParabolicOptimizationProblems

LetusfirstintroducetheHilbertspacesVandH,whereViscontinuouslyembeddedand
:HindenseV→dH.
WeidentifytheHilbertspaceHwithitsdualspaceH∗,andtogetherwiththedualspaceof
V,V∗,weretrievetheGelfandtriple
V→dH=∼H∗→dV∗.
Furthermore,thedualitypairingofV∗andVisdenotedby∙,∙V∗×V,andthescalarproduct
onHisgivenby(∙,∙)H.Inthefollowing,weconsiderthecontinuouscontinuationof(∙,∙)H
onto∙,V∗×VasanewrepresentationforthefunctionalsinV∗.Thiscanbedoneduetothe
mark:rewingfolloRemark2.1.LettheinjectionofVintoV∗bedenotedbyi:V→H.Thedualmappingof
iistheinjectionofH∗intoV∗andisdenotedbyi∗:H∗→V∗.Fromthedefinitionofi∗
thefollowingidentityholdsforeveryh∈H=∼H∗:
i∗(h),vV∗×V=(h,i(v))H∀v∈V,
andwecanconsiderhasalinearcontinuousfunctionalonV.Duetothedenseembeddingof
H∗intoV∗,everyfunctionalv∗,∙V∗×Vcanbeuniformlyapproximatedbyscalarproducts
(h,i(∙))H.Thereforeitisreasonabletoconsiderthecontinuouscontinuationof(∙,∙)Honto
∙,V∗×VasanewrepresentationforthefunctionalsinV∗.Adetaileddescriptionofthis
conceptcanbefoundforexampleinLions[28]andWloka[43].
Now,letRbeaspatialHilbertspaceforthecontrolq(t).Weassumethatthetimedependent
functionsuandfhavetemporalvaluesu(t)∈Vandf(t)∈V∗,andtheinitialvalueofour
stateuisgivenbyu(0)=u0∈H.OnatimeintervalI=(0,T),0<T<∞,weconsider
parabolicoptimizationproblemsofthefollowingabstracttype:
∂tu(t)+A(u(t))+B(q(t))=f(t),(2.2)
u(0)=u0.
Remark2.2.(Moregeneralnonlinearequations)Thedecouplingofuandqin(2.2)isdone
forthepurposeofnotationalsimplification.ThegeneralcaseofanoperatorC:X×Q→V∗
PDEnonlinearondingcorrespwith∂tu(t)+C(u(t),q(t))=f(t),
u=(0)u0canbetreatedanalogously.Therefore,allresultspresentedinthisthesiscanbeappliedto
o.tocase,thisHere,Bisassumedtobea(nonlinear)operator,withB:R→V∗,givenbyasemi-linear
formb¯:R×V→Ras
B(q¯),v¯V∗×V=b¯(q¯)(v¯)∀¯v∈V.

10

2.2FormulationofAbstractParabolicOptimizationProblems

TheellipticspatialdifferentialoperatorA:V→V∗isgiveninweakformulationbythe
semi-linearform¯a:V×V→Ras
A(u¯),v¯V∗×V=a¯(u¯)(v¯)∀v¯∈V.
Fortheweakformulationofproblem(2.2)weintroduceanotherHilbertspaceXforthetime
dependentstates,
X:=W(I):={v|v∈L2(I,V)and∂tv∈L2(I,V∗)},
forwhichwehave(see,forexample,[16])acontinuousembed2dinginC(I¯,H).Furthermore,
weassumethatthespaceQofthecontrolsisasubspaceofL(I,R),
Q⊆L2(I,R).
Itsscalarproductandnormaredenotedby(∙,∙)Qand∙Q.
Now,wehavethemathematicaltoolsathandtoposethestateequation(2.2)inaweak
form:Foragivencontrolq∈Qfindastateu∈Xsuchthatforallϕ∈X
I(∂tu(t),ϕ(t))Hdt+Ia¯(u(t))(ϕ(t))dt+Ib¯(q(t))(ϕ(t))dt=I(f(t),ϕ(t))Hdt,
u(0)=u0.
Inthefollowing,weomittheindexHatthescalarproduct,(∙,∙),andforthesakeofbrevity
weadditionallyintroducethefollowingnotation:
((v,w)):=(v(t),w(t))dt,
Ia(u)(v):=Ia¯(u(t))(v(t))dt,
b(q)(v):=b¯(q(t))(v(t))dt.
IBycouplingtheinitialconditiontothestateequation,weretrievebyvirtueoftheabbreviatory
notationthefollowingcompactformofthestateequation:
((∂tu,ϕ))+a(u)(ϕ)+b(q)(ϕ)+(u(0),ϕ(0))=((f,ϕ))+(u0,ϕ(0))∀ϕ∈X.(2.3)
TheobjectiveorcostfunctionaloftheoptimizationproblemisdenotedbyJ:Q×X→R.
WedefineJasthesumoftwofunctionals,J1:X→RandJ2:H→R,andaregularization
ybtermJ(q,u)=α1J1(u)+α2J2(u(T))+α23q−qˆ2Q,(2.4)
wherewedemandαi≥0,i=1,2,3andqˆ∈Q.Furthermore,weassume,thatthereisa
functionalF:V→Rsuchthat
J1(u)=IF(u(t))dt.(2.5)
Weneedthisassumptionfortheconsiderationofthemultipleshootingapproach.Inthis
context,wedecomposethetimedomainIintosmallersubintervalsandwanttoconsiderthe
restrictionofthecostfunctionalJ1toeachofthesubintervals.

11

2FormulationandTheoryofPDEConstrainedOptimizationProblems

Rmindemarkthat2.3.thegeneralThroughoutcasethisfollowsthesis,wstraighesettforwqˆ=ard0forandtheiseaseoftenofofpresenrelevancetation,inbutkapplicationeepin
problemswhereaprioriinformationonthecontrolisavailable.
Remark2.4.Lateron,wemainlyconsidercostfunctionalsofthefollowingstructure:
J1(u):=21Iu(t)−u¯(t)2dtandJ2(u(T)):=21u(T)−u¯T2
whereu¯∈Xandu¯T∈H.
Remark2.5.Inthecontextofaposteriorierrorestimation,weassumethatJ1andJ2are
threetimesGâteauxdifferentiable,whichhastobeverifiedforeachconcretefunctionalanew.
InthecaseofJ1andJ2havingthestructurestatedinRemark2.4thisassumptionclearly
holds.ThegoaloftheoptimizationproblemisnowtominimizeJ(q,u)undertheconstraining
demandthatqandufulfillthestateequation(2.3).Thus,theoptimizationproblemreads

(q,u)∈minQ×XJ(q,u)subjectto(2.3).(2.6)
Beforediscussingexistenceanduniquenessofsolutionstoparabolicoptimizationproblems,
letusfirstpresentthreeexamplesforproblemsofthistype.Weconsiderexamplesfortwo
differenttypesofquadraticfunctionalswithlinearandnonlinearconstrainingequations.
LetusfirstreconsiderExample2.1inthisabstractframework:
LipscExamplehitz2.2.domaininR(Distributedd.Theconoptimaltrolofconatrolterminalproblemtimeisfgivenbunctional)yLetΩbeabounded
(q,u)∈minQ×XJ(q,u):=α22u(T)−u¯TL22(Ω)+α23q(t)L22(Ω)dt
Isubjecttothelinearheatequation
∂tu−Δu=qinΩ×I,
u=0on∂Ω×I,
u=0inΩ×{0}.
Thisexamplecanberegardedinthepreviousabstractcontextbychoosingthespaces
H=L2(Ω),V=H01(Ω),andQ=L2(I,L2(Ω)).
Wehavechosenα1=0,andJ2isgivenby
1J2(u(T))=2u(T)−u¯TL22(Ω).
Thesemi-linearformsarechosenas
a(u)(ϕ)=((u,ϕ)),
b(q)(ϕ)=−((q,ϕ)).
Andfinallytheright-handsideandinitialconditionaregivenby
f=0andu0=0.

12

2.2FormulationofAbstractParabolicOptimizationProblems

Whereasinthepreviousexamplewewantedtofitagivenfunction¯uTattheterminaltime
point,wemightfurthermorebeinterestedinmatchingagiventimedependentfunctionu¯(t):
Example2.3.(Distributedcontrolofadistributedfunctional)
αα(q,u)∈minQ×XJ(q,u):=21Iu(t)−u¯(t)L22(Ω)dt+23Iq(t)L22(Ω)dt
subjecttothenonlinearparabolicequation
∂tu−Δu+u3=qinΩ×I,
u=0on∂Ω×I,
u=0inΩ×{0}.
Forthisexamplewehaveintheabstractformulation
H=L2(Ω),V=H01(Ω),andQ=L2(I,L2(Ω)).
Withα2=0,theremainingpartofthecostfunctionalisgivenby
J1(u)=21u(t)−u¯(t)L22(Ω)dt,
Iandthesemi-linearformsarechosenas
a(u)(ϕ)=((u,ϕ))+((u3,ϕ)),
b(q)(ϕ)=−((q,ϕ)).
Andfinallytheright-handsideandinitialconditionaregivenby
f=0andu0=0.
Finally,weconsideranexampleofNeumannboundarycontrol:
Example2.4.(DistributedNeumannboundarycontrolofadistributedfunctional)
(q,u)∈minQ×XJ(q,u):=α21u(t)−u¯(t)L22(Ω)dt+α23q(t)L22(∂Ω)dt
IIsubjecttothenonlinearparabolicequation
∂tu−Δu+u3−u=0inΩ×I,
∂nu=qon∂Ω×I,
u=0inΩ×{0}.
Forthisexamplewehaveintheabstractformulation
H=L2(Ω),V=H01(Ω),andQ=L2(I,L2(∂Ω)).
Here,α2=0,andforJ1weobtain
J1(u)=1u(t)−u¯(t)L22(Ω)dt.
2I

13

2FormulationandTheoryofPDEConstrainedOptimizationProblems

Thesemi-linearformsarechosenas
a(u)(ϕ)=((u,ϕ))+((u3,ϕ)),
b(q)(ϕ)=−(q,ϕ)L2(∂Ω)dt,
Iandfinallytheright-handsideandinitialconditionaregivenby
f=0andu0=0.

Weproceedwiththediscussionofexistenceanduniquenessofsolutions.

2.3ExistenceandUniquenessofSolutions

Thematterofexistenceanduniquenessofsolutionstooptimizationproblemsaspresented
[ab18o],veandhasTröltzscextensivh[ely41].bIneentheliterdiscussedaturefortwoexamplediffereninttecthehniqutextbesoareoksofusedLionsfor[pro28],vingFursikresultsov
ontheestatesxistenceareandconsiunderediqueness.asaOnfunctiontheofonethehand,controltheqr.educOnedtheapprotheroachishand,appliedthenon-rsucheducthated
apprreducedoachtreatsapproachtheforstatestheandtheoreconticaltrolsinvexplicitlyestigationofcoupled.existenceIntheandfollowing,uniqueness.werefertothe
Letusfirstrecallsomeabstractresultsonexistenceanduniqueness.Weassumetheexistence
ofasolutionoperatorS:Q→Xwhichmapsthecontrolqontothesolutionu(q)ofthe
constrainingstateequation(2.3).Thevalidityofthisassumptiononlydependsontheunique
solvabilityoftheparabolicequation(2.3)andhastobeverifiedforeachproblemindetail.
Withinthereducedapproach,thereducedcostfunctionalj:Q→Risintroducedas
j(q):=J(q,S(q)),
andtheoptimizationproblem(2.6)isreformulatedasanunconstrainedoptimizationproblem
q∈minQj(q),q∈Q.(2.10)
Weapplytheclassicaltheoremonexistencefromthecalculusofvariations:
Theorem2.1.Letthereducedfunctionalj:Q→Rbeweaklylowersemi-continuous,that
isnlim→∞infj(qn)≥j(q)wheneverqnqinQ
andcoerciveoverQ,thatis
j(q)≥αqQ+β
foreveryq∈Qandforsomeα>0,β∈R.Thenproblem(2.10)hasatleastonesolution
.Qq∈Proof.SeeforexamplethetextbookofDacorogna[15].

14

2.3ExistenceandUniquenessofSolutions

Furthermore,fortheuniquenessofthesolutionwehavetodemandstrongerrestrictionson
:jfunctionalreducedtheTheorem2.2.LetthereducedfunctionaljfulfilltherequirementsofTheorem(2.1).Ifin
additionjisstronglyconvexonQ,thatis
j(λq1+(1−λ)q2)<λj(q1)+(1−λ)j(q2)
forallλ∈(0,1)andallq1,q2∈Q,q1=q2,thenproblem(2.10)hasauniquesolution.
Proof.Letusassumethatq1andq2,q1=q2,aresolutionsof(2.10).Forλ∈(0,1)the
followinginequalityholdsduetothestrongconvexityofj:
j(λq1+(1−λ)q2)<λj(q1)+(1−λ)j(q2)=q∈minQj(q).
Thisisincontradictiontotheoptimalityofq1andq2.

Fortheapplicationofthesetheoremstoarbitrarynonlinearproblems,therequirementsonj
havetobeverified.Weshowuniquesolvabilitywithintheabstractframeworkoftheprevious
sectiononlyforthesimplecaseofExample2.2.
Letusfirststatetheuniquesolvabilityofthelinearheatequation:
Theorem2.3.LetIbeaboundedtimeintervalandΩbeaboundedLipschitzdomain.Set
H=L2(Ω)andV=H01(Ω).Thelinearparabolicequation
∂tu−Δu=finΩ×I,
u=0on∂Ω×I,
u=u0inΩ×{0}
hasauniquesolutionu∈Xforf∈L2(I,V∗)andu0∈H.Additionally,udepends
continuouslyonthedata:
(f,u0)→u
isacontinuousmappingfromL2(I,V∗)×HintoX.
Proof.SeeforexamplethetextbookofLions[28].

Now,wecanstatethefollowingtheorem:
Theorem2.4.LetIbeaboundedtimeintervalandΩbeaboundedLipschitzdomain.Set
H=L2(Ω)andV=H01(Ω),Q=L2(I,L2(Ω)).Furthermoreletα1,α3>0.Thenthe
oblemproptimization(q,u)∈minQ×XJ(q,u):=α22u(T)−u¯L22(Ω)+α23q(t)L22(Ω)dt(2.11a)
I

15

2FormulationandTheoryofPDEConstrainedOptimizationProblems

(2.11b)

subjecttothelinearheatequation
∂tu−Δu=qinΩ×I,
u=0on∂Ω×I,(2.11b)
u=0inΩ×{0}
hasauniquesolution(q,u)∈Q×X.
isProknoof.wnFromtobeTheoremcontin2.3uoustheandsolutionlinear.opTheeratorcontinS:uitQy→andX,conSvqexit=yuofoftheequationreduced(2.11b)cost
functionalj:Q→Rcandirectlybeseenfromitsdefinition
j(q):=J(q,S(q))=α2S(q(T))−u¯L22(Ω)+α3q(t)L22(Ω)dt,
22Iandthusitisweaklylowersemi-continuous.ApplicationofTheorem(2.1)yieldstheexistence
ofatleastonesolution,andα3>0ensuresstrongconvexityofjandthusuniquenessofthe
solution.

Forexistencemoreandgeneral,uniquenessnonlinearofsolutionsparabisolicquitesimilaroptimizationtotheproblemsonepresenthetedproforcedurethecaseofofprolinearving
quadraticoptimalcontrolproblems.Nevertheless,theproofsaremorecomplicated,andfor
thisfurtherthesis,detailsweweassumerefertothattheourliteratureoptimizationcitedattheproblembofeginninginterestofthis(2.6)section.admitsa(loThroughoutcally)
uniquesolution.Furthermore,inthecontextofmultipleshooting,wealsoassumethatthe
intervalwiseproblemsadmita(locally)uniquesolution.

2.4ConditionsyOptimalit

In(2.6)thisbysection,meanswofethepresentreducedfirstorderapproach.necessaryBeforeandrsufficieecallingntapoptimalitpropriateyconditionstheorems,forweproblemshortly
reviewthestandarddefinitionsofdifferentiabilityinnormedvectorspaces.
Definition2.1.(Directionalderivative)LetXandYbenormedvectorspacesandUbea
neighborhoodofapointx∈X,andletf:U→Y.Ifforanyh∈Xthereexiststhelimit
f(x)(h):=limf(x+th)−f(x),
t0tthenf(x)(h)iscalledthedirectionalderivativeoffatxindirectionh.Ifthislimitexists
forallh∈X,thenfiscalleddirectionallydifferentiableatx.
Definition2.2.(Gâteauxderivative)LetXandYbenormedvectorspacesandUbea
theneighdireborhoctionalodofderivapativointefx(∈x)Xis,aandconletintfuous:U→linearYbemappingdirectionallyfromXtoYdifferen,thentiablefisincx.alledIf
Gâteauxdifferentiableandf(x)iscalledtheGâteauxderivativeoffatx.

16

ditionsConyOptimalit2.4

neighDefinitionborhood2.3.ofa(Fprécoinhettx∈derivX,ativande)LetletfX:Uand→YYb.eIfnormedtherevecexiststoraconspacestinanduousUblinearea
mappingf(x):X→Ysuchthat
limf(x+h)−f(x)−f(x)(h)Y=0,
hX→0hX
thenfiscalledFréchetdifferentiableatxandf(x)iscalledtheFréchetderivativeoffatx.

Withthesepreparationsathand,wecanstatethefirstandsecondordernecessaryand
secondordersufficientoptimalityconditions.Thetheoremsandproofsinamoredetailed
explanationaregiveninthebookofTroeltzsch[41].
Theorem2.5.(Firstordernecessaryoptimalitycondition)Letthereducedfunctionaljbe
GâteauxdifferentiableonanopensubsetQ0⊆Q.Ifq∈Q0isalocaloptimalsolutionofthe
optimizationproblem(2.10),thenqfulfillsthefirstordernecessaryoptimalitycondition
j(q)(δq)=0∀δq∈Q.(2.12)

Proof.WithQ0openandgivendirectionδq∈Qthereexistsduetolocaloptimalityof
q∈Q0apositiveλ∈Rsuchthatq+λδq∈Q0andj(q+λδq)<j(q).Therefore,wehave
tquotiendifferencetheforj(q+λδq)−j(q)≥0.
λWithλ0weobtaininthelimit
j(q)(δq)≥0.
DuetolinearityoftheGâteauxderivativeandwith−δqafeasibledirection,weobtain
analogouslyj(q)(δq)≤0
andthusthestatedcondition.
Remark2.6.(Additionalconvexityofj)Inthespecialcasethatthefunctionaljisadditionally
convex,thatisforallλ∈[0,1]andallq1,q2∈Qthereholds
j(λq1+(1−λ)q2)≤λj(q1)+(1−λ)j(q2),
condition(2.12)isnotonlyanecessarybutalsoasufficientoptimalityconditionof(2.10).
Theorem2.6.(Secondordernecessaryoptimalitycondition)Letthereducedfunctionaljbe
twotimescontinuouslyFréchetdifferentiableonanopensubsetQ0⊆Qofq.Ifq∈Q0is
alocaloptimalsolutionoftheoptimizationproblem(2.10),thenitholdsthesecondorder
necessaryoptimalitycondition

j(q)(δq,δq)≥0∀δq∈Q.

17

2FormulationandTheoryofPDEConstrainedOptimizationProblems

Proof.WithQ0openandgivendirectionδq∈Qthereexistsapositiveλ∈Rsuchthat
q+λδq∈Q0.FromlocaloptimalityofqweobtainbyTaylorexpansion
20≤j(q+λδq)−j(q)=λj(q)(δq)+λj(q)(δq,δq)+r2j(q,λδq),
2wherer2jisaremaindertermofsecondorder.Fromthefirstordernecessaryoptimality
conditionanddivisionbyλ2/2weobtain
0≤j(q)(δq,δq)+2r2j(q,λδq)
2λandinthelimitforλ0
0≤j(q)(δq,δq)
whichcompletestheproof.

Finally,werecallthesecondordersufficientoptimalitycondition.
Theorem2.7.(Secondordersufficientoptimalitycondition)Letthereducedfunctionaljbe
twotimescontinuouslyFréchetdifferentiableonaneighborhoodQ0⊆Qofq.Assumethatq
fulfillsthefirstordernecessaryoptimalitycondition(2.12)andthatthereexistsapositive
γ∈Rsuchthatthesecondordersufficientoptimalityconditionholds:
j(q)(δq,δq)≥γδq2Q∀δq∈Q.
Thenthereexistsapositiveconstantρ∈Rsuchthatthefollowingquadraticgrowthcondition
holds:j(q+δq)≥j(q)+γδq2Q
4forallδq∈Qwithδq2Q≤ρ.Thus,qisalocalsolutionoftheoptimizationproblem(2.10).
Proof.TheproofisperformedbyapplicationofTaylorexpansion.Withθ∈(0,1)weobtain
forρsmallenoughsuchthatq+δq∈Q0
j(q+δq)=j(q)+j(q)(δq)+1j(q+θδq)(δq,δq)
2=j(q)+21j(q+θδq)(δq,δq)
=j(q)+1j(q)(δq,δq)+1[j(q+θδq)−j(q)](δq,δq).
22Withtheassumedcontinuityofjathand,weretrieveforsmallδqQ≤ρ
j(q+θδq)−j(q)(δq,δq)≤γδq2Q.
2Insertingthesecondpropositionofthetheorem,wefinallyobtainthestatedresult
j(q+δq)≥j(q)+γδq2Q−γδq2Q=j(q)+γδq2Q.
442

Thesetheoreticalresultshavebeenpresentedratherforthesakeofcompletenessthanforthe
investigationofourproblemofinterest.Whenevernecessary,wehintatthetheoremsinthe
xt.teconactual

18

3HistoricalBackgroundoftheMultiple
roachAppotingSho

Thischapterpresentsahistoricalmotivationofmultipleshootingandgivesanoverviewon
certainpropertiesofmultipleshootinginthecontextofODEoptimization.Wehaveseen
inChapter1thatmultipleshootingforPDEconstrainedoptimizationisamongstothers
motivatedbytheapplicationtohighlyinstableproblems.Thehistoricalbackground,however,
isduetotheapplicationtoODEboundaryvalueproblems.Therefore,inSection3.1we
giveastandardexampleofanODEboundaryvalueproblemtoillustratetheinsufficiencyof
singleshootingmethodsandsummarizeshortlytheideaandapplicationofmultipleshooting
toODEboundaryvalueproblemsinSection3.2.Theideaofcondensingispresentedin
Section3.3,andefficientderivativegenerationisbrieflydiscussedinSection3.4.Finally,in
Section3.5theusualdirectmultipleshootingapproachforODEconstrainedoptimizationis
duced.troin

3.1TheSingleShootingApproachforODEBoundaryValue
Problems

AcommonapproachforthesolutionofboundaryvalueproblemsinODEsisthesocalled
singleshootingmethod.Thisapproachsuggestsitselfandisconsequentlyofsimplestructure.
Itisintroducedinmostofthestandardtextbooksonordinarydifferentialequationnumerics
asthebeforementionedbookofStoerandBulirsch[39]fromwhichweborrowthenotation.
Remark3.1.Inthefollowing,wemakeuseoftheclassicalODEnotationandfurthermoreof
thecommonnotationusedinthecontextofsingleandmultipleshootingforODEproblems.
Firstofall,wedenotethederivativewithrespecttotimebyu(t)andthesolutionofthe
correspondingdifferentialequationbyu(t).Thesolutionofaninitialvalueproblemwhichis
dependingontheinitialvalues∈Rn,iswrittenasu(t;s).
InanODEboundaryvalueproblemwewanttofindafunctionu:(a,b)→Rnwith
u(t)=f(t,u(t)),(3.1a)
whereuandfconsistofthecomponents
u1(t)f1(t,u1(t),...,un(t))
u(t)=...,f(t,u(t))=...fort∈(a,b),
un(t)fn(t,u1(t),...,un(t))

19

3HistoricalBackgroundoftheMultipleShootingApproach

)(3.2a

suchthatwithr:Rn×Rn→Rn,
r1(u1,...,un,v1,...,vn)
r(u,v)=...,
rn(u1,...,un,v1,...,vn)
ufulfillstheboundarycondition
r(u(a),u(b))=0.(3.1b)
Existenceanduniquenessofsolutionstoproblemsofthistypehaveextensivelybeenstudied
intheliteraturementionedintheintroduction.Weskipthesetheoreticalaspectsforthe
sakeofbrevityandgoonwiththesolutionofproblem(3.1)byapplicationofsingleshooting.
Theideaofthesingleshootingmethodisthereformulationofproblem(3.1)asaninitial
valueproblemwithanadditionalparameterσ∈Rnfortheinitialvalue.Thisparameteris
determinediterativelyduringthesolutionprocess:
Findσ=(σ1,...,σn)∈Rnsuchthatforuwith
u(t;σ)=f(t,u(t;σ)),u(a;σ)=σ(3.2a)
thefollowingboundaryconditionholds:
F(σ):=r(σ,u(b;σ))=0.(3.2b)
Theequivalenceofproblem(3.2)and(3.1)iseasilyshownbyelementarycalculusandisnot
repeatedinthisoverview.Thisproblemoffindingazeroσofequation(3.2b)subjectto
theODEinitialvalueproblem(3.2a)canforexamplebesolvedbyapplicationofNewton’s
d:methoσ0∈Rn,σk+1:=σk−DF(σk)−1F(σk)k=0,1,....
Thisiterationneedstheevaluationofthefunctionanditsderivative,
F∂F(σk)and(DF(σk))ij=∂σi(σk),
jineachstep.ForthedeterminationofF(σk)=r(σk,u(b;σk))theinitialvalueproblem(3.2a)
hastobesolvedwithinitialvalueσ=σk,andthecalculationofthederivativecaneitherbe
performedbystraightforwardapplicationofdifferencequotients,
DF(σk)≈(Fη1(σk),...,Fηn(σk)),
whereFηj(σk):=F(σ1,k,...,σj,k+ηj,...,σn,k)−F(σ1,k,...,σj,k,...,σn,k),
ηjorbymoresophisticatedderivativegenerationtechniques.Wegointothisinmoredetail
lateroninthischapter.
However,thesingleshootingapproachoftenlacksstabilitywithrespecttothesolutionof
theinitialvalueproblem(3.2a).Astandardexampleforthispropertycanbefoundinthe
textbookofStoerandBulirsch[39]andshallnotberepeatedinthisoverview.Summarizing,

20

3.2TheDirectMultipleShootingApproachforODEBoundaryValueProblems

theexamplethereinoutlinesthatthecomputationalsolutionofboundaryvalueproblems
withsingleshootingisassortedwithdifficulties:forageneralsolutionofaninitialvalue
problemofthetypeu(t)=f(t,u(t)),u(a;s)=swithLipschitzcontinuousfunctionfand
LipschitzconstantL>0,thefollowingwellknownestimateonthesensitivitytoerrorsin
holds:datainitialtheu(t;s1)−u(t;s2)≤eL|t−a|s1−s2.
Obviously,theinfluenceoftheerrorcanbebounded,whentheintervalofinterestischosen
smallenough.ThispropertyleadstotheideaofmultipleshootingforthesolutionofODE
problems.aluevoundaryb

3.2TheDirectMultipleShootingApproachforODEBoundary
VProblemsueal

ThemultipleshootingmethodforthesolutionofODEtwopointboundaryvalueproblemsis
outerbasedoniterativtheeideamethoofdissolvingapplieinitialdtovaluematchtheproblemsintervonsmallalwisetratimesubjectoriesdomainsontheinedgesparallel.oftheAn
timesubdomains.Wereconsidertheboundaryvalueproblem(3.1)forwhichatimedomain
decompositionoftheintervalischosen
a=τ0<τ2<...<τm=b,
andm+1additionalvariables,themultipleshootingvariables,s0,...,sm∈Rnareintroduced.
Now,considerjtheintervalwisenrestrictedinitialvalueproblemswhichdetermineintervalwise
functionsu:(τj,τj+1)→Rwith
(uj)=f(t,uj),u(τj)=sj,j=0,...,m−1.(3.3)
Thesolutionujdependsontheinitialvaluesjandisdenotedbyuj(∙;sj)inthefollowing.The
goalofmultipleshootingisthedeterminationofthemultipleshootingvariabless0,...,sm
suchthatthecorrespondingpiecewisetrajectoriesof(3.3)fittogetherattheedgesofthe
intervalsandtheboundaryconditionisfulfilled.
Inmathematicalformulation:forj=0,...,mfindsjsuchthat
F0(s0,s1)s1−u0(τ1;s0)
F1(s1,s2)s2−u1(τ2;s1)
F(s):=...:=...=0(3.4)
Fm−1(sm−1,sm)sm−um−1(τm;sm−1)
Fm(s0,sm)r(s0,sm)
withuj(∙;sj)solutionof(3.3),ands:=(s0,...,sm).Newtontypemethodsareappropriate
choicesforthesolutionofequation(3.4).Forinstance,theapplicationofNewton’smethod
iterationtheyieldss0∈Rn,sk+1:=sk−DF(sk)−1F(sk)fork=0,1,....

21

3HistoricalBackgroundoftheMultipleShootingApproach

0s

1s

3s2s

1−ms

ms

τ0τ1τ2τ3τm−1τm
Figure3.1:Ideaofmultipleshootingforboundaryvalueproblems.

bWithoundarythisvbasicaluedescproblemsriptionatofhand,themweultiplearesablehotootingpresentapproactwohimpfortheortantsolutiontechniquofteswoforpointhet
weefficienaretinsolutionneedofofthesolvingmtheultiplesholinearizedotingsystemforminulationNewton’softhemethoproblem.d.OnOnthetheotheronehandhand,,
ofmostthetimelinearoftheproblem,multiplewhicshohisotingequivmethoalentdtoissptheendderivinativtheeassemgenerationblingof∂ujjtheonlefteachhandofsidethe
s∂subintervals.Therefore,condensingtechniquesandefficientderivativegenerationplayan
importantroleinmultipleshooting.

Condensing3.3echniquesT

Thisparagraphmainlyequipsuswiththeideaofcondensingforthesimplestcaseofmultiple
shooting.Nevertheless,thesetechniquescanbeextendedtomorecomplicatedcases,especially
toODEconstrainedoptimizationproblems.Fortheeaseofpresentation,weomittheindexk
oftheNewtonstepinthesequel.Then(m+1)×n(m+1)JacobianDF(s)=(∂∂sFji(s))i,j=0,...,m
inhas,tervduealwisetotheJacobiansspecialandstructureidentities,ofthematchingconditions,asparseblockstructureof
−G0I00
0−G1I...
DF(s)=.........0.
0...−Gm−1I
B00AtheTherefore,JacobiansfurtherGk,A,B∈simplificationsRn×narecanbedeterminedperformedbyondifferenthetiationsystem.oftheConsideringstateequationtheblocandks,

22

Condensing3.3hniquesecT

(3.5)

theboundaryconditionwithrespecttothemultipleshootingvariables:
jGj:=DsjFj(s)=∂∂suj(τj+1;sj),j=0,...m−1,
A:=Ds0Fm(s)=Ds0r(s0;sm),
B:=DsmFm(s)=Dsmr(s0;sm).
IftheNewtonupdateforsj∈RnisdenotedbyΔsj∈Rn,j=0,...,m,equation(3.4)turns
intoG0Δs0−Δs1=−F0,
G1Δs1−Δs2=−F1,
..(3.5).Gm−1Δsm−1−Δsm=−Fm−1,
AΔs0+BΔsm=−Fm.
Simpletransformationandrecursiveinsertionoftheequationsyields
Δs1=G0Δs0+F0,
...m−1m−1(3.6)
Δsm=Gm−1Gm−2...G0Δs0+GlFj.
j=0l=j+1
Fromthelast0identityof(3.5)wefinallyobtainthedeterminingequationfortheremaining
asΔstincremen(A+BGm−1Gm−2...G0)Δs0=w.(3.7)
Thereby,therighthandsidewisdeterminedvia
w=−(Fm+BFm−1+BGm−1Fm−2+∙∙∙+BGm−1Gm−2...G1F0).
Hence,problem(3.5)fortheevaluationoftheNewtonincrementΔs∈R(m+1)∙nisreduced
tothesolutionofthelinearsystem(3.7)forΔs0∈Rnandsuccessivebackwardsubstitution
accordingto(3.6)forthecalculationoftheremainingmincrements.Forfurtherdetailson
theconvergenceofNewton’smethodforthisproblemandtheinvertabilityofthematrix
A+BGm−1Gm−2...G0werefertotheliteraturementionedintheintroductionofthis
hapter.cItiseasilyverifiedbyelementarycalculusthatineachNewtonstepminitialvalueproblems
havetobesolvedforthecalculationoftheresidual,whilefortheexplicitassemblingofthe
lefthandsidematrixweneedtosolvem×nadditionalinitialvalueproblems–theassembling
ofGjrequiresthesolutionofninitialvalueproblems,oneforeachdirection.Furthermore,
toensureconvergenceofNewton’smethod,acertainaccuracyofthenumericalsolution
mustbeguaranteed.Therefore,efficienttechniquesforthegenerationofthederivatives
areindispensable.Thenextsectionisdevotedtoabriefreviewoftwotechniquesthatare
commonlyused.First,wediscussthe(inefficient)applicationofdifferencequotients,and
second,wepresenttheinternalnumericaldifferentiation(IND)whichisstableandhighly
t.efficien

23

3HistoricalBackgroundoftheMultipleShootingApproach

Derivative3.4Generation

Thefirstandsimplestapproachforthecalculationofthederivativesistheapplication
ofdifferencequotients.InthecontextoftheODEexampleproblem,thismeanstosolve
theinitialvalueproblemrepeatedlywithslightlyperturbedinitialvalues.Thisprocedure
resultsinafirstorderapproximationofthederivativebyfinitedifferences.Indetail,an
approximationofthen×nWronskian
ju∂Wj:=∂sj(τj+1;sj)
isobtainedbyperturbingtheinitialvalueineachcomponentslj,l=1,...,n,bythe
perturbationηl,andperformingthisprocedureforeverycomponentofthesolution.Therefore,
wedefinetheperturbedvectoras
sηjl:=(s1j,...,slj−1,slj+ηl,slj+1,...,snj)
andobtaintheWronskianas
jjηjjjj
Wilj=(∂uj(τj+1;sj))il≈(∂uj(τj+1;sj))il:=ui(τj+1;sηl)−ui(τj+1;s).
∂s∂sηl
Thismethod,sometimesalsodenotedasexternalnumericaldifferentiation,hascertain
disadvantages.Ontheonehandtheapplicationofintegrationschemeswithvariableorder
andstepsizeisdifficult–slightperturbationsintheinitialvaluemightleadtodifferent
integrationsteps,andareliablecalculationofthederivativesisnotguaranteed.Ontheother
hand,theapplicationofintegrationschemeswithfixedorderandstepsizerequiresamuch
highercomputationaleffortandisthusinefficient.Theaccuracyfortheintegrationscheme
wouldhavetobesetonahighlevel,√andeventhenthebestachievableaccuracyforthe
calculationofthederivativeisεD=ε.Hereεdenotestheaccuracytowhichtheoriginal
trajectoryiscalculated.Ifwedenoteby
∂ujη,h
∂sj(τj+1;sj)
iltheinexactapproximationofthedifferencequotientobtainedbynumericalintegration,we
errortheforwritecanjη,h2
εD=∂uj(τj+1;sj)−∂uj(τj+1;sj)=O(ε)+O(η).
∂sil∂silη
AminimalboundofthisexpressionisgivenεD=√εforη2=O(ε).
Apromisingalternativewasfirstpresentedin[8].Thesocalledinternalnumericaldiffer-
entiation(IND)isbasedontheideaofdifferentiatingthediscretizationschemeitself.In
thecaseofavailableexactderivativesoff(forexamplebyautomaticdifferentiation),this
isequivalenttothesolutionofthevariationaldifferentialequationwiththeintegration
schemeusedforthesolutionoftheoriginaldifferentialequation.Thus,thedifferencequotient

24

3.5TheMultipleShootingApproachforODEConstrainedOptimizationProblems

isapproximated2bytheintegratorwithaccuracyO(ε).Summarizingthis,theachievable
accuracyisforη=O(εmach)givenbytheidentity
εD=O(ε)+O(√εmach),
whereεmachdenotesthemachineprecision.Internalnumericaldifferentiationallowsus
toreusethestepsizesandmatricesoftheintegrationschemeusedforthecalculationof
uj(τj+1;sj).Thatis,thecalculationofthenominaltrajectoryujandthecorresponding
Wronskianisperformedsimultaneously,whichmeansthateachtimestepisperformedfirst
forthenominaltrajectoryandthenwiththesameparametersandmatrixforthederivatives.
Thisreduces,accordingto[9],theeffortincomparisontoexternalnumericaldifferentiation
80%.toupyb

3.5TheMultipleShootingApproachforODEConstrained
OptimizationProblems

TheideatoapplymultipleshootingtechniquestoODEconstrainedoptimizationproblems
goesbacktoBulirsch[13]wheretheindirectmultipleshootingapproachisdevelopedfor
ODEconstrainedoptimalcontrolproblems.Amorepopularapproach,thedirectapproach,
wasintroducedinthelateseventiesandearlyeighties,forexamplebyPlitt[33]andBock[11].
Wegiveashortintroductiontothedirectapproach,butdonotgointodetailconcerningthe
solutionofthemultipleshootingproblem.Fordetails,werefertotheliteraturementionedin
duction.trointhennisLettheI=con(0,trol.T)Fdenoteurthermorethetimeweindefineterval,theu:Icost→RufunctionalistheJ:stateRnvq×ariable,Rnu→andRqnJ:Iand→Rtheq
functionf:I×Rnu×Rnq→Rnu.Theoptimizationproblemofinterestreads
qmin,uJ(q,u)(3.8a)
thathsucu(t)=f(t,u(t),q(t)),u(0)=u0.(3.8b)
Mostcommonapproachesforthesolutionofthisproblemmakeuseofacombinationof
iterativeoptimizationtoolsandforwardsolversfortheconstrainingequation(3.8b).The
asmaultiplemultishopoinotingtboundaryapproach,valuforeexampleprobleminwhic[9],husescanbethesolvideaedofbyinmultipleterpretingshotheoting.problemAsbefore,(3.8)
themultipleshootingapproachexploitsthestablesolutionofsmallerinitialvalueproblems
onsubintervalswithadditionalmatchingconditions.Thetimedomaindecompositionand
m0ultiplemshonotinguvariablesarechosenasbefore,0=τ0<τ1<...τm−1<τm=Tand
s,...,s∈R,andthecontrolisparameterizedintervalwiseby
qj:=q(τj,τj+1)∈Rnq,j=0,...,m−1.
Theintervalwiseinitialvalueproblemsaregivenby
uj(t)=f(t,uj(t),qj),u(τj)=sj,forj=0,...,m−1.

25

3HistoricalBackgroundoftheMultipleShootingApproach

Again,additionalmatchingconditionshavetobeposedontheedgesoftheintervalsto
ensurecontinuityofthemultipleshootingsolution.Thesematchingconditionsaregivenby
equations,nonlinearofsystemaF(s0,...,sm,q0,...qm−1)=0,
wherethefunctionFonthelefthandsideisdefinedas
s1−u0(τ1;s0,q0)
s2−u1(τ2;s1,q1)
..
F(s0,...,sm,q0,...qm−1):=m−1m−2.m−2m−2.
s−u(τm−1;s,q)
0sm−um−1(τm;sm−1,qm−1)
us0−Thereformulationof(3.8)intermsofthemultipleshootingvariablesandtheparameterized
controlsreadsasfollows:

)(3.9a

s0,...,sm,qmin0,...qm−1J˜(s0,...,sm,q0,...qm−1)(3.9a)
suchthatthefollowingequalityconstraintsholds:
F(s,˜q˜)=0.(3.9b)
Here,thecostfunctionalJ˜isobtainedstraightforwardbyinsertingtheparameterizedstates
candhoiceconoftrolstheinmtoultipletheshooriginalotingcostnodesfunctionalensuresJ.Inthe[9],itexistencehasbandeenbshooundwnedthatnessanofujappropriateonthe
intervals.Additionally,underappropriateassumptionsonJandF,thewellposednessand
differentiabilityofproblem(3.9)canbeshown.Furtheron,theLagrangianissetupandthe
optimalitysystemhastobesolved,usuallybyapplicationofNewtontypemethods,likeSQP
methodsandreducedSQPmethodswhicharecurrentlystateoftheart.Mostapproaches
derivapplyativereductiongenerationandteccondensinghniquesliktecebachniqueskwardsimilardifferentotiationequationform(3.6)ulasandincom(3.7)andbinationefficienwitht
IND.ForadescriptionofNewtontypemethodsforthesolutionoftheseproblems,wereferto
thetextbookonnumericaloptimizationofNocedalandWright[32],andadetaileddescription
ofderivativegenerationcanbefoundinthediplomathesisofAlbersmeyer[1].
theMultiplelastyshoears.otingforNonlinearODEmodelconstrainedpredictiveconoptimizationtrolhasproblemsbeenasdevwellelopasedPDEquitefarconstrainedduring
optimizationproblemshavebeensolvedbythisapproach,andalotofefforthasbeenputinto
thedevelopmentofmoreefficientderivativegenerationtechniques,condensingtechniquesand
solvsolveders,byfordiscretizingexampleinthe[1],[PDEs35],orwith[10].thePDEmethodofconstrainedlinesandoptimizationapplyingusualproblemsODEhamvebultipleeen
shootingtechniquesfortheresultingsystemofODEs.Thisapproachislimitedtofixedcoarse
spatialmeshessuchthatthesolutionslackspatialaccuracy.Ourapproachwillovercome
thesedifficultiesbyapplyingmultipleshootingdirectlytothePDEprobleminfunctionspace.
ThenextchapterintroducestheideaofmultipleshootingforPDEconstrainedoptimization
andprovidesanappropriatenotationalandtheoreticalframework.

26

4TheMultipleShootingApproachforPDE
OptimizationConstrained

Inthischapterwedevelopandinvestigatethebasicideasofmultipleshootingfornonlinear
optimalcontrolproblemsconstrainedbypartialdifferentialequations.Westartwiththe
discussionofthegeneraldifferencesbetweenmultipleshootingforODEconstrainedand
4.2PDEwithaconstraineddetailedoptdevimizationelopmentofproblemsthesoincalledSection4.1.indirectAfterwmultipleards,wshoeprootingceedapproacinhSectionfor
PDEconstrainedoptimizationproblems.Thechapterisclosedbythedescriptionofthe
directmultipleshootingapproachinSection4.3.Forbothapproaches,thenotationaland
theoreticalframeworkisderived,andtherealizationoftheapproachisoutlinedbymeansof
problems.examplesimple

4.1FromODEstoPDEs–DifferencesandChallenges

ThegeneralizationofmultipleshootingapproachesforODEconstrainedtoPDEconstrained
optimizationproblemsassignsuswithavarietyofnewchallenges.Thefollowingcomposition
describesthedifferentchangesandresultingtasksindevelopingamultipleshootingapproach
forPDEconstrainedoptimization.Forsimplicity,weconsiderasystemofonlyonecomponent
listing.wingfollothein

1.Firstofall,inthecaseofPDEconstrainedoptimizationwehavetoconsidernotonlya
ftime∈C(depI)theendentsolutionstate,isbutininC1(I),additionwenothewsphavatialetodepconsiderendencea.solutionWhereasinforW(OI)DE.swith
2.Consequently,forPDE2constrainedproblemsthemultipleshootingvariablesareno
longerinR,butinL(Ω).
3.IntobtheediscrcaseofetizedPDEfortheconstrainednumericaloptimization,solution.theinfinitedimensionalspatialspacehas
4.ofForODEappropriateconstrainedtimestepproblemspingscahemeshighofaccuracysufficiencantlyeasilyhighbeorder.obtainedPDEbyconstrainedapplication
effortoptimizationneededtorequiresobtainspaatialcertainmeshaccuracyadaptation.inordertoreducethecomputational
5.aForODEmanageablenproblems,umbermatrofenicestries,ofinterestwhereas(formatricexampleesoforccurringsystemsintheofcontextequations)ofhaPDEsve
usuallytendtobecomequitehugeduetoafinespatialdiscretization.

27

4TheMultipleShootingApproachforPDEConstrainedOptimization

6.Spatialmeshadaptationpossiblyleadstodifferentadjacentmeshesatthemultiple
shootingnodes.Thus,weneedanappropriateformulationofthematchingconditions,
forexamplebymeansofprojectionoperators.
Inthefollowingsections,wepresenttwopossiblemultipleshootingapproacheswhichovercome
thesedifficulties.Ontheonehand,weconsidertheindirectmultipleapproachinwhich
weapplymultipleshootingtotheoptimalitysystemoftheoriginalproblem.Ontheother
hand,thedirectmultipleshootingapproachispresentedwhichfollowstheclassicalODE
approachandparameterizestheconstrainingequationbymultipleshootingbeforeapplying
theLagrangeformalismonthecostfunctionalandmatchingconditions.Summarizing,we
differentiatebetweenafirst-optimize-then-multiple-shootingandafirst-multiple-shooting-
then-optimizeapproach.Wewillfinallyseehowtheseapproacheshaveagreatdealin
commonbutatthesametimebearseveralvarietieswithrespecttotheperformance.

4.2TheIndirectMultipleShootingApproach

TheindirectmultipleshootingapproachforPDEconstrainedoptimizationisarecenttopic
ofresearchandwasfirstintroducedforlinearquadraticoptimalcontrolproblemsin[21].
Thederivationofthisapproachisgiveninthefollowing.Incontrasttothedirectapproach
wheremultipleshootingisappliedtotheconstrainingequation,thebasicideaofindirect
multipleshootingistheapplicationofmultipleshootingtotheoptimalitysystemofthe
optimizationproblem.Multipleshootingisappliedtotheprimalanddualvariable.The
controlisnolongerpartofthemultipleshootingsystem,butiscoveredbythecouplingof
primal,dual,andcontrolvariablebyboundaryvalueproblemswiththesamestructureas
system.yoptimalitoriginaltheRemark4.1.Inthefollowing,weassumethatthestateequationandtheintervalwisestate
equationshaveuniquesolutions.Furthermore,werequesttheuniquesolvabilityofall
boundaryvalueproblemsunderconsideration.
First,letusrecalltheconstrainedoptimizationproblem(2.6):
Theconstrainingparabolicpartialdifferentialequationisgivenby
((∂tu,ϕ))+a(u)(ϕ)+b(q)(ϕ)+(u(0),ϕ(0))=((f,ϕ))+(u0,ϕ(0))∀ϕ∈X.(4.1)
WithJasdefinedin(2.4),theoptimizationproblemofinterestreads
(q,u)∈minQ×XJ(q,u)subjectto(4.1).(4.2)
ApplicationoftheLagrangeformalismyieldstheLagrangianL:Q×X×X→Rofthe
problemwithLagrangemultiplierz∈Xwhichwerefertoasthedualvariableinthesequel.
L(q,u,z):=J(q,u)−{((∂tu,z))+a(u)(z)+b(q)(z)+(u(0),z(0))−((f,z))−(u0,z(0))}.
(4.3)Bydifferentiationwithrespecttothestatesandthecontrol,weretrievethefirstorder
optimalitysystemconsistingofthreeequations,namelyprimal,dualandcontrolequation:

28

4.2TheIndirectMultipleShootingApproach

equation:Primal((∂tu,ϕ))+a(u)(ϕ)+b(q)(ϕ)+(u(0),ϕ(0))=((f,ϕ))+(u0,ϕ(0))∀ϕ∈X.(4.4a)
equation:Dual−((∂tz,ψ))+au(u)(ψ,z)+(z(T),ψ(T))−Ju(q,u)(ψ)=0∀ψ∈X.(4.4b)
equation:trolConbq(q)(χ,z)−Jq(q,u)(χ)=0∀χ∈Q.(4.4c)
ReplacingJin(4.4)bythedefinitionfromequation(2.4),wegetthefollowingformulation:
equation:Primal((∂tu,ϕ))+a(u)(ϕ)+b(q)(ϕ)+(u(0),ϕ(0))=((f,ϕ))+(u0,ϕ(0))∀ϕ∈X.(4.5a)
equation:Dual−((∂tz,ψ))+au(u)(ψ,z)+(z(T),ψ(T))−α1J1(u)(ψ)−α2J2(u(T))(ψ(T))=0∀ψ∈X.
(4.5b)equation:trolConbq(q)(χ,z)−α3(q,χ)Q=0∀χ∈Q.(4.5c)
Multipleshootingisnowappliedtothisoptimalitysystemasdescribedinthesequel:
Letus,asintheapproachforordinarydifferentialequations,decomposethetimeinterval
I=(0,T)intommultipleshootingintervalsIj:=(τj,τj+1)with
0=τ0<τ1<...<τm−1<τm=T.
Forthepurposeofmultipleshootinginfunctionspace,weadditionallyintroducetheinter-
valwisespaces
IjXj:=W(Ij)andQj:=Q(Ij):=qq∈Q
andthescalarproductsandnorms((∙,∙))jonXj,(∙,∙)Qjand∙QjonQj.Theintervalwise
restrictionofthestatesandcontrolsu,zandqshallbedenotedby
jjjqj:=qI,uj:=uI,zj:=zIforj=0,...,m−1.
Remark4.2.(Choiceofthecontrolspace)Theproblemformulationconsideredinthisthesisin
combinationwithintervalwiseconsiderationofoptimalcontrolproblemsyieldsarestrictionof
thesuitablecontrolspaces.Anequivalentreformulationoftheoriginalprobleminthedirect
orindirectmultipleshootingapproachisonlypossible,iftheintervalwisecalculatedcontrol
isinthecontrolspaceQ,thatisforarbitraryq0∈Q0,...,qm−1∈Qm−1thecomposition
Ijq:I→Rwithq=qjfulfillstheinclusionq∈Q.Forclarification,considerthefollowing
cases:concrete

29

4TheMultipleShootingApproachforPDEConstrainedOptimization

WhileforQ=L2(I,R)thestatedrequirementforQisnorestrictionatall,wearenotable
toconsiderthecaseofacontrolwhichisconstantonthewholetimeinterval:
(4.6)Q=v∈L2(I,R)v(t)=c∈R
Qj=v∈L2(Ij,R)v(t)=cj∈R
andthereforeqj=cjbutnotnecessarilyc0=c1=...=cm−1=c.Consequently,the
appropriatechoiceofQinthecontextofmultipleshootingis
IjQ(I)=v∈L2(I,R)v=cj∈R,(4.7)
thespaceofintervalwiseconstantfunctionsintime.Nevertheless,manycontrolproblems
requireapiecewiseconstantorpiecewiselinearcontrolintimesuchthatthisrestrictioncan
feasible.consideredebjOnthemultipleshootingnojdesτj,j=0,j..+1.,m,weintroducethemultipleshojotingvariables
s∈Hasinitialvalueforuinτjandλ∈Hasterminaltimevalueforzinτj+1.Thus,
wecannowconsiderintervalwisetwopointboundaryvalueproblemsofthesamestructure
as(4.5)oneachintervalIj.Fortheformulationoftheseproblems,jweintroduceaccording
tothenotationinequation(2.5)theintervalwisefunctionalsJ1:X(Ij)→Rby
J1j(uj):=F(u(t))dt.
IjInthisnotation,theintervalwiseboundaryvalueproblemsreadasfollows:
equation:Primal((∂tuj,ϕ))j+a(uj)(ϕ)+b(qj)(ϕ)+(uj(τj)−sj,ϕ(τj))−((f,ϕ))j=0∀ϕ∈Xj.(4.8a)
equation:Dual−((∂tzj,ψ))j+au(uj)(ψ,zj)+(zj(τj+1)−λj+1,ψ(τj+1))−α1J1j(uj)(ψ)=0∀ψ∈Xj.
(4.8b)ation:quetrolConbq(qj)(χ,zj)−α3(qj,χ)Qj=0∀χ∈Qj.(4.8c)
Next,wegivethereformulationoftheoptimalitysystem(4.5)intermsofthemultiple
shootingformulation.Therefore,werequestadditionalmatchingconditionsatthemultiple
shootingnodesandassumethattheintervalwisestatesandcontrolssolve(4.8):
Finds0,...,sm,λ0,...,λmsuchthat
(s0−u0,v)=0∀v∈H,
(sj+1−uj(τj+1),v)=0∀v∈H,j=0,...,m−1,
(4.9)(λj−zj(τj),v)=0∀v∈H,j=0,...,m−1,
(λm,v)−α2J2(sm)(v)=0∀v∈H.
andqj,uj,zjsolvesystem(4.8)forj=0,...,m−1.

30

(4.9)

4.2TheIndirectMultipleShootingApproach

Remark4.3.Theintroductionofs0,λ0,smandλmisartificial,andthevariablescould
beremovedfromthesystembyinsertingtheirdefinitions.Thus,itwouldbesufficientto
introducemultipleshootingvariablesontheinteriornodes.Nevertheless,theirintroduction
allowsustohandletheequationsonallintervalsidentically,whichturnsouttobehelpful
withrespecttothesimplicityoftheimplementation.
Letusconsiderforj=0,...,m−1thesolutionoperators
H×H→XjH×H→Xj
Σuj:(sj,λj+1)→ujandΣzj:(sj,λj+1)→zj
whichmaptheboundaryvaluessjandλj+1ontothesolutionsujandzj.
Furthermoreweintroducetheoperators
¯H×H→H¯H×H→H
Σuj:(sj,λj+1)→uj(τj+1)andΣzj:(sj,λj+1)→zj(τj)
mappingtheboundaryvaluessjandλj+1ontouj(τj+1)andzj(τj).Wecannowinsertthese
operatorsintothesystemofmatchingconditions(4.9)andintroducethefollowingnotation:
Accordingtothenumberofmultipleshootingvariablesintheindirectapproach,wedefine
thefollowingabbreviationsfortheCartesianproductofvectorspaces:
times+1mH˜:=H×∙∙∙×H,Q˜:=Q0×∙∙∙×Qm−1,X˜:=X0×∙∙∙×Xm−1.(4.10a)
Furthermoreweabbreviatethevectorsofmultipleshootingvariablesby
s˜:=(s0,...,sm)∈H˜,u˜:=(u0,...,um−1)∈X˜,
λ˜:=(λ0,...,λm)∈H˜,z˜:=(z0,...,zm−1)∈X˜,(4.10b)
q˜:=(q0,...,uq−1)∈Q˜
andintroduceontheproductspacesthestandarddefinitionforthenorm.Forexample,for
H˜thisstandardnormisgivenby
mvH˜:=vi2.
=0iFrom(4.9),wefinallyobtainthemultipleshootingformulation:Find(s,˜λ˜)∈H˜×H˜such
that(s0−u0,v)=0∀v∈H,
(sj+1−Σ¯uj(sj,λj+1),v)=0∀v∈H,j=0,...,m−1,
(4.11)(λj−Σ¯zj(sj,λj+1),v)=0∀v∈H,j=0,...,m−1,
(λm,v)−α2J2(sm−1)(v)=0∀v∈H.
Thisformulationisequivalenttotheoriginalproblem(4.5)asstatedinthefollowinglemma.

31

4TheMultipleShootingApproachforPDEConstrainedOptimization

Lemma4.1.Let(q,u,z)∈Q×X×Xbeasolutiontoproblem(4.5)anddefinefor
j=0,...,m−1
sj:=u(τj),λj+1:=z(τj+1).(4.12)
Then(s,˜λ˜)∈H˜×H˜isasolutiontoproblem(4.11).
Let(s,˜λ˜)∈H˜×H˜beasolutiontoproblem(4.11)andletqjbedefinedthroughtheboundary
valueproblem(4.8)forj=0,...,m−1.Then(q,u,z)∈Q×X×Xdefinedby
uIj:=Σuj(sj,λj+1),
zIj:=Σzj(sj,λj+1),j=0,...,m−1,(4.13)
jqI:=qj,j=0,...,m−1
isasolutiontoproblem(4.5).

Proof.Let(q,u,z)∈Q×X×Xbeasolutiontotheboundaryvalueproblem(4.5)andfor
j=0,...,m−1letsj,λj+1bedefinedthroughequation(4.12).Nowletforsj,λj+1onIj
thesolutionsoftheboundaryvalueproblems(4.8)begiventhroughqj,uj,zj.Forreasons
ofuniquesolvabilityoftheboundaryvalueproblem(4.5)anditsrestriction(4.8)wedirectly
retrievetheidentities
jjjqj=qI,uj=uI,zj=zI.
Forreasonsjofcjontinuityofq,u,andzwecaneasilyseethatsjandλj+1andthecorresponding
solutionsu,zfulfillthecontinuityconditions(4.11).
Now,let(s,˜λ˜)∈H˜×H˜beasolutiontoproblem(4.11)andlet(q,u,z)∈Q×X×Xbe
definedbytheequations(4.13).Dueto(4.11)q,uandzarecontinuous.Bysummingupthe
intervalwiseboundaryvalueproblems(4.8)wecandirectlyseethatq,uandzaresolutions
oftheboundaryvalueproblem(4.5).

Tomaketheseratherabstractresultseasierunderstandable,weshortlypresenttheindirect
multipleshootingapproachforExample2.3.
Example4.1.(Indirectmultipleshootingfor3Example2.3)Theconstrainingequationis
theheatequationwithnonlinearreactivetermu:
((∂tu,ϕ))−((u,ϕ))+((u3,ϕ))+(u(0)−0,ϕ(0))=((q,ϕ))∀ϕ∈X,
Hand=inL2(theΩ),Vcost=H1(functionalΩ),Rw=eL2set(Ωα)1and=1,Qα=2L=2(I0,,Lα23(Ω=)).1.ItisTheeasilyspacesvareerified,definedthattheas
0optimalitysystemforthisproblemreadsasfollows:
((∂tu,ϕ))+((u,ϕ))+((u3,ϕ))+(u(0)−0,ϕ(0))=((q,ϕ))∀ϕ∈X,
−((∂tz,ψ))+((z,ψ))+((3u2z,ψ))+(z(T),ψ(T))=((u−¯u,ψ))∀ψ∈X,(4.14)
((q,χ))=−((z,χ))∀χ∈X.
pWecositionhose0the=τ0time<τin1terv<τal2I==1(0of,1)mand=2mconsiderultipleamshoultipleotinginshotervotingals.timeInthedomainframewdecom-ork

32

4.3TheDirectMultipleShootingApproach

ofmultipleshootingforthisproblemwedenotes˜:=(s0,s1,s2)andλ˜:=(λ0,λ1,λ2)and
q˜:=(q0,q1)andretrievetheintervalwiseboundaryvalueproblemsforj=0,1:
Forallϕ,ψ∈Xj,χ∈Qj:
((∂tuj,ϕ))j+((uj,ϕ))j+(((uj)3,ϕ))j+(uj(τj)−sj,ϕ(0))=((qj,ϕ))j,
−((∂tzj,ψ))j+((zj,ψ))j+((3(uj)2zj,ψ))j+(zj(τj+1)−λj+1,ψ(T))=((uj−¯u,ψ))j,
((qj,χ))j=−((zj,χ))j.
(4.15)Andfinallythematchingconditionsforprimalanddualsolutionareeasilyderivedas
(s0−0,ϕ)=0∀ϕ∈H,
(λ0−Σ¯z0(s0,λ1),ϕ):=(λ0−z0(τ0;s1,λ1),ϕ)=0∀ϕ∈H,
(s1−Σ¯u0(s0,λ1),ϕ):=(s1−u0(τ1;s0,λ1),ϕ)=0∀ϕ∈H,
(λ1−Σ¯z1(s1,λ2),ϕ):=(λ1−z1(τ1;s1,λ2),ϕ)=0∀ϕ∈H,
(s2−Σ¯u1(s1,λ2),ϕ):=(s2−u1(τ2;s1,λ2),ϕ)=0∀ϕ∈H,
(λ2−0,ϕ)=0∀ϕ∈H.
Theoptimizationproblemintermsoftheindirectmultipleshootingformulationrequests
thesolutionofthematchingconditions(4.11)subjecttotheintervalwiseboundaryvalue
problems(4.8).Wediscussdifferenttechniquestosolvethisconstrainedproblemoffinding
azerotothesystemofmatchingconditionsinChapter6.Inwhatfollows,weintroduce
andgeneralizethedirectmultipleshootingapproach,whichhasfirstbeenaddressedby
HeinkenschlossandComasin[20]and[14]forlinearquadraticoptimalcontrolproblems.

4.3TheDirectMultipleShootingApproach

anThissectionappropriateisdevotednotationaltodevframewelopmenork,tofwhicthehdirectdiffersmfromultipletheshootingnotationapproacusedihn.[W14e]inandtro[duce20],
andproblemswewithgeneralizearbitrarytheapproaccosthfunctionals.presentedWinetheseconsideragainpublicationsproblemto(4.2)nonlinearanddecompoptimizationose
thedenotetimetheinintervtervalIalwise=(0,T)spacesintobymXmjandultipleQjshowithotingtheintervscalaralsproIj:=ducts(τj,τandj+1).normsAs(b(∙,∙efore))wone
jXj,(∙,∙)Qjand∙QjonQj.Furthermore,theintervalwiserestrictionsofthestateuand
controlqaregivenby
qj:=qIjanduj:=uIjforj=0,...,m−1.
Incontrasttotheindirectapproach,whichreformulatestheproblemintermsofmatching
isbconditionsasedonandtheinformtervalwiseulationbofinoundarytervvalwisealueinitialproblems,valuethedirectproblemsmwithultipleshoadditionalotingmatcapproachingh
conditions.Thetransformationsperformedontheoptimizationproblem(4.2)aresimilar

33

4TheMultipleShootingApproachforPDEConstrainedOptimization

totheproceduresneededforthedirectmultipleshootingapproachforODEconstrained
optimization.Weintroduceonthemultipleshootingnodeτjthemultipleshootingvariablesj∈Hfor
j=0,...,m−1asinitialvaluefortheinitialvalueproblem(4.1)restrictedtoIj:
((∂tuj,ϕ))j+a(uj)(ϕ)+b(qj)(ϕ)+(uj(τj),ϕ(τj))=((f,ϕ))j+(sj,ϕ(τj))∀ϕ∈Xj.(4.16)
Thestatevariableujcannowbetreatedasafunctionofthecontrolqjandtheinitialvalue
sj.Wedefineoneachintervalthe(nonlinear)solutionoperatorSjandtheoperatorS¯jwhich
mapstheinitialvalueandthecontrolontotheterminaltimevalueu(τj+1)ofthesolution:
S:H×Qj→XjandS¯:H×Qj→H.
j(sj,qj)→ujj(sj,qj)→Sj(sj,qj)(τj+1)
Forgivenqjandsj,j=0,...,m−1,thestateujisaccordingtoRemark4.1uniquely
determinedonIj.Byposingadditionalmatchingconditionsontheedgesoftheintervals
toensurecontinuityoftheglobalstate,weareabletoreformulateproblem(4.2)asan
equivalentequalityconstrainedoptimizationprobleminthemultipleshootingvariables
s0,...,sm−1,q0,...,qm−1.Webrieflyrecapitulatetheabbreviatorynotationforthevectors
ofmultipleshootingvariablesandtheCartesianproductsofthespaces:
timesm(s,˜q˜):=(s0,...,sm−1,q0,...,qm−1),H˜:=H×∙∙∙×H,Q˜:=Q0×...×Qm−1.
Inthisnotation,wedefinethereformulatedcostfunctionalJ¯:H˜×Q˜→Rby
J¯(s,˜q˜):=α1J1j(Sj(sj,qj))+α2J2(S¯m−1(sm−1,qm−1))+23qj2Qj.
m−1m−1α
=0j=0jNow,anequivalentformulationofproblem(4.2)intermsofthemultipleshootingvariables
readstrolsconparameterizedand(s,˜q˜)∈minH˜×Q˜J¯(s,˜q˜)(4.17a)
suchthatujsolves(4.16)andthefollowingmatchingconditionsarefulfilled:
(s0−u0,v)=0∀v∈H,
(sj+1−S¯j(sj,qj),v)=0∀v∈H,j=0,...,m−2.(4.17b)
Remark4.4.If(s,˜q˜)∈H˜×Q˜fulfillstheequalityconstraints(4.17b)itiscalledafeasible
pointofproblem(4.17).Ifwedefineforafeasiblepointthesolutionsuj:=Sj(sj,qj)∈Xj
forj=0,...,m−1and(q,u)∈Q×XwithqIj:=qjanduIj:=ujthenthefollowing
identityholdsforthereformulatedcostfunctionalin(4.17a):
J(q,u)=J¯(s,˜q˜).
ThisidentitycandirectlybederivedfromthedefinitionofJ¯.

34

4.3TheDirectMultipleShootingApproach

Beforeproceedingwiththedescriptionofthedirectmultipleshootingapproach,letusstate
theequivalenceofproblem(4.2)andproblem(4.17)intermsofthefollowinglemma:
Lemma4.2.Let(q,u)∈Q×Xbeanoptimalsolutiontoproblem(4.2).Then(s,˜q˜)∈H˜×Q˜,
byintervalwiseddefineqj:=qIjandsj:=u(τj)forj=0,...,m−1(4.18)
isanoptimalsolutiontoproblem(4.17).
Let(s,˜q˜)∈H˜×Q˜beanoptimalsolutionto(4.17),then(q,u)∈Q×X,definedby
jjqI:=qjanduI:=Sj(sj,qj)forj=0,...,m−1
isanoptimalsolutionofproblem(4.2).
Proof.Weonlyprovethefirstpartofthelemma,theproofofthesecondpartisanalogous.
Assume(q,u)∈Q×Xtobeanoptimalsolutiontoproblem(4.2)andletqj∈Qjand
sj∈H,j=0,...m−1,bedefinedbyequation(4.18).Duetotheuniquesolvabilityof
equation(4.16)wecandirectlyderivetheidentityuIj=Sj(sj,qj)forj=0,...,m−1.For
all(qˆ,uˆ)∈Q×Xwehaveduetotheoptimalityof(q,u)theinequality
J(q,u)≤J(qˆ,uˆ).(4.19)
Nowsupposethatthereexistsafeasiblepoint(δ˜s,δ˜q)∈H˜×Q˜withJ¯(δ˜s,δ˜q)<J¯(s,˜q˜).
Then(δq,δu),definedbyδqIj:=δqjandδuj:=Sj(δsj,δqj)forj=0,...,m−1isafeasible
pointforproblem(4.2).Thus,duetoRemark4.4,wecannowwrite
J(δq,δu)=J¯(δ˜s,δ˜q)<J¯(s,˜q˜)=J(q,u),
whichisincontradictiontoassumption(4.19).
Thefirstordernecessaryoptimalityconditionsforproblem(4.17)areobtainedbyapplication
oftheLagrangeformalism.LetusintroducetheLagrangemultiplierspj∈H,j=0,...,m−1,
anddefinep˜:=(p0,...,pm−1)∈H˜.Withthisnotation,theLagrangianL:Q˜×H˜×H˜→R
ofproblem(4.17)isgivenby
m−2
=0jL(q˜,s,˜p˜):=J¯(q˜,s˜)−(s0−u0,p0)+(sj+1−S¯j(sj,qj),pj+1),
andthefirstordernecessaryoptimalityconditionisobtainedastheKarushKuhnTucker
system)(KKTsystemL(q˜,s,˜p˜)(ϕ,˜ψ˜,χ˜)=0∀(ϕ,˜ψ˜,χ˜)∈Q˜×H˜×H˜.
Explicitcalculationofthederivativeonthelefthandsideyieldsasystemofequations.We
omitthearguments(sj,qj)ofthederivativesofthesolutionoperatorsforthepurposeof
y:brevitnotationalSjsj(ψj):=Sjsj(sj,qj)(ψj),Sjqj(ϕj):=Sjqj(sj,qj)(ϕj),
S¯jsj(ψj):=S¯jsj(sj,qj)(ψj),S¯jqj(ϕj):=S¯jqj(sj,qj)(ϕj).

35

4TheMultipleShootingApproachforPDEConstrainedOptimization

)(4.20a(4.20b)

Withthisnotationweretrievethefollowingoptimalitysystem:
Differentiationwithrespecttopj:
Forj=0andforallϕ∈H:
(s0−u0,ϕ)=0.(4.20a)
Forj=1,...,m−1andforallϕ∈H:
(sj−S¯j−1(sj−1,qj−1),ϕ)=0.(4.20b)
Differentiationwithrespecttosj:
Forj=0,...,m−2andforallψ∈H:
α1J1j(uj)(Sjsj(ψ))−(ψ,pj)+(S¯jsj(ψ),pj+1)=0.(4.20c)
Forj=m−1andforallψ∈H:
α1J1m−1(um−1)(Sm−1sm−1(ψ))+α2J2(um−1(τm))(S¯m−1sm−1(ψ))−(ψ,pm−1)=0.(4.20d)
Differentiationwithrespecttoqj:
Forj=0,...,m−2andforallχ∈Qj:
α1J1j(uj)(Sjqj(χ))+α3(qj,χ)Qj+(S¯jqj(χ),pj+1)=0.(4.20e)
Forj=m−1andforallχ∈Qm−1:
α1J1j(um−1)(Sm−1qm−1(χ)+α2J2(um−1(τm))(S¯m−1qm−1(χ))+α3(qm−1,χ)Qm−1=0.(4.20f)
Forabetterunderstandingletusbrieflydescribehowthederivativesofthesolutionoperators
havetobeinterpretedinthisformulation.
Lemma4.3.(Derivativesofthesolutionoperators)Letforj=0,...,m−1thesolution
operatorsfortherestrictedproblem(4.16)begivenasbeforeby
H×Qj→XjH×Qj→H
Sj:(sj,qj)→ujandS¯j:(sj,qj)→Sj(sj,qj)(τj+1).
Furthermoreletthefollowingproblemsforthedeterminationofv∈Xjandw∈XjonIjbe
given:((∂tv,ϕ))j+au(uj)(v,ϕ)+(v(τj)−s,¯ϕ(τj))=0∀ϕ∈Xj,(4.21)
((∂tw,ϕ))j+au(uj)(w,ϕ)+bq(qj)(q¯,ϕ)+(w(τj),ϕ(τj))=0∀ϕ∈Xj.(4.22)
Forthederivativesofthesolutionoperatorsthefollowingidentitieshold:
H→XjH→H
Sjsj:s¯→vandS¯jsj:s¯→v(τj+1),
Qj→XjQj→H
Sjqj:q¯→wandS¯jqj:q¯→w(τj+1).

36

4.3TheDirectMultipleShootingApproach

Proof.Theprooffollowsdirectlybydifferentiationoftherestrictedstateequation(4.16)
withrespecttotheinitialvaluesjandtherestrictedcontrolqjforj=0,...,m−1.
Remark4.5.Lemma4.3yieldsthatSjsjisthesolutionoperatorofproblem(4.21)whichis
obtainedbylinearizationoftheoriginalproblem(4.16)withrespecttotheinitialvalueinto
thedirections¯.Therefore,theoperatorismappingtheinitialvalueofthelinearizedproblem,
s¯,tothesolutionvofequation(4.21).Furthermore,Sjqjisthesolutionoperatorofproblem
(4.22).Again,theequationisobtainedasthelinearizationof(4.16),herewithrespectto
thecontrolintothedirectionq¯.Thus,thesolutionoperatormapsq¯ontothesolutionw.
Analogously,S¯jsjresp.S¯jqjevaluatethesolutionvresp.wattheterminaltimepointτj+1
oftheintervalIj.
Forfurthertransformationsofsystem(4.20),weadditionallyintroduceadjointoperatorsand
describetheirinterpretationassolutionoperatorsoftheadjointequationsinLemma4.4.
Sj∗sj:=Sj∗sj(sj,qj):Xj∗→H∗,
Sj∗qj:=Sj∗qj(sj,qj):Xj∗→Qj∗,
S¯j∗sj:=S¯j∗sj(sj,qj):H∗→H∗,
S¯j∗qj:=S¯j∗qj(sj,qj):H∗→Qj∗.
Weremindthat,accordingtoRemark2.1,theHilbertspaceHwasidentifiedwithitsdual
∗.HspaceLemma4.4.Forj=0,...,m−1letthesolutionoperatorsfortheequations(4.21)and
(4.22)begivenasbeforeby
H→XjH→H
Sjsj:s¯→vandS¯jsj:s¯→v(τj+1),
Qj→XjQj→H
Sjqj:q¯→wandS¯jqj:q¯→w(τj+1).
Furthermoreletthefollowingproblemsforthedeterminationofx∈Xjandy∈XjonIjbe
given:−((∂tx,ψ))j+au(uj)(ψ,x)+(x(τj+1),ψ(τj+1))=r,ψXj∗×Xj∀ψ∈Xj,(4.23)
−((∂ty,ψ))j+au(uj)(ψ,y)+(y(τj+1)−ζ,ψ(τj+1))=0∀ψ∈Xj.(4.24)
Fortheadjointoperatorsthefollowingidentitieshold:
(Sj∗sj(r),ξ)=(x(τj),ξ)∀ξ∈H,
Sj∗qj(r),χQj∗×Qj=−bq(qj)(χ,x)∀χ∈Qj,(4.25)
(S¯j∗sj(ζ),ξ)=(y(τj),ξ)∀ξ∈H,
S¯j∗qj(ζ),χQj∗×Qj=−bq(qj)(χ,y)∀χ∈Qj.

(4.25)

37

4TheMultipleShootingApproachforPDEConstrainedOptimization

Proof.Weproveonlythefirstidentity,theremainingequationscanbeshownanalogously.
BydefinitionoftheadjointoperatorandtheidentificationH=∼H∗,thefollowingequation
istrueforSj∗sj:
(Sj∗sj(r),ξ)=r,Sjsj(ξ)Xj∗×Xj∀ξ∈H,∀r∈Xj∗.(4.26)
Fromequation(4.23)wehaveforallξ∈H:
j
r,Sjsj(ξ)Xj∗×Xj=−((∂tx,Sjsj(ξ)))j+au(u)(Sjsj(ξ),x)+(x(τj+1),Sjsj(ξ)(τj+1))
j=((∂tSjsj(ξ),x))j+au(u)(Sjsj(ξ),x)+(x(τj),Sjsj(ξ)(τj)).
)(4.27Equation(4.21)togetherwiththedefinitionofSjsj(ξ)yields:
((∂tSjsj(ξ),x))j+au(uj)(Sjsj(ξ),x)+(x(τj),Sjsj(ξ)(τj))=(x(τj),ξ)∀ξ∈H.
Now,togetherwithequation(4.26)and(4.27)weretrievetheassertion
∗(Sjsj(r),ξ)=(x(τj),ξ)∀ξ∈H.

Withtheseidentitiesathand,weareabletoperformfurthersimplificationsoftheKKT
system(4.20).Weprovethefollowingtheorem:
Theorem4.5.Letthedualandcontrolequations(4.20c),(4.20d),(4.20e)and(4.20f)be
givenasbefore:Determinep˜∈X˜jandq˜∈Q˜jsuchthatforallψ∈Xj,χ∈Qj
α1J1j(uj)(Sjsj(ψ))−(ψ,pj)+(S¯jsj(ψ),pj+1)=0,j=0,...,m−2,(4.28a)
α1J1j(uj)(Sjsj(ψ))+α2J2(uj(τj+1))(S¯jsj(ψ))−(ψ,pj)=0,j=m−1,(4.28b)
α1J1j(uj)(Sjqj(χ))+α3(qj,χ)Qj+(S¯jqj(χ),pj+1)=0,j=0,...,m−2,(4.28c)
α1J1j(uj)(Sjqj(χ))+α2J2(uj(τj+1))(S¯j,qj(χ))+α3(qj,χ)Qj=0,j=m−1.(4.28d)
Anequivalentformulationoftheseequationsisgivenbythesystem
(zj(τj)−pj,ψ)=0∀ψ∈H,j=0,...,m−1,
α3(qj,χ)Qj−bq(qj)(χ,zj)=0∀χ∈Qj,j=0,...,m−1
wherezjisobtainedasthesolutionofthefollowingintervalwiseinitialvalueproblems:
−((∂tzj,ϕ))j+au(uj)(ϕ,zj)+(zj(τj+1)−pj+1,ϕ(τj+1))
=α1J1j(uj)(ϕ)∀ϕ∈Xj,j=0,...,m−2,(4.29a)

38

−((∂tzj,ϕ))j+au(uj)(ϕ,zj)+(zj(τj+1),ϕ(τj+1))
=α1J1(uj)j(ϕ)+α2J2(uj(τj+1))(ϕ(τj+1))∀ϕ∈Xj,j=m−1.(4.29b)

4.3TheDirectMultipleShootingApproach

Proof.Theproofismainlybasedontheinterpretationoftheadjointoperators(4.25).We
transformtheequations(4.28)byapplyingthedefinitionoftheadjointoperatorssuchthat
thetestfunctionsarenolongerargumentsoftheoperatorsitself.Theobtainedequations
allowtheapplicationofLemma4.4.Thevariablescanbeinterpretedassolutionsofthe
partialdifferentialequations(4.23),(4.24),andthepropositionfollowsbylinearcombination
ofthesolutions.Indetail,weproceedasfollows:
DuetotheassumptionsonthepartsofthefunctionalinRemark2.5thelinearfunctional
J1(uj):Xj→RiscontinuousandthusboundedonXj.Wewriteforauniquelydetermined,
butunknown,elementg1∈Xj∗therepresentation
α1J1(uj)(v)=g1,vXj∗×Xj∀v∈Xj.
Accordingly,J2(uj(τj+1)):H→RisaboundedlinearfunctionalonH,andwecanwritefor
anelementg2∈HtheRieszrepresentation
α2J2(um−1(τm))(w)=(g2,w)∀w∈H.
Now,wehaveeverythingathandtotransformequation(4.28)intoamorecompactformula-
tion.First,weinsertg1andg2intotheequations.Inthenewformulation,wesearchp˜∈X˜j
andq˜∈Q˜jsuchthatforallψ∈H,χ∈Qj
g1,Sjsj(ψ)Xj∗×Xj−(pj,ψ)+(pj+1,S¯jsj(ψ))=0,j=0,...,m−2,
g1,Sjsj(ψ)Xj∗×Xj+(g2,S¯jsj(ψ))−(pj,ψ)=0,j=m−1,
g1,Sjqj(χ)Xj∗×Xj+α3(qj,χ)Qj+(pj+1,S¯jqj(χ))=0,j=0,...,m−2,
g1,Sjqj(χ)Xj∗×Xj+(g2,S¯j,qj(χ(τm)))+α3(qj,χ)Qj=0,j=m−1.
Second,weinsertthedefinitionoftheadjointoperatorsandobtainthefollowingsystemof
equations:(Sj∗sj(g1),ψ)−(pj,ψ)+(S¯j∗sj(pj+1),ψ)=0,j=0,...,m−2,(4.30a)
(Sj∗sj(g1),ψ)+(S¯j∗sj(g2),ψ)−(pj,ψ)=0,j=m−1,(4.30b)
Sj∗qj(g1),χQj∗×Qj+α3(qj,χ)Qj+S¯j∗qj(pj+1),χQj∗×Qj=0,j=0,...,m−2,(4.30c)
Sj∗qj(g1),χQj∗×Qj+S¯j∗,qj(g2),χ(τm)Qj∗×Qj+α3(qj,χ)Qj=0,j=m−1.(4.30d)
Weexploit(4.25)togetherwiththelinearpartialdifferentialequations(4.23)and(4.24).
Introducingtheintervalwisestatesxj,yj∈Xjassolutionsofthepartialdifferentialequations

−((∂txj,ϕ))j+au(uj)(ϕ,xj)+(xj(τj+1),ϕ(τj+1))=g1,ϕXj∗×Xj
=α1J1j(uj)(ϕ)∀ϕ∈Xj,j=0,...,m−1,

−((∂tyj,ϕ))j+au(uj)(ϕ,yj)+(yj(τj+1)−pj+1,ϕ(τj+1))=0∀ϕ∈Xj,j=0,...,m−2,

39

4TheMultipleShootingApproachforPDEConstrainedOptimization

−((∂tyj,ϕ))j+au(uj)(ϕ,yj)+(yj(τj+1),ϕ(τj+1))=(g2,ϕ(τj+1))
=α2J2(uj(τj+1))(ϕ(τj+1)))∀ϕ∈Xj,j=m−1,
thenonlinearsystemofequations(4.30)transformsinto
(xj(τj),ψ)−(pj,ψ)+(yj(τj),ψ)=0∀ψ∈H,j=0,...,m−1,
−bq(qj)(χ,xj)−bq(qj)(χ,yj)+α3(qj,χ)Qj=0∀ψ∈Qj,j=0,...,m−1.(4.32)
Now,wecandefinezj:=xj+yj∈Xjforj=0,...m−1,andbylinearcombinationofthe
linearPDEsforxjandyjweobtainthefinalformulationintermsofthedualvariablezj:
Letzjbedeterminedbythepartialdifferentialequation
−((∂tzj,ϕ))j+au(uj)(ϕ,zj)+(zj(τj+1)−pj+1,ϕ(τj+1))=g1,ϕXj∗×Xj
=α1J1j(uj)(ϕ)∀ϕ∈Xj,j=0,...,m−2,

−((∂tzj,ϕ))j+au(uj)(ϕ,zj)+(zj(τj+1),ϕ(τj+1))=g1,ϕXj∗×Xj+(g2,ϕ(τj+1))
=α1J1j(uj)(ϕ)+α2J2(uj(τj+1))(ϕ(τj+1)))∀ϕ∈Xj,j=m−1.
Thenonlinearsystemofequations(4.32)canberewrittenas
(zj(τj)−pj,ψ)=0∀ψ∈H,j=0,...,m−1,
α3(qj,χ)Qj−bq(qj)(χ,zj)=0∀ψ∈Qj,j=0,...,m−1,
whichcompletestheproof.
Rduceemarkan4.6.additionalInorderfutonctionpmsymmetrize∈Honthethenotationrightbforounthedaryofprimaltheandtimeduinalvterval.ariables,Fweurthermore,intro-
weremark,thatthevariablep0isobviouslyredundant.Thematchingconditionsanddefining
equationsforzjtransforminto
(zj(τj)−pj,ψ)=0,j=1,...,m−1,
α2J2(um−1(τm))(ψ)−(pm,ϕ)=0,
wherethedualvariablezj∈Xjisobtainedfromthepartialdifferentialequation
−((∂tzj,ϕ))j+au(uj)(ϕ,zj)+(zj(τj+1)−pj+1,ϕ(τj+1))=((g1,ϕ))j
=α1J1j(uj)(ϕ)∀ϕ∈Xj,j=0,...,m−1.(4.34)
Next,weintroducesolutionoperatorsforthedualequations.Thevariablezjmustbe
consideredasafunctionnotonlyofthecontrolqj,butalsooftheterminaltimevaluepj+1
andviathedependenceonuofsj.Thesolutionoperatorsforthedualequationaredefined
bythefollowingidentities:
H×H×Qj→XjH×H×Qj→H
Ξj:(pj+1,sj,qj)→zjandΞ¯j:(pj+1,sj,qj)→Ξj(pj+1,sj,qj)(τj).

40

4.3TheDirectMultipleShootingApproach

Summarizing,wecanwritethemultipleshootingformulationofthedirectapproachas
follows.InsertingthereformulatedmatchingconditionsduetoTheorem4.5,considering
Remark4.6andapplyingthedefinitionofthesolutionoperators,thedirectmultipleshooting
formulationisgivenbytheformulation:Finds˜∈H˜,p˜∈H˜,q˜∈Q˜suchthatthefollowing
hold.titiesiden(s0−u0,v)=0∀v∈H,
(sj+1−S¯j(sj,qj),v)=0∀v∈H,j=0,...,m−2,
(pj−Ξ¯j(pj+1,sj,qj),ψ)=0∀ψ∈H,j=1,...,m−1,.(4.35)
(pm,ϕ)−α2J2(um−1(τm))(ψ)=0∀ψ∈H,
α3(qj,χ)Qj−bq(qj)(χ,Ξj(pj+1,sj,qj))=0∀χ∈Qj,j=0,...,m−1.
Finally,bymeansofTheorem4.5wecanpointouttherelationbetweenthedirectandthe
indirectmultipleshootingapproach.Wehavealreadyseenthatbothapproachesreformulate
theoriginalproblemintermsofa(nonlinear)systemofequationswhichallowstheapplication
ofanonlineariterativesolveraswewillseeinChapter6.Thesesystemsofequationsfordirect
andindirectmultipleshootingarecloselyrelatedasisdepictedinFigure4.1.Theequations

otinghoSMultipleDirect

ProblemsalueVInitialEquationPrimalEquationDualConditionshingMatcariableVPrimalDualariableVariableVtrolCon

ShoMultipleIndirectoting

ProblemsalueVBoundaryEquationPrimalEquationDualEquationtrolConConditionshingMatcariableVPrimalariableVDual

Figure4.1:Relationbetweendirectandindirectmultipleshooting.
forprimalanddualintervalwisestatescoincideforthedirectandindirectmultipleshooting
approach.KeepinginmindtheredundancyofλmandsmduetoRemark4.3,thiscanbe
seenfromequations(4.16),(4.34)and(4.8a),(4.8b).Additionallythematchingconditions
formatcthehingprimalconditionanddualcorrvespariablesondingtocoincidetheconduetroltointheequationsdirect(4.35)case,andthe(4.9)last.Fequationurthermore,of(4.35)the,

41

4TheMultipleShootingApproachforPDEConstrainedOptimization

andthecontrolequation(4.8c)areidentical.Thus,weseethatequationsandmatching
conditionsarecloselyrelated,andtheessentialdifferenceistheeliminationofthecontrol
fromthesystemofmatchingconditionsintheindirectmultipleshootingapproach.This
yieldsintervalwisefeasiblestatesandcontrolsduetotheintervalwisesolutionofboundary
valueproblems.Theseboundaryvalueproblemscanbeinterpretedasoptimalitysystems
ofintervalwiseoptimalcontrolproblems.Incontrast,forthedirectapproachfeasibilityis
obtainednotuntilconvergenceofthemultipleshootingapproach.Weseetheadvantagesand
disadvantagesofbothofthesepropertieslateron,whendiscussingthenumericalsolutionof
themultipleshootingformulations.
Example4.2.(DirectmultipleshootingforExample2.3)Forabetterunderstandingof
thisabstractformulationofthedirectmultipleshootingapproach,letusnowtakealook
atthecaseofExample2.3withα1=1,α2=0,andα3=1.Wechosethetimeinterval
I=(0,1)andconsideramultipleshootingtimedomaindecomposition0=τ0<τ1<τ2=1
ofm=2multipleshootingintervals.
Theconstrainingequationistheheatequationwithnonlinearreactivetermu3:
((∂tu,ϕ))−((u,ϕ))+((u3,ϕ))+(u(0)−0,ϕ(0))=((q,ϕ))∀ϕ∈X.
WehaveH=L2(Ω),V=H01(Ω),R=L2(Ω),andQ=L2(I,L2(Ω)).Intheframework
ofmultipleshootingforthisproblemwedenotes˜:=(s0,s1)andq˜:=(q0,q1),andthe
readsfunctionalcostulatedreform11J¯(s,˜q˜):=1Sj(sj,qj)(t)−u¯(t)2dt+1qj(t)2dt.
j=02Ijj=02Ij
Theintervalwisestateequationsreadforj=0,1
((∂tuj,ϕ))j−((uj,ϕ))j+(((uj)3,ϕ))j+(uj(τj)−sj,ϕ(0))=((qj,ϕ))j∀ϕ∈Xj.(4.36)
Thematchingconditionsaregivenby
(s0−0,ϕ0)=0∀ϕ0∈H,
(s1−S¯0(s0,q0),ϕ1)=0∀ϕ1∈H.
NowwehaveeverythingathandtosetuptheLagrangianoftheproblemwithp˜=(p0,p1),
p0,p1∈H:
11L(q˜,s,˜p˜):=2ISj(sj,qj)(t)−u¯(t)2Hdt
j=0j1+1qj(t)2dt−(s0−0,p0)+(s1−S¯0(s0,q0),p1).
2Ij=0j

42

Thecorrespondingoptimalitysystemreads

4.3TheDirectMultipleShootingApproach

(s0,ϕ)=0∀ϕ∈H,
(s1−S¯0(s0,q0),ϕ)=0∀ϕ∈H,
((S0s0(ψ),S0(s0,q0)−u¯))0−(ψ,p0)+(S¯0s0(ψ),p1)=0∀ψ∈H,
((S1s1(ψ),S1(s1,q1)−u¯))1−(ψ,p1)=0∀ψ∈H,
((S0q0(χ),S0(s0,q0)−u¯))0+((q0,χ))0+(S¯0q0(χ),p1)=0∀χ∈Q0,
((S1q1(χ),S1(s1,q1)−u¯))1+((q1,χ))1=0∀χ∈Q1.
Wecaneasilyverifybydifferentiationoftheintervalwisestateequation(4.36)thatthe
directionalderivativesofthesolutionoperatorsaredeterminedbythefollowingequations.
Forgivenj=0,1,sj∈Handqj∈Qj–andthusgivenuj∈Xj–andgivendirections
ϕ∈Handψ∈Qjthefollowingidentitieshold:
((∂tSjsj(ϕ),v))j+((Sjsj(ϕ),v))j+((3(uj)2Sjsj(ϕ),v))j
+(Sjsj(ϕ)(τj)−ϕ,v(τj))=0∀v∈Xj,
((∂tSjqj(ψ),v))j+((Sjqj(ψ),v))j+((3(uj)2Sjqj(ψ),v))j
+(Sjqj(ψ)(τj),v(τj))=((ψ,v))j∀v∈Xj.
TheseequationscorrespondtotheabstractinterpretationoftheoperatorsinLemma4.3.
Now,theintroductionoftheadjointoperatorsallowsustoreformulatetheoptimalitysystem
ws:folloas(s0,ϕ)=0∀ϕ∈H,
(s1−S¯0(s0,q0),ϕ)=0∀ϕ∈H,
(S0∗s0(S0(s0,q0)−u¯),ψ)−(p0,ψ)+(S¯0∗s0(p1),ψ)=0∀ψ∈H,
)(4.37(S1∗s1(S1(s1,q1)−u¯),ψ)−(p1,ψ)=0∀ψ∈H,
S0∗q0(S0(s0,q0)−u¯),χQ0∗×Q0+((q0,χ))0+(S¯0∗q0(p1),χ)=0∀χ∈Q0,
S1∗q1(S1(s1,q1)−u¯),χQ1∗×Q1+((q1,χ))1=0∀χ∈Q1.
Forthesakeofcompleteness,letusshortlypresenttheequationsthatallowustocalculate
theevaluationsoftheadjointoperators.Letforgivenj=0,1,uj∈Xjandr∈Xj∗,ξ∈H
thefunctionsx∈Xjandy∈Xjsolvethefollowingequationsforallϕ∈Xj:
−((∂tx,ϕ))j+((x,ϕ))j+((3(uj)2x,ϕ))j+(x(τj+1),ϕ(τj+1))=r,ϕXj∗×Xj,
−((∂ty,ϕ))j+((y,ϕ))j+((3(uj)2y,ϕ))j+(y(τj+1)−ξ,ϕ(τj+1))=0.
Fortheadjointoperatorsthefollowingequationshold:
(Sj∗sj(r),ρ)=(x(τj),ρ)∀ρ∈H,
Sj∗qj(r),χQj∗×Qj=(x,χ)Qj∀χ∈Qj,
(S¯j∗sj(ξ),ρ)=(y(τj),ρ)∀ρ∈H,
S¯j∗qj(ξ),χQj∗×Qj=(y,χ)Qj∀χ∈Qj.

43

4TheMultipleShootingApproachforPDEConstrainedOptimization

ApplicationofTheorem4.5finallyyieldsthesimpledirectmultipleshootingformulation
(u0(τ0)−s0,ϕ)=0∀ϕ∈H,
(u0(τ1)−s1,ϕ)=0∀ϕ∈H,
(u1(τ2)−s2,ϕ)=0∀ϕ∈H,
(z0(τ0)−p0,ψ)=0∀ψ∈H,
(z1(τ1)−p1,ψ)=0∀ψ∈H,
(q0,χ)Q0+(z0,χ)=0∀χ∈Q0,
(q1,χ)Q1+(z1,χ)=0∀χ∈Q1,
wherethedualstatesz0andz1andareobtainedfromthefollowingequations:Forallϕ∈X0
1:Xresp.−((∂tz0,ϕ))0+((z0,ϕ))0+((3(u0)2z0,ϕ))0+(z0(τ1)−p1,ϕ(τ1))=((u0−¯u,ϕ))0,
−((∂tz1,ϕ))1+((z1,ϕ))1+((3(u1)2z1,ϕ))1+(z1(τ2)−p2,ϕ(τ2))=((u1−¯u,ϕ))1.

Sofar,optimizationwehaveproblemsdevelopinedtwfunctionodifferenspace.tmWeultiplehaveshodiscuotingssedtheapproachessimilaritiesforPDEandconstraineddifferences
oftimedirectandandspacehaindirectvetombeultiplediscretizedshootingappinroptheriatelypreviousasitissection.discussedFortheinntheumericalnextchaptersolution,.

44

DiscretizationElementFiniteSpace-Time5

Thischapterisdevotedtothediscretizationofstatesandcontrolsintimeandspace.Tobe
precise,wediscussGalerkinfiniteelementdiscretizationswhicharecrucialforthedevelopment
oftheaposteriorierrorestimatorlateron.
InthefirstSection5.1weexplainthediscontinuousGalerkinmethodofdegreer(dG(r)
method)forthetemporaldiscretizationofthestatesoftheintervalwiseproblems(4.8).
Thereafter,Section5.2pointsoutthespacediscretizationofthesemidiscreteproblems
obtainedbythepreviouslymentionedtimediscretization.ItisfollowedbySection5.3which
dealsontheonehandwiththediscretizationofthemultipleshootingvariablesinSubsection
5.3.1andontheotherhandwiththetime-spacediscretizationofthestates.Inthiscontext,
weconsiderdynamicallychangingmeshes(cf.Subsection5.3.2)andintervalwiseconstant
meshes(cf.Subsection5.3.3).Furthermore,Section5.4dealsshortlyandratherabstract
withthediscretizationofthecontrolspaceQ.Finally,Section5.5specifiesaconcretetime
steppingschemewhichwasusedfortheimplementation.
Remark5.1.(Discretizationfordirectandindirectmultipleshooting)Fromtheprevious
chapter,andespeciallyFigure4.1,wehaveseenhowtheequationsofdirectandindirect
approacharecloselyrelated.Infact,theequationsoftheoptimalitysystem(4.8)ofthe
indirectapproachcovertheequationsofthedirectmultipleshootingapproach.Therefore,
weconsideronlythediscretizationoftheindirectmultipleshootingapproachinthefollowing
tation.presenRemark5.2.(Commutabilityofdiscretizationandoptimization)Galerkindiscretization
schemeshavethepleasantpropertythatdiscretizationanddualizationinterchange.Therefore,
weareallowedtoconsiderthediscretizationoftheoptimalitysystem(4.8)directlyinstead
ofdiscretizingtheoptimalcontrolproblem(4.2)firstandcalculatingthediscreteoptimality
systemafterwards.Inotherwords,forGalerkindiscretizationsthefirst-optimize-then-
discretizeapproachisequivalenttothefirst-discretize-then-optimizeapproach.

DiscretizationTime5.1

InthissectionweexplainthediscontinuousGalerkinmethodofdegreer(dG(r)method)
whichisdefinedbytheuseofdiscontinuoustestandtrialfunctionsofdegreer.Adetailed
derivationanddescriptionofdiscontinuoustimediscretizationmethodscanbefoundinthe
textbookofEriksson,Estep,Hansbo,andJohnson[17].Forthefollowingdescriptionofthe
dG(r)method,weneedsomepreliminariesandnotationalframework.First,letuschosea
partitionoftheclosureofthetimeintervalI¯j=[τj,τj+1]:
I¯j={τj}∪Ij,1∪Ij,2∙∙∙∪Ij,nj−1∪Ij,nj,

45

5Space-TimeFiniteElementDiscretization

withIj,l=(tj,l−1,tj,l],
τj=tj,0<tj,1<∙∙∙<tj,nj−1<tj,nj=τj+1
andkj,l:=|Ij,l|.Thediscretizationparameterisdenotedbyk,andisapiecewiseconstant
,ljfunctiondefinedthroughthelengthofthesubintervalskI:=kj,l.
Withthesubintervalsathand,letusdefinethefollowingspaces,whichweneedforthe
definitionofthetrialandtestfunctionsinthefollowingsections.LetPr(Ij,l,V)denotethe
spaceofpolynomialsonIj,luptoorderrwithvaluesinV.Wedefinethespacesofpiecewise
polynomialsasfollows:
X˜jrk:=vkj∈L2(Ij,H)vkjIj,l∈Pr(Ij,l,V),l=1,...njandvkj(τj)∈H.
Remark5.3.WeshouldremarkthatX˜kj,r⊂Xj,becausethepiecewisedefinedfunctionsof
thisspacearenotnecessarilycontinuous.
ThediscontinuousGalerkinmethodofdegreercalculatesapiecewisepolynomialsolutionin
thespaceX˜kj,r.Thediscontinuitiesofthefunctionsvkj∈X˜kj,rontheinteriortimenodesof
Ijareconsideredbymeansofthefollowingnotation:
vkj,l+:=limvkj(tj,l+t),vkj,l−:=limvkj(tj,l+t)=vkj(tj,l),[vkj]l:=vjk+,l−vkj,l−.
t0t0
ForexampleforthedG(0)methodthisnotationcanbeinterpretedasfollows:vkj,l+isthe
limit“fromabove”,vkj,l−isthelimit“frombelow”and[vkj]listhe“jump”invkj(t)atthe
nodetj,l.WeillustratethisinFigure5.1.WecannowformulatethedG(r)methodforthe
+jv,lk

[vkj]lvkj
−jv,lk

Ij,l−1Ij,l
tj,l−1tj,ltj,l+1
Figure5.1:NotationforthedG(0)method.
jdiscretizationjofjthein˜j,rtervalwisestateequations(4.16)asfollows:Findforgivencontrol
qk∈Qastateuk∈Xksuchthat
((∂tukj,ϕk))j,l+a(ukj)(ϕk)+b(qkj)(ϕk)+([ukj]l,ϕk+,l)+(ukj,−0,ϕk−,0)
njnj−1
=0l=1l=((f,ϕk))j+(sj,ϕk−,0)∀ϕk∈X˜kj,r,(5.1)

46

DiscretizationTime5.1

where((v,w))j,l:=Ij,l(v(t),w(t))dt.
Existenceanduniquenessofthetimediscretesolutionukj∈X˜kj,rof(5.1)isprovedinthe
bookofThomee[40].Inthisbook,uniquenessisshownbystandardarguments,andthe
existenceofasolutionisconcludedbyreducing(5.1)toafinitedimensionalproblemby
applicationofaneigenspacedecompositionoftheoperatorAasdefinedin(2.2).
Remark5.4.ThedG(r)approachasstatedin(5.1)incorporatestheinitialvalueintothe
definitionofX˜kj,r.Amorecommon,equivalentdescription,likein[40]and[17],eliminates
theinitialvaluefromthedefinitionofX˜kj,r:
njnj−1
((∂tukj,ϕk))j,l+a(ukj)(ϕk)+b(qkj)(ϕk)+([ukj]l,ϕk+,l)+(ukj,+0,ϕk+,0)
=1l=1l=((f,ϕk))j+(sj,ϕk+,0)∀ϕk∈X˜kj,r.
Theequivalenceofbothschemescanbeshownbyelementarycalculus.
Considerationofthetimediscretizationoftherestrictedoptimalitysystem(4.8)yieldsthe
equations:discretesemiwingfolloequation:Primal((∂tukj,ϕk))j,l+a(ukj)(ϕk)+b(qkj)(ϕk)+([ukj]l,ϕk+,l)+(ukj,−0,ϕk−,0)
njnj−1
=0l=1l=((f,ϕk))j+(sj,ϕk−,0)∀ϕk∈X˜kj,r.(5.2a)
equation:Dualnjnj−1
−((∂tzkj,ψk))j,l+au(ukj)(ψk,zkj)−([zkj]l,ψk−,l)+(zkj,n−j,ψk−,nj)
=0l=1l=α1J1j(ukj)(ψk)+(λj+1,ψk−,nj)∀ψk∈X˜kj,r.(5.2b)
equation:trolConbq(qjk)(χk,zkj)−α3(qkj,χk)Qj=0∀χk∈Qj.(5.2c)
Remark5.5.ThisratherabstractdiscreteformulationforthegeneralcaseofthedG(r)
discretizationmethodisspecifiedinSection5.5forr=0whichistheimplicitEulertime
heme.cssteppingRemark5.6.(Evaluationoftheintegrals)Theevaluationoftheintegralsofprimalanddual
equationsisdonebyapplicationofaquadratureformula,forexampletheboxrule.
Remark5.7.(Semidiscreteformulationof(4.5))Forpurposeoflateruse,letusshortly
discuss,howthesemidiscretizationofthenonrestrictedproblemonthewholetimeinterval
canbeformulatedintermsoftheaboveintroducednotation.Letusdenotetheintervalwise
restrictionofthestatesandcontrolsasbefore,andletusaccordinglydenotethetimediscrete
spaceonthetimeintervalIby
,ljX˜kr:=vk∈L2(I,H)vkI∈Pr(Ij,l,V),j=0,...m−1,l=1,...nj,andvk(0)∈H.

47

5Space-TimeFiniteElementDiscretization

System(4.5)readsintimediscreteformulationasfollows:
equation:Primalm−1njnj−1
j=0l=1l=1
((∂tukj,ϕkj))j,l+a(ukj)(ϕkj)+b(qkj)(ϕkj)+([ukj]l,ϕkj,l+)
2−m+([uk0]0,ϕk,0+0)+(uk(,j0+1)+−ukj−,nj,ϕk(,j0+1)+)+(uk0,−0,ϕk0,−0)
=0j=((f,ϕkj))+(u0,ϕk0,−0)∀ϕkj∈X˜kr.(5.3a)
equation:Dualm−1njnj−1
j=0l=1l=1
−((∂tzkj,ψkj))j,l+au(ukj)(ψkj,zkj)−([zkj]l,ψkj,l−)
2−m−([zk0]0,ψk0,−0)−(zk(,j0+1)+−zkj,n−,ψkj,−0)+(zk(,nm−1)−,ψk(,nm−1)−)
j=0jm−1m−1
1−m=α1J1j(ukj)(ψkj)+α2J2(um−1(T)(ψk(,nmm−−1)1−)∀ψk∈X˜kr.(5.3b)
=0jControlequation:
bq(qk)(χk,zk)−(qk,χk)Q=0∀χk∈Q.(5.3c)
InthenextsectionwedevelopthespatialdiscretizationofthecontinuousspacesVandH
whichstillremaininthedefinitionofthesemidiscretespacesX˜jk,r.

DiscretizationSpace5.2

ThespatialsdiscretizationoftheHilbertspaceVisdonebydefiningfinitedimensional
maysubspacesdifferVhfor⊆eacVhinconsistingtervalIjof,l,finitewhichwelementsetakeofintomaximalaccounortderbys.addingThespatialadditionaldiscindicesretizationsj
anandlinarbitrarythediscretenotationofspacetheVs⊆discreteVforspaceagivenwhenevertriangulation.necessary.Fornow,wemerelyconsider
hdInandthisdevelopsectionweappropriateintroducefinitetheelementtriangulationspacesofonthethisboundedtriangulation.spatialWedomainassumeΩ⊂theRb,d=oundary2,3,
ofbuttheisfordomain,example∂Ω,toderivbeedofpintheolygonalbookshapofe.BraessThe[12general].Depcaseisendingnotontheconsidereddimensioninthisofthesisthe
space,wedecomposethedomainΩintoquadrilaterals(ford=2)orhexalaterals(ford=3)
denotedbyKwhichcoverthewholedomainΩ.Thecorrespondingtriangulationislabeled
Th={K},wheretheparameterhisacellwiseconstantfunctionandgivesinformationon
thediameterofthecurrentcell,hK=hK:=diamK.
Amathematicalformulationofthiscontextisgivenin[12]intermsofthefollowingdefinition:

48

DiscretizationSpace5.2

Definition5.1.(Regulartriangulation)AtriangulationTh={K}ofΩiscalledregularif
thefollowingpropertieshold:
1.Ω¯=K∈Th.
2.IffortwocellsK1andK2,K1=K2,theintersectionK1∩K2=p,p∈Ω,thenpisa
verticeofbothcellsK1andK2.
3.IffortwocellsK1andK2,K1=K2,theintersectionK1∩K2consistsofmorethan
onepoint,thenK1∩K2isafaceofbothcellsK1andK2.
Remark5.8.TheparameterhisoftenconsideredasthemaximumdiameterofallcellsinTh,
thatish:=maxK∈ThdiamK.

Weareespeciallyinterestedinfamiliesoftriangulationsarisingfromthesuccessiverefinement
ofcells.Therefore,werequestthefollowingpropertiesforthetriangulations:
Definition5.2.(Quasiuniformfamilyoftriangulations)Afamilyofregulartriangulations
{Th}withh0iscalledquasiuniform,ifthereexistκ>0suchthateachK∈hTh
containsacircleofradiusρKwith
ρK≥hK.
κDefinition5.3.(Uniformfamilyoftriangulations)Afamilyoftriangulations{Th}with
h0iscalleduniform,ifthereexistκ>0suchthateachK∈hThcontainsacircleof
withρradiusKh,ρK≥κwherehhastobeinterpretedinthesenseofRemark5.8.
Remark5.9.Thedefinitionsofquasiuniformityanduniformityabovearemerelyusedfor
theoreticalpurposewhenconsideringapproximationpropertiesofthefiniteelementspaces
andperformingapriorierroranalysis.Wedonotpersecutethistheoreticalinvestigations,
butrefertotheliteraturementionedbefore.Nevertheless,ourtriangulationsfulfillquasi
practice.inyuniformit

ForthepurposeoflocalrefinementwehavetoweakenthelastconditioninDefinition5.1.
Weallowhangingnodes,thatarenodesthatlieinthemiddleofafaceofacellK,and
assumefurthermorethatthetriangulationThisobtainedfromthetriangulationT2hby
patchwiserefinement.Apatchdenotesasetoffourcells(ford=2)oreightcells(ford=3)
obtainedbyrefinementofacoarsercell.Thispatchwiseorganizationisofimportancewhen
weconsiderpatchwiseinterpolationforhigherorderapproximationoftheexactsolution.
ThisisneededfortheevaluationofaposteriorierrorestimatorsinChapter7.Anexample
foratriangulationwithpatchwiseorganizationofthecellsisgiveninFigure5.2.
Inthecontextoffiniteelements,weconsiderthecellsofatriangulationastransformations
ofareferencecellKˆ,forexampletheunitsquareinthecaseofquadrilaterals.Wedenote

49

5Space-TimeFiniteElementDiscretization

hh2TTFigure5.2:Triangulationwithpatchwisecellorganization(right)obtainedfromacoarser
triangulation(left)byglobalrefinement.

thetransformationthatmapsthereferencecellontothecurrentcellKbyϕK:Kˆ→K,see
Figure5.3,anddefinethepolynomialspaceonthereferencecellKˆby
d
=1kQˆs(Kˆ):=spanxkαkαk∈{0,1,...,s}.
ThenthecorrespondingspaceofpolynomialsonKisdefinedbyuseofthetransformationϕ:
Qs(K):=vv◦ϕ∈Qˆs(Kˆ).

ˆKK

Figure5.3:Transformationofthereferencecellintoanarbitrarycell.

Thefiniteelementspaceconsistsofcontinuouspiecewisepolynomialfunctionsoforderupto
s,thatis
KVhs:=vh∈Vvh∈Qs(K),K∈Th.
Remark5.10.Theadmissionofhangingnodesleadstoproblemswithrespecttothecontinuity
onthefaces.Inordertomakeafiniteelementfunctiongloballycontinuous,wehavetomake
surethatthehangingnodeshavevaluesthatarecompatiblewiththeadjacentnodeson
thevertices,suchthatthefunctionhasnojumpwhencomingfromtherefinedcellstothe
adjacentcoarsecell.Therefore,weeliminatethedegreesoffreedomcorrespondingtothe
hangingnodesanddeterminethevaluesatthehangingnodesfromtheadjacentdegreesof
freedombyaninterpolationprocedureafterthesolutionprocess.

50

5.3DiscretizationofTimeandSpace

5.3DiscretizationofTimeandSpace

InthissectionwefinallymergethediscretizationschemesfortimeandspacefromSections
5.1and5.2inordertoobtainatime-spacediscreteformulationofproblem(4.8)withfully
discretizedstates.Wepresenttwodifferentdiscretizationschemes,oneofwhichhasbeen
proventobemoreefficientwithrespecttotheimplementationandthecomputationaleffort.
First,weconsiderthediscretizationofthemultipleshootingvariables.Thiscanbedone
independentlyfromthediscretizationofthestatesontheintervalsandoffersalargevarietyof
possibilitiesforthechoiceofthemesh.Second,wepresentthediscretizationwithdynamically
changingmeshesineverytimestepwhichareonlyrelatedbyacommoncoarsegrid.And
finally,aspecialcaseofthedynamicallychangingmeshes,namelythecaseofintervalwise
constantmeshes,isderived.

5.3.1DiscretizationoftheMultipleShootingVariables

Theintervalwiseconsiderationofthetimespacediscreteintervalwiseproblemspossiblyyields
differentadjacentspatialmeshesonthemultipleshootingnodes.Thisisduetothefact,
thatthelocalmeshsizeofThj,lisdeterminedbyrefinementindicatorsontheintervalIj.
Therefore,thefinalmeshofIjandthefirstmeshofIj+1areingeneraldifferentandonly
relatedbyacommoncoarsegrid.Asaconsequence,wehaveacertainfreedomtochoosethe
discretizationfortheshootingvariablesconnectingthetwo.Inparticular,weendupwith
differentdiscretizationsofHinτjeachshootingnodeτj.WedenotethesespacesbyHhjand
thecontinucorrespousondingGalerkinmeshesapproacbyhThof.degreeThes.spatialThespatialdiscretizationdiscreteis,asmbultipleefore,shopeotingrformedvwithariablesa
aredenotedbysjh,λhj∈Hhj.
Remark5.11.(Choiceofthenodemeshes)WhilethemeshusedforHjcanbechosen
independentoftheneighboringintervals,weusethetracemeshofeitherthehleftintervalor
therightforpracticalpurposesinthesectiononnumericalexperiments.

MeshesSpatialChangingDynamically5.3.2

Aspatialdiscretizationwithdynamicallychangingmeshesintimesuggestsitself.The
triangulationcorrespondingtothetimepointtj,lisdenotedbyThj,landtheappropriatefinite
elementspacebyVhj,l,sasillustratedinFigure5.4.
Withthisnotationathand,weareabletodefinethefollowingfullydiscretespaces:
I,ljX˜kjh,r,s:=vkh∈L2(Ij,H)vkh∈Pr(Ij,l,Vhj,l,s),l=1,...nj,andvkh(τj)∈Vhj,0,s.
ItcandirectlybeseenthatduetotheconformityVhj,l,s⊆VtheinclusionX˜kjh,r,s⊆X˜kj,rholds
forthefullydiscretespace.Fordiscretizationsofthistype,thenotationcG(s)dG(r)hasbeen
introducedin[17]becauseofthecG(s)discretizationinspaceandthedG(r)discretization
intime.ThecG(s)dG(r)formulationofproblem(4.16)isconsequentlyobtainedfromthe

51

5Space-TimeFiniteElementDiscretization

Thj,l−1

+1,ljT,l−1Thj,lh
I+1,ljIj,lttj,l+1
,ljt1−,ljFigure5.4:Dynamicallychangingspatialmeshesintime.

semidiscreteformulation(5.1)byreplacingthespaceX˜kj,r,sbyitsfullydiscreteequivalent
X˜kjh,r,sandaddinganindexhtothevariablesandtestfunctions:
Findforgivencontrolqkjh∈Qjastateukjh∈X˜kjh,r,ssuchthat
((∂tukh,ϕkh))j,l+a(ukh)(ϕkh)+b(qkh)(ϕkh)+([ukh]l,ϕkh,l)+(ukh,0,ϕkh,0)
njjjjnj−1j+j−−
=0l=1l=((f,ϕkh))j+(shj,ϕk−h,0)∀ϕkh∈X˜kjh,r,s.
Bythesameprocedure,weobtainthecG(s)dG(r)formulationfromthesemidiscreteopti-
(5.2):ystemsymalitequation:Primal((∂tukh,ϕkh))j,l+a(ukh)(ϕkh)+b(qkh)(ϕkh)+([ukh]l,ϕk+h,l)+(ukh,0,ϕk−h,0)
njjjjnj−1jj−
l=1l=0jj,r,s
=((f,ϕkh))j+(sh,ϕk−h,0)∀ϕkh∈X˜kh.(5.4a)
equation:Dualnjnj−1
−((∂tzkjh,ψkh))j,l+au(ukjh)(ψkh,zkjh)−([zkjh]l,ψk−h,l)+(zkj−h,nj,ψk−h,nj)
l=1m−1l=0
=α1J1j(ukjh)(ψkh)+(λhj+1,ψk−h,nj)∀ψkh∈X˜kjh,r,s.(5.4b)
=0jation:quetrolCon

52

jjjbq(qkh)(χkh,zkh)−α3(qkh,χkh)Qj=0∀χkh∈Qj.

(5.4c)

5.3DiscretizationofTimeandSpace

Remark5.12.(Time-spacediscreteformulationof(4.5))Tocompletethispresentationand
forpurposeoflaterusage,thetimespacediscreteformulationofproblem(4.5)isgainedby
thesameprocedureappliedtothesemidiscretesystem(5.3).Weintroducethefullydiscrete
spaceonthetimeintervalIby
I,ljX˜kr,hs:=vk∈L2(I,H)vk∈Pr(Ij,l,Vhj,l,s),j=0,...m−1,l=1,...nj,vk(0)∈Vhj,0,s
andwritetheoptimalitysystemintime-spacediscreteformulationasfollows:
equation:Primalm−1njnj−1
j=0l=1l=1
((∂tukjh,ϕkjh))j,l+a(ukjh)(ϕkjh)+b(qkjh)(ϕkjh)+([ukjh]l,ϕkj+h,l)
2−m+([uk0h]0,ϕk0+h,0)+(uk(jh,0+1)+−ukj−h,nj,ϕk(jh,0+1)+)+(uk0h,−0,ϕk0h,−0)
=0j=((f,ϕkjh))+(u0,ϕk0h,−0)∀ϕkjh∈X˜krh,s.(5.5a)
equation:Dualm−1njnj−1
j=0l=1l=1
−((∂tzkjh,ψkjh))j,l+au(ukjh)(ψkjh,zkjh)−([zkjh]l,ψkj−h,l)
2−m−([zk0h]0,ψk0h,−0)−(zk(jh,0+1)+−zkj−h,nj,ψkjh,−0)+(zk(mh,n−m1)−−1,ψk(mh,n−m1)−−1)
=0j1−m=α1J1j(ukjh)(ψkjh)+α2J2(um−1(T)(ψ(kmh,n−m1)−−1)∀ψkh∈X˜krh,s.(5.5b)
=0jequation:trolConbq(qkh)(χkh,zkh)−(qkh,χkh)Q=0∀χkh∈Q.(5.5c)
Remark5.13.(Discretematchingconditions)Thediscretematchingconditionsarefinally,
underconsiderationofRemark5.11,givenas
(sh0−u0,vh)=0∀vh∈Hh0,
(shj+1−ukj−h,nj(τj+1),vh)=0∀vh∈Hh(j+1),j=0,...,m−1,
(λhj−zkjh,−0(τj),vh)=0∀vh∈Hhj,j=0,...,m−1,
(λhm,vh)−α2J2(shm)(vh)=0∀vh∈Hhm.
Remark5.14.(Efficiency)Theapplicationofdynamicallychangingmeshesformultiple
shootingmethodsisverytimeconsuminginpractice.Thenumberofmatrixassemblationsis
extremelylarge,andmostofthecalculationtimeisthusspentbycalculatingthesystem
matricesandright-handsides.Apossibleimprovementistheconsiderationofintervalwise
constantmeshesaspresentedinthefollowingsubsection.

53

5Space-TimeFiniteElementDiscretization

MeshesSpatialConstantIntervalwise5.3.3cThehoiceofspatialdynamicallydiscretizationchanginwithgintervmeshes.alwiseAsconstandepictedtinmeshesFigureisa5.5,promisingthespatialalternativmesheistothethe
sameforalltimestepsofonemultipleshootinginterval.
Thj−1ThjThj+1

Ij−1τjIjτj+1Ij+1
Figure5.5:Intervalwiseconstantmeshesintime.

Ontheonehand,thissimplifiesthenotationbecausewenolongerneedthetimestepindex
lforthediscretizationofV,ontheotherhandweseelateron,thatthisideaprovidesusa
highlyefficientframeworkfortheimplementationandsolutionofthediscretizedproblems.
Asthisapproachisonlyaspecialcaseofthegeneralideaofdynamicallychangingmeshes,the
formulationofthediscretizedintervalwiseoptimalitysystemsandthematchingconditionsis
thesameasintheprevioussubsection.

ControlstheofDiscretization5.4jXInjwtheaspreviousdiscretizedinsections,timetheandconspacetrolbyspacetheQcG(hass)bdGe(enr)keptmethod.Inundiscretizedthewsequel,hilewetheshortlyspace
discusspossiblediscretizations,andthediscretizationmethodforQjisillustratedbymeans
oftheExampleoptimalcon2.3.trolBeforeproblemweproispceedossible,weshouldwithoutremarkdiscretizithatngtheaccordingcontroltospace,[22]theevenifsolutionitisofof
dimension.infiniteThecontrolspacesQjhavebeenintroducedbytheinclusion
Qj⊆L2(Ij,R)
withaHilbertspaceR.Concerningthecontrol,wegenerallychosethetimediscretizationof
themesh.conFtrolorthethespacesameasfordiscretizationthestates,ofRthatweisaallowdG(ar)cG(p)discretizationdiscretizationonthemethosamedoftemploworaler

54

5.4DiscretizationoftheControls

degreethanchosenforthestates,thatisp≤s.Altogether,wechoseafinitedimensional
subspaceQdj⊆QjwhichisdefinedthroughacG(p)dG(r)discretizationmethod.Inthe
followingwedenotethespatialdiscreteequivalentofRintj,lbyRhj,l,p.Incombinationwith
thediscretizationofthestates,weobtainafullydiscreteformulationoftheoptimization
problem.OnthebasisofExample2.3wecanseethatthischoiceisreasonable:
Example5.1.(DiscretizationofQjforExample2.3)Inthisexamplethecontrolspaceis
ybengivR=L2(Ω)andthusQj=L2(Ij,L2(Ω)),
whereasthespaceforthestateswasdefinedthrough
H=L2(Ω)andV=H01(Ω).
Fromthecontrolequationin(4.14)weretrievetherelationbetweenthedualstatezjand
thecontrolqjas
((q,χ))j=−α1((z,χ))j∀χ∈Qj.
3Fromthiscontext,itisconsequentialtochosethediscretizationofthecontroleitherthe
sameasforthestatezjoracoarseroneoradiscretizationoflowerdegree.Thechoiceof
acoarsertriangulationinspaceortimeisnotconsideredinthisthesis.Nevertheless,the
extensionofthehereinmentionedmultipleshootinganderrorestimationmethodstothese
ard.tforwstraighiscasesOurdiscretizationsQdjofQj,aspresentedforExample2.3,alwaysfulfillconformity,such
thatthefullydiscreterestrictedoptimizationproblemcanbederivedstraightforwardfrom
(5.4)byaddinganadditionaldiscretizationindextothestates,controlsandthecontrol
space.FollowingthenotationofMeidner[29],wedenotethefullydiscretevariablesby
qkjhd,ukjhd,zkjhdandabbreviatetheindexkhdbyσ.Withthisnotation,thefullydiscretized
stateequationreadsasfollows:
Findforgivencontrolqσj∈Qdjastateuσj∈X˜kjh,r,ssuchthat
((∂tuσj,ϕσ))j,l+a(uσj)(ϕσ)+b(qσj)(ϕσ)+([uσj]l,ϕσ+,l)+(uσj,−0,ϕσ−,0)
njnj−1
=1l=0l=((f,ϕσ))j+(shj,ϕσ−,0)∀ϕσ∈X˜jk,rh,s,
andanalogouslyweobtainthefullydiscreteformulationoftheoptimalitysystem(5.2):
equation:Primalnjnj−1
((∂tuσj,ϕσ))j,l+a(uσj)(ϕσ)+b(qσj)(ϕσ)+([uσj]l,ϕσ+,l)+(uσj,−0,ϕσ−,0)
=0l=1l=((f,ϕσ))j+(shj,ϕσ−,0)∀ϕσ∈X˜kjh,r,s.(5.6a)

55

5Space-TimeFiniteElementDiscretization

equation:Dualnjnj−1
−((∂tzσj,ψσ))j,l+au(uσj)(ψσ,zσj)−([zσj]l,ψσ−,l)+(zσj,n−j,ψσ−,nj)
=0l=1ljj+1j,r,s
=α1J1(uσj)(ψσ)+(λh,ψσ−,nj)∀ψσ∈X˜kh.(5.6b)
Controlequation:
jbq(qσj)(χσ,zσj)−α3(qσj,χσ)Qj=0∀χσ∈Qd.(5.6c)
Remark5.15.(Fullydiscreteformulationof(4.5))Thefullydiscreteformulationof(4.5)is
derivedanalogouslyfrom(5.5):
equation:Primal+jm−1njnj−1
j=0l=1l=1
((∂tuσj,ϕσj))j,l+a(uσj)(ϕσj)+b(qσj)(ϕσj)+([uσj]l,ϕσ,l)
2−m+([uσ0]0,ϕσ,0+0)+(uσ(,j0+1)+−uσj−,nj,ϕσ(,j0+1)+)+(uσ0,−0,ϕσ0,−0)
=0j=((f,ϕjσ))+(u0,ϕσ0,−0)∀ϕσj∈X˜σr,s.(5.7a)
equation:Dualm−1njnj−1
−((∂tzσj,ψσj))j,l+au(uσj)(ψσj,zσj)−([zσj]l,ψjσ,l−)
j=0l=1l=1
m−2(j+1)+
−([zσ0]0,ψσ0,−0)−(zσ,0−zσj,n−j,ψσj,−0)+(zσ(,nmm−−1)1−,ψσ(,nmm−−1)1−)
=0j1−m=α1J1j(uσj)(ψσj)+α2J2(um−1(T)(ψσ(,nm−1)−)∀ψσ∈X˜σr,s.(5.7b)
1−m=0jControlequation:
bq(qσ)(χσ,zσ)−(qσ,χσ)Q=0∀χσ∈Qd.(5.7c)
Tocompletethissectiononthefullydiscreteformulation,westatethematchingconditions
forthefullydiscretecaseinthefollowingremark.
Remark5.16.(Fullydiscreteformulationofthematchingconditions)Thediscretematching
conditionsarefinallygivenas
(sh0−u0,vh)=0∀vh∈Hh0,
(shj+1−uσj−,nj(τj+1),vh)=0∀vh∈H(hj+1),j=0,...,m−1,
(5.8)(λhj−zσj,−0(τj),vh)=0∀vh∈Hhj,j=0,...,m−1,
(λhm,vh)−α2J2(shm)(vh)=0∀vh∈Hhm.

56

5.5TheImplicitEulerTimeSteppingScheme

Remark5.17.(Notation)Inviewofthesubsequentchapters,weintroduceanadditional
notationfortheCartesianproductofdiscretespacesasgivenbelow:
X˜kh:=X˜k0h,r,s×∙∙∙×X˜kmh−1,r,s,
Q˜d:=Qd0×∙∙∙×Qdm−1,
H˜h:=Hh0×∙∙∙×Hhm.
5.5TheImplicitEulerTimeSteppingScheme

Inthissection,weshortlydescribeaconcretetimediscretizationmethodforr=0,whichwe
usethroughoutourcomputations.ThetimesteppingschemeforthecG(s)dG(0)method
correspondstotheimplicitEulerschemeaspresentedinthesequel,whereweapproximate
thearisingintegralsbytheboxrule.Theparameterchoiceofr=0meanstohave
piecewiseconstantfunctionsintimesuchthatitisstraightforwardtointroducethefollowing
notation:alwisetervsubinQσj,l:=qσj,l−,Uσj,l:=uσj,l−,Zσj,l−:=zσj,l−.(5.9)
WeremarkthatthefunctionalJ1wasdefinedsuchthatanintervalwisesplittingaccording
toequation(2.5)ispossible.
ThecG(s)dG(0)schemeforproblem(4.8)infullydiscreteformulationisstatedasfollows:
forallϕ∈Vhj,l,s,χ∈Rhj,l,pletthefollowingequationsbefulfilled
equation:Primal:0=l(U0j,ϕ)=(sσj,ϕ),
l=1,...,nj:
(Uσj,l,ϕ)+kj,la¯(Uσj,l)(ϕ)+kj,lb¯(Qσj,l)(ϕ)=(Uσj,l−1,ϕ)+kj,l(f(tj,l),ϕ),
equation:Dual:n=lj(Znjj,ϕ)=(λσj+1,ϕ),
l=1,...,nj−1:
(Zσj,l,ϕ)+kj,la¯u(Uσj,l)(ϕ,Zσj,l)=(Zσj,l+1,ϕ)+α1kj,lF(Uσj,l)(ϕ),
:0=l(Z0j,ϕ)=(Z1j,ϕ),
equation:trolConb¯q(Qjσ,l)(χ,Zσj,l)=α3(Qσj,l,χ)R.
Weproceedwiththeinvestigationofsolutiontechniquesforthemultipleshootingproblem
hapter.cnextthein

57

6SolutionTechniquesfortheMultiple
roachAppotingSho

Thischapterisdevotedtothedevelopmentanddiscussionofdifferentsolutiontechniques
forthemultipleshootingapproach.Multipleshootingaspresentedinthepreviouschapters
exploitstheideaofapplyingstandarditerativesolverstothesystemofequations,treatingthe
routine.underlyingInbthisouncdaryhapter,orweinitialderivvaleueforbproblemsothdirectassolvandedinbydireact(notmyetultipleconcshoretized)otingfirstsolutionthe
iterativesolversforthesystemofequationsandsecondthesolutiontechniquesforthe
boundaryandinitialvalueproblems.
Consequently,thefirstSection6.1dealswiththesolutiontechniquesfortheindirectmultiple
shootingapproach.Firstofall,thesolutionofthediscretizedsystemofmatchingconditions
(5.8)byNewton’smethodisconsideredinSubsection6.1.1.Inthiscontext,thelinearized
asystemKrylovissubderivspaceed,andmethoadshortisinpsveudoestigatedmatrixinthenotationsubsequenisinttroSduced.ubsectionIts6.1.2.solutionThebymeansnecessitofy
ofapreconditionerisoutlinedbynumericalexamples,andconsequentlythreedifferent
preconditionersarepresentedinthesequel.Itwillturnout,thatmostofthecomputational
effortproblems.isspThendus,inthedifferentsolutionsolutionoftectheinhniquestervforalwisetheinnonlineartervalwiseandlineproblemsarbareoundarydiscussedvalue
bintheoundaryvSubsectionsalueproblems6.1.3,by6.1.4,applicandation6.1.6.ofWeinNewton’svestigatemethothedonideatheofwholesolvingintervthealnonlinearproblem
in6.1.3andlineoutthelimitationsofthisapproachinSubsection6.1.5.
Section6.2isengagedwiththesolutiontechniquesforthedirectmultipleshootingapproach.
WestartwithananalogousdiscussionofNewton’smethodforthesolutionofthediscrete
ofanonlinearpreconditionedsystemofmatcKrylohingvsubspaceconditionsinmethoSdubsectionforthe6.2.1solutionandproofceedthewithlineariztheedinvsystemestigationin
isinSubsectiontroduced6.2.2.inFSubsectionurthermore,6.2.an3.Thisefficientprocedurecondensingallotecwsahniqueremarforkabthelelinearizedreductionofsystemthe
solutionspace.ThecondensingtechniqueissimilartotheODEcondensingapproach,and
approacconsequenhtotlywPDEsedevisotesubSubjectsecto.tion6.2.4totherestrictionswhichthetransferoftheODE
byFinallymeans,weofnclosethisumericalchapterexampleswithinthenSectionumerical6.3.comparisonofthedifferentsolutiontechniques
Remark6.1.(Notation)Throughoutthischapterweconsiderwithafewexceptionsthe
fullysolutionopdiscretizederatorsproblemwithinandhaveaccortobdeinglyunderstotheofullydasdiscrethoseteofmatcthehingdiscretizedconditions.problems.Thus,theAs

59

6SolutionTechniquesfortheMultipleShootingApproach

dofromnotthedenoteargumenthetsgivdiscreteentothesolutionopsolutioneratorsoperators,differentlytherefromisnothewayconoftinuousmisunderstandones.ingwe

6.1SolutionTechniquesfortheIndirectMultipleShooting
roachApp

Thissectiondealswiththedevelopmentoftwodifferentsolutiontechniquesfortheindirect
multipleshootingapproach.BothideasstartwiththeapplicationofNewton’smethodtothe
thesystemsolutionofmatcofthehinglinearizedconditionssystem.andtheapNewton’splicationmethoofdaniterrequiresativetheKevrylovaluationsubspofacethemethoresdidual,for
andleft-handthesideiterativofethelinearlinearsolverproblem.needstheClearlycalculation,theliofnethearsolverdirectionaldependsderivonativtheesformingsolutiontheof
intervalwiselinearboundaryvalueproblems(BVPs)whiletheevaluationoftheresidual
skneedsetchedtheinsolutionFigureof6.1.nonlinearproblems.Theinterrelationofthedifferentsubproblemsis

hingMatcConditionsNewton

SystemLinearized

BVPNonlinearBVPLinear

ResidualesativDerivDirectional

Figure6.1:Nestingofthedifferentsolvers.

6.1.1SolutionoftheMultipleShootingSystem
TheconsiderationofNewton’smethodasasolverforthesystemofmatchingconditions(5.8)
yieldsthefollowinglinearsystemforthecalculationoftheNewtonincrement(Δs˜h,Δλ˜h):
Find(Δs˜h,Δλ˜h)∈H˜h×H˜hsuchthatforallj=0,...,m−1
(Δsh0,v)=−(sh0−u0,v)∀v∈Hh0,
(Δshj+1−Σ¯uj,sj(Δshj)−Σ¯uj,λj+1(Δλhj+1),v)=−(shj+1−Σ¯uj(shj,λhj+1),v)∀v∈Hhj+1,
(Δλhj−Σ¯zj,sj(Δshj)−Σ¯zj,λj+1(Δλhj+1),v)=−(λhj−Σ¯zj(shj,λhj+1),v)∀v∈Hhj,
(Δλhm,v)−α2J2(shm)(v,Δshm)=−(λhm,v)−α2J2(shm)(v)∀v∈Hhm,
(6.1)

60

6.1SolutionTechniquesfortheIndirectMultipleShootingApproach

forwhichweusedtheabbreviations
Σ¯uj,sj(Δshj):=Σ¯uj,sj(shj,λhj+1)(Δshj),
Σ¯uj,λj+1(Δλhj+1):=Σ¯uj,λj+1(shj,λhj+1)(Δλhj+1),
Σ¯zj,sj(Δshj):=Σ¯zj,sj(shj,λhj+1)(Δshj),
Σ¯zj,λj+1(Δλhj+1):=Σ¯zj,λj+1(shj,λhj+1)(Δλhj+1).
ThedirectionalderivativesofthesolutionoperatorscanbeinterpretedintermsofLemma
6.1,andtheirdiscretizationsareobtainedaccordinglyforthediscretizedequations.
Lemma6.1.OnIj,j=0,...,m−1weconsiderthetwopointboundaryvalueproblems:
((∂tws,ϕ))j+au(uj)(ws,ϕ)+bq(qj)(xs,ϕ)+(ws(τj)−ρ,ϕ(τj))=0∀ϕ∈Xj,
−((∂tys,ψ))j+auu(uj)(ws,ψ,zj)+au(uj)(ψ,ys)+(ys(τj+1),ψ(τj+1))
−α1J1j(uj)(ws,ψ)=0∀ψ∈Xj,
bqq(qj)(xs,χ,zj)+bq(qj)(χ,ys)−α3(xs,χ)Qj=0∀χ∈Qj.
and((∂twλ,ϕ))j+au(uj)(wλ,ϕ)+bq(qj)(xλ,ϕ)+(wλ(τj),ϕ(τj))=0∀ϕ∈Xj,
−((∂tyλ,ψ))j+auu(uj)(wλ,ψ,zj)+au(uj)(ψ,yλ)+(yλ(τj+1)−ξ,ψ(τj+1))
−α1J1j(uj)(wλ,ψ)=0∀ψ∈Xj,
bqq(qj)(xλ,χ,zj)+bq(qj)(χ,yλ)−α3(xλ,χ)Qj=0∀χ∈Qj.
Thenthefollowingidentityholdsforthesolutionoperators:
¯H→H,¯H→H,
Σuj,sj:ρ→ws(τj+1)andΣzj,sj:ρ→ys(τj),
¯H→H,¯H→H,
Σuj,λj+1:ξ→wλ(τj+1)andΣzj,λj+1:ξ→yλ(τj).

Proof.Thestatementsofthelemmafollowdirectlybydifferentiationoftheboundaryvalue
problem(4.8)withrespecttotheboundaryvaluesandfromthedefinitionofthesolution
operatorsΣ¯ujandΣ¯zjforj=0,...,m−1.
Remark6.2.Lemma6.1impliesthatinthecontextofNewton’smethodadditionalintervalwise
linearboundaryvalueproblemshavetobesolvedforthecalculationofthedirectional
derivativesofthestatevariables.

LetusassumethataccordingtoRemark2.4thefunctionalJ2hasthespecialstructure
J2(sm)=21sm−u¯(T)2.ThegeneralcaseforarbitraryJ2followsanalogouslybutismore

61

6SolutionTechniquesfortheMultipleShootingApproach

complicatedwithrespecttothenotation.Forreasonsofbrevity,weintroducethefollowing
notation:Aj:=I0,j=0,...m−1,
I011120−Σ¯
Bj:=BBj21BBj22:=Bj:1Bj:2:=0−Σ¯zj,sj,j=0,...m−1,
jjuj,sj(6.2)
C11C12−Σ¯0
Cj:=Cj21Cj22:=Cj:1Cj:2:=−Σ¯zj,λj+10,j=0,...m−1,
jjuj,λj+1
F:=0−I,
0+1jj+1jrI:=sh−u0,
rj:=shj−Σ¯uj(jsh,λjh+1),j=0,...m−1,(6.3)
λh−Σ¯zj(sh,λh)
rF:=λhm−(shm−u¯).
Here,Bj:1,Bj:2,Cj:1,andCj:2denotethecolumnsofthepseudomatrices.Thelinearized
systemofmatchingconditions(6.1)cannowbewritteninshortpseudomatrixnotation:
IΔsh0rI
B0:2A0C0(Δλh0,Δsh1)Tr0
B1A1C1(Δλh1,Δsh2)Tr1
............=−....(6.4)

Bm−2Am−2Cm−2(Δλhm−2,Δshm−1)Trm−2
mBm−1Am−1Cm:1−1(Δλhm−1,Δshm)Trm−1
FIΔλhrF
=:K=:Δx=:b
ThesolutionofthemultipleshootingproblembyapplicationofNewton’smethodtothe
systemofmatchingconditionreadsconsequentlyinalgorithmicformulationaspresentedin
6.1.AlgorithmAlgorithm6.1Newton’smethodfortheindirectapproach
1:Seti=0.
2:Startwithinitialguessformultipleshootingvariablesxi=(s0h,i,...,smh,i,λ0h,i,...,λmh,i)T.
eatrep3:4:Calculatetheresidualsbi=(rI,i,r0,i,...,rm−1,i,rF,i)Taccordingto(6.3).
5:Solvethelinearsystem(6.4)fortheNewtonupdate,
KiΔxi=−bi,withΔxi=(Δs0h,i,...,Δsmh,i,Δλ0h,i,...λmh,i)T.
rateitenextCalculate6:xi+1=xi+Δxi.
7:Seti=i+1.
8:untilbiH˜×H˜:=rI,i2+jm=0−1rj,i2+rF,i2<tol.

62

6.1SolutionTechniquesfortheIndirectMultipleShootingApproach

Remark6.3.(Matrix-vectorbasedformulationofNewton’smethod)Theconsiderationof
Newton’smethodinthediscretecaseneedsthedeterminationofthecoefficientvectorsof
thesolutionsfromalinearsystemofequations.Therefore,weneedtheconcretizationofthe
linearsystem(6.4)forgivenbasesofthediscretizedspacesonthemultipleshootingnodes.
Wedothisexemplarilyforonlyoneblockrow(BjAjCj)ofthepseudoblockmatrixK.
Wememorizethecorrespondingequationsofthelinearizedsystem(6.1):
(Δλhj−Σ¯zj,sj(Δshj)−Σ¯zj,λj+1(Δλhj+1),v)=−(λhj−Σ¯zj(shj,λhj+1),v)∀v∈Hhj,
(Δshj+1−Σ¯uj,sj(Δshj)−Σ¯uj,λj+1(Δλhj+1),v)=−(shj+1−Σ¯uj(shj,λhj+1),v)∀v∈Hhj+1.
Now,weassumethatϕ0j,...,ϕnjjisabasisofHhj,andthecoefficientvectorsoftheNewton
incrementsΔshjandΔλhjwithrespecttothisbasisaregivenbyΔξj,s∈RnjandΔξj,λ∈Rnj.
Withthisnotationathand,weareabletorewritethelinearequationsas
G0Δξj,λ−G1sΔξj,s−G1λΔξj+1,λ=−d0,
G2Δξj+1,s−G3sΔξj,s−G3λΔξj+1,λ=−d1,
wherethematricesG0,G1s∈Rnj×nj,G1λ∈Rnj×nj+1,G2,G3λ∈Rnj+1×nj+1,G3s∈Rnj+1×nj
havetobeunderstoodaccordingtothefollowingidentities:
(G0)li=(ϕij,ϕlj),(G1s)li=(Σ¯zj,sj(ϕij),ϕlj),(G1λ)li=(Σ¯zj,λj+1(ϕij+1),ϕlj),
(G2)li=(ϕij+1,ϕlj+1),(G3s)li=(Σ¯uj,sj(ϕij),ϕlj+1),(G3λ)li=(Σ¯uj,λj+1(ϕij+1),ϕlj+1)
andtheright-handsidevectorsd0∈Rnjandd1∈Rnj+1accordingto
(d0)l=(λkj−Σ¯zj(skj,λkj+1),ϕlj),
(d1)l=(skj+1−Σ¯uj(skj,λkj+1),ϕlj+1).
Weremark,thatd0andd1areusuallydifferentfromthecoefficientvectorsfortheright-hand
sides.Withthisinterpretationathand,thetransferofthefunctionspacemethodtoa
matrix-vectorbasedoneisstraightforwardbysettingupthediscretelinearsystemforthe
determinationofthecoefficientvectorsoftheNewtonupdates.Inaddition,itisoften
convenientandsuitabletoreplacetheHilbertspacenormonHbythel2-normonthefinite
dimensionalvectorspaceRnj.
Forreasonsofvividness,letusconsiderNewton’smethodforExample2.3:
Example6.1.(Newton’smethodforExample2.3)Reconsideringthemultipleshooting
approachforExample2.3whichwasintroducedinExample4.1,weretrievethefollowing
system:linearized(Δsh0,ϕ)=−(sh0−u0,ϕ)∀ϕ∈Hh0,
(Δλh0−Σ¯z0,s0(Δsh0),ϕ)=−(λh0−Σ¯z0(sh0),ϕ)∀ϕ∈Hh0,
(Δsh1−Σ¯u0,λ1(Δλh1),ϕ)=−(sh1−Σ¯u0(λh1),ϕ)∀ϕ∈Hh1,
(Δλh1−Σ¯z1,s1(Δs1),ϕ)=−(λh1−Σ¯z1(sh1),ϕ)∀ϕ∈Hh1,
(Δsh2−Σ¯u1,λ2(Δλh2),ϕ)=−(s2−Σ¯u1(λh2),ϕ)∀ϕ∈Hh2,
(Δλh2,ϕ)=−(λh2,ϕ)∀ϕ∈Hh2.

63

6SolutionTechniquesfortheMultipleShootingApproach

Differentiationofthe(undiscretized)boundaryvalueproblems(4.15)yieldsforthederivatives
ofthesolutionoperatorsthefollowingidentities:
OnIjlettheboundaryvalueproblembegivenasfollows:forallϕ,ψ,χ∈X:
((∂tw,ϕ))j+((w,ϕ))j+((3(uj)2w,ϕ))j+(w(τj),ϕ(0))=((x,ϕ))j,
−((∂ty,ψ))j+((y,ψ))j+((3(uj)2y,ψ))j+(y(τj+1)−ξ,ψ(T))=((w,ψ))j−((6ujwzj,ψ))j,
((x,χ))j=−((y,χ))j.
ThenΣ¯uj,λj+1(ξ)=w(τj+1).
Forallϕ,ψ,χ∈X:
((∂tw,ϕ))j+((w,ϕ))j+((3(uj)2w,ϕ))j+(w(τj)−ρ,ϕ(0))=((x,ϕ))j,
−((−∂ty,ψ))j+((y,ψ))j+((3(uj)2y,ψ))j+(y(τj+1),ψ(T))=((w,ψ))j−((6ujwzj,ψ))j,
((x,χ))j=−((y,χ))j.
ThenΣ¯zj,sj(ρ)=y(τj).
Forthediscretederivativesofthesolutionoperatorsthisholdsanalogouslyforthediscretized
equations.

Throughoutthischapter,weconsiderthisexampleforpointingoutthesetupofthedifferent
methods.Inthefollowing,wediscussthesolutionofthelinearizedsystem(6.4)byapplication
ofaniterativesolverfromtheclassofKrylovsubspacemethods.

6.1.2TheGMRESMethodfortheSolutionoftheLinearizedSystem

Thepseudomatrixnotationofproblem(6.4)revealsasparsetridiagonalblockstructure
oftheproblemwithidentityoperatorsonthediagonal.Thispropertyprovesusefulwhen
subspconsideringacemethothed.solutionTheofsolutiontheofalinearizedlinearizedsystembysystemwithapplicationaofsimilaraprecstructureonditioneasdinKrylov(6.4)
issubspaceextensivelymethodstudiedisinsuggested.[20]andOur[c14],hoiceiswhereaprtheeconditioneapplicationdgenerofaalizedpreconditionedminimumrKryloesidualv
method(GMRES),andasareasonablepreconditionertheforwardbackwardblockGauss
Seidelpreconditionerischosen.Tounderlineontheonehandthenecessityofapreconditioner
andontheotherhandtheadvantageofforwardbackwardblockGaussSeidelpreconditioning,
simplestdifferentcasepreofcondanitioneoptimalrshavconebtroleenproblemtestedataconstrainedlinearbyaexample.parabThisolicPDEexampleandservpresenestsquitethe
wellfortheoutlineofthebasicpropertiesoftheGMRESmethod.
Example6.2.LetΩ=(−1,1)×(−1,1)beasquareshapeddomain,I=(0,5),andletthe
optimalcontrolproblembegivenby
120.0012
(q,u)∈minQ×XJ(q,u):=2u(T)−0.5L2(Ω)+2Iq(t)L2(Ω)dt

64

6.1SolutionTechniquesfortheIndirectMultipleShootingApproach

subjecttotheconstrainingheatequation
∂tu−Δu=qinΩ×I,
ππu=0on∂Ω×I,
u=cos2xcos2yinΩ×{0}.
Weconsidertheproblemonafivetimesgloballyrefinedmesh,thatisameshof1024cells.
Duetothelinearityoftheproblem,theouterNewtonmethodconvergesinonestepwhich
resultsinonlyoneapplicationoftheGMRESmethodforthesolutionofthelinearized
systemofmatchingconditions.InFigure6.2,wecomparethenumberofiterativestepsuntil
convergencefordifferentpreconditionersandvariousnumbersofintervals.Theresultsofthe
calculationsillustratethatpreconditioningisessential.TheforwardbackwardblockGauss
Seidelpreconditioneryieldsbyfarthebestresultsoftheconsideredpreconditioners,and
additionallythenumberofGMRESiterationsuntilconvergenceisapproximatelyconstant
77stepsforthedifferentnumbersofintervals.Theassumption,thattheforwardbackward
GaussSeidelpreconditionerleavesthenumberofGMRESiterationsconstantfordifferent
timedomaindecompositionswassubstantiatedbycalculationofotherexamplesnotpresented
thesis.thisin350GSFW-BWGSFW300GSBWID250

200

150

100

50510152025303540

Figure6.2:ResultsfortheforwardbackwardGaussSeidelpreconditioner(red),theforward
tionerGauss(blue)Seidelandpreconditionerwithout(darkpreconditionerblue),the(magenbacta)kwardcalculatedGaussforSeidelExampleprecondi-6.2.
(x-axis:numberofintervals,y-axis:numberofiterations)

Remark6.4.(Parallelizability)Asacrucialdrawbackofthisapproachweshouldmention
thatthepreconditionerpresentedaboveisnotparallelizable.Asamatteroffact,several
preconditionershavebeenstudiedin[14].Numericalexperimentsshowthatsofarconsidered

65

6SolutionTechniquesfortheMultipleShootingApproach

goodpreconditionersforproblem(6.4)arenotparallelizablewhileparallelizableprecondi-
tionersdonotyieldthedesiredimprovementwithrespecttotheconvergenceoftheKrylov
d.methosubspaceForthepurposeofpresentingtheblockGaussSeidelpreconditioner,thefollowingpseudo
blockmatrixdecompositionintoalowertriangularpseudoblockmatrixL,anuppertriangular
pseudoblockmatrixUandadiagonalpseudoblockmatrixDisintroduced:
I...−UB0:2AC0
D:=..........
−L...Bm−1ACm:1−1
IFThepreconditioningpseudomatricesfortheforwardblockGaussSeidelandthebackward
blockGaussSeidelpreconditionerread
Pfwd:=D−LandPbwd:=D−U,
andfortheforwardbackwardblockGaussSeidelpreconditionerthepreconditioningpseudo
asdefinedismatrixP:=(D−L)D−1(D−U).(6.5)
Thus,theapplicationoftheforwardbackwardblockGaussSeidelpreconditionerinthe
GMRESmethodforthesolutionof(6.4)isequivalenttothesolutionofthesystem
P−1Kx=P−1b(6.6)
bytheunpreconditionedGMRESmethod.Duetothespecialstructureofthematrices,and
duetoD=D−1,thecalculationofP−1v,v∈H˜×H˜,canbeperformedbyforwardand
backwardsubstitutionasoutlinedinthefollowing:
Iy0vI
B0:2A(x0,y1)TTv0
B1A(x1,y2)v1
.........=...
Bm−2A(xm−2,ym−1)Tvm−2
Bm−1A(xm−1,ym)Tvm−1
FIxmvF
D−Lzv
y0=vI
(x0,y1)T=v0−B0:2y0
.(x1,y2)T=v1−B2(x0,y1)T
⇔...
(xm−2,ym−1)T=vm−2−Bm−2(xm−3,ymT−2)
(xm−1,ym)T=vm−1−Bm−1(xm−2,ym−1)T
xm=vF−F(xm−1,ym)T

66

6.1SolutionTechniquesfortheIndirectMultipleShootingApproach

TheinversionofD−Uistreatedanalogously:
xm=vF
(xm−1,ym)T=vm−1−Cm:1−1xm

(xm−2,ym−1)T=vm−2−Cm−2(xm−1,ym)T
(D−U)z=v⇔....
(x1,y2)T=v1−C1(x2,y3)T

(x0,y1)T=v0−C02:(x1,y2)T
y0=vI
Here,thepseudomatrixvectorproducts,thatistheevaluationofthedirectionalderiva-
tives,andtheresidualsarecalculatedaccordingtoLemma6.1fromintervalwiseboundary
valueproblems.The(unpreconditioned)GMRESmethod(seeforinstancethetextbookof
Saad[34])isproposedinAlgorithm6.2.Therefore,inadditiontothealreadyintroduced
pseudovectorsΔx,b∈H˜h×H˜h,weredefiner∈H˜h×H˜hastheresidualofthelinearized
system,andintroduceadditionalauxiliarypseudovectorsv,w∈H˜h×H˜handrealnumbers
β,yi,zi,hil,τ,ν,ci,cs∈Rfortheformulationofthealgorithm.
Algorithm6.2GMRESforthesolutionof(6.4)
tol.Chose1:2:Seti=−1.
3:StartwithinitialguessfortheNewtonupdateΔx0.
4:Calculater0:=b0−KΔx0,β:=r0,v1:=r0/β,z0:=β.
eatrep5:6:i=i+1.
7:wi=Kvi.
8:forl=0toido
9:Sethli:=(vl,wi)H˜×H˜.
10:Setwi:=wi−hlivl.
11:Sethi+1,i:=wiH˜×H˜.
12:Setvi+1:=wi/hi+1,i.
hsch14:Sethl+1li,i:=−sllsllhl+1li,i.
13:forl=0toi−1do
15:Setτ:=|hii|+|hi,i+1|.
16:Setν:=τ∙(hii/τ)2+(hi,i+1/τ)2.
17:Setci:=hii/ν,andsi:=hi,i+1/ν.
18:Setzi+1:=−sizi.Setzi:=−cizi.
19:until|zi|/β<tol.
20:yi:=zi/hii.
i22:Setyl=(zl−j=l+1hljyj)/hll.
21:forl=i−1downto0do
23:Δxi=Δx0+li=0ylvl.

67

6SolutionTechniquesfortheMultipleShootingApproach

Remark6.5.(Matrix-vectorbasedformulationoftheGMRESmethod)Asbefore,thepractical
implementationofthemethodisbasedonthedeterminationofthecoefficientvectorsof
thesolutions.TheinterpretationofthoseisduetoRemark(6.3)andthusnotrepeated
d.methothisforparticularlyFromthediscussionabove,ithasbecomeclearthatallpreconditionerscanonlybeapplied
atadditionalcosts.Indetail,fortheforwardbackwardblockGaussSeidelpreconditioner
wehavetosolve2additionallinearboundaryvalueproblemsoneachinterval,whichmakes
atotalamountof4linearboundaryvalueproblemsperintervalforeveryiterationofthe
GMRESmethod.Overall,mostoftheeffortisspendforthepreconditionediterativesolver,
outlinedbyasimpleanalysisofExample6.1.InFigure6.3thetotalnumberofsolved
linearandnonlinearproblemsispresentedforthedifferentnumberofintervals.Wehave
previouslyseen,thatforthisexamplethepreconditionediterationneeds77iterativesteps,
whichmakesatotalnumberof4∙77linearboundaryvalueproblemsperinterval,oroverall
m∙308problemsuntilconvergence.Ontheotherhand,wehavetoevaluatethematching
conditionstwice,suchthat2∙mnonlinearboundaryvalueproblemshavetobesolved.Half
ofthesolvedlinearboundaryvalueproblemsareduetothepreconditioning,theotherhalf
aresolvedduringthecalculationofthedirectionalderivativesthemselves.Nevertheless,we
seethatdespitetheadditionaleffort,thepreconditionediterationisbyfarbetterthanthe
d.methoGMRESunpreconditioned3.0∙104
2.5∙104nonlinear
(preconditioned)linear2.0∙104linear(notpreconditioned)
1.5∙104
1.0∙104
5.0∙103
0510152025303540

Figure6.3:Numberofsolvedlinearandnonlinearboundaryvalueproblemsfordifferent
numbersofshootingintervals,calculatedwithandwithoutpreconditioning.
(x-axis:numberofintervals,y-axis:numberofsolvedproblems)

Randemarkthus6.6.theWesolutionshouldktimeeepinforthemindintethatrvforalwisedifferenbtoundarynumbveralueofintervproblems,als,theisintervdifferenalt,length,too.
butTherefore,notontheFigure6.3absolutecontainseffortonlyandtimeinformationneededonforthetherelativpeerformanceefficiencyofofthethemethods.preconditioners,

68

6.1SolutionTechniquesfortheIndirectMultipleShootingApproach

Next,weinvestigateefficientsolutiontechniquesfortheintervalwiseboundaryvalueproblems,
linearandnonlinearones.Thisefforthasresultedintwodifferentapproacheswhichwe
explaininthefollowingparagraphs.

6.1.3SolutionoftheIntervalProblems–Newton’smethod
ThecalculationoftheNewtonresidualsin(6.1)fortheouterNewtonmethodrequiresthe
solutionoftheintervalwisenonlinearboundaryvalueproblems(5.6).Inthissubsection
wepresentasolutionstrategywhichismotivatedbytheideaoflinearizingthenonlinear
boundaryvalueproblemswithinaninnerNewtonmethod.Ithasthegreatadvantageof
solvingbothlinearandnonlinearboundaryvalueproblemsbyonlyonesolver.Therefore,
thewholeproblemisbroughtdowntothesolutionoflinearboundaryvalueproblemsby
standardtechniques.WesketchthisideaintheflowdiagraminFigure6.4.
ByapplicationofNewton’smethodonthenonlinearboundaryvalueproblemsweobtain
anothersetoflinearboundaryvalueproblemsofthesamestructureasthosepresentedin
6.1.Lemma

LinearintervalBVP

matrixsystemSameersolvlinearSame

BVPaltervinNonlinear

Newton

inLinearBVPalterv

Figure6.4:Transformationofthenonlinearboundaryvalueproblems.

Remark6.7.Wesetupthisapproachfortheundiscretizedproblemonlyandremark,thatin
theequations,discretizedsolutioncases,theandcorrespspacesbyondingtheirdiscrealgorithmteisequivobtainedalents.byreplacingtheundiscretized
Werecalltheformulationoftheintervalwisenonlinearboundaryvalueproblemsinthe
following,beforeweproceedwiththedescriptionofNewton’smethodfortheirsolution:
equation:Primal((∂tuj,ϕ))j+a(uj)(ϕ)+b(qj)(ϕ)+(uj(τj)−sj,ϕ(τj))−((f,ϕ))j=0∀ϕ∈Xj.(4.8a)
equation:Dual−((∂tzj,ψ))j+au(uj)(ψ,zj)+(zj(τj+1)−λj+1,ψ(τj+1))−α1J1j(uj)(ψ)=0∀ψ∈Xj.
(4.8b)equation:trolConbq(qj)(χ,zj)−α3(qj,χ)Qj=0∀χ∈Qj.(4.8c)

69

6SolutionTechniquesfortheMultipleShootingApproach

TheapplicationofNewton’smethodtothewholenonlinearsystemresultsinthelinearized
system(6.8)fromwhichtheNewtonupdate(δqlj,δulj,δzlj)∈Qj×Xj×Xjinsteplis
calculated.equation:Primal((∂tδulj,ϕ))j+au(ulj)(δulj,ϕ)+bq(qlj)(δqlj,ϕ)+(δulj(τj),ϕ(τj))=−((ru,ϕ))j∀ϕ∈Xj.
)(6.8aequation:Dual−((∂tδzlj,ψ))j+auu(uj)(δulj,ψ,zj)+au(uj)(ψ,δzlj)+(δzlj(τj+1),ψ(τj+1))
−α1J1j(ulj)(δulj,ψ)=−((rz,ψ))j∀ψ∈Xj.(6.8b)
ation:quetrolConbqq(qlj)(δqlj,χ,zlj)+bq(qlj)(χ,δzlj)−α3(δqlj,χ)Qj=−(rq,χ)Qj∀χ∈Qj.(6.8c)
Here,theright-handsideisgivenbytheresidualofthenonlinearboundaryvalueproblem
((ruj,ϕ))j:=((∂tulj,ϕ))j+a(ulj)(ϕ)+b(qlj)(ϕ)+(ulj(τj)−sj,ϕ(τj))−((f,ϕ))j,
((rzj,ψ))j:=−((∂tzlj,ψ))j+au(ulj)(ψ,zlj)+(zlj(τj+1)−λj+1,ψ(τj+1))−α1J1j(uj)(ψ),
(rqj,χ)Qj:=bq(qlj)(χ,zlj)−α3(qlj,χ)Qj.
(6.9)ThesubsequentNewtoniterateisthencalculatedasusualbyupdating
qlj+1=qlj+νlδqlj,ulj+1=ulj+νlδulj,zlj+1=zlj+νlδzlj,
wherethedampingparameterνl∈(0,1]isdeterminedbystandardbacktrackingtechniques
suchthatadescentintheNewtondirectionisguaranteed.Thepseudocodenotationofthe
algorithmisgiveninAlgorithm6.3.
Algorithm6.3Newton’smethodforthesolutionof(4.8)
1:Setl=0.
2:Startwithaninitialguessq0j∈Qj,u0j∈Xj,z0j∈Xj.
eatrep3:4:Calculatetheresidualsrju,l,rzj,landrqj,laccordingto(6.9).
5:Solvethelinearboundaryvalueproblem(6.8)forδqlj,δulj,δzlj.
6:Chooseνlsuchthatforthenextiterateadescentintheresidualisobtained.
rateitenextCalculate7:(qlj+1,ulj+1,zlj+1)=(qlj,ulj,zlj)+νl(δqlj,δulj,δzlj).
8:Setl=l+1.
9:untilrju,l2+rzj,l2+rqj,l2Q<tol
Acomparisonshowsthatthelinearboundaryvalueproblemsdevelopedabovecoincidewith
thoseforthedirectionalderivativesofthestatesinLemma6.1exceptfortheright-hand

70

6.1SolutionTechniquesfortheIndirectMultipleShootingApproach

sidesandtheinitialvalues.Asaconvenientconsequence,weareabletoapplythesame
linearsolverforbothlinearandnonlinearboundaryvalueproblems.
Summarizing,thisapproachrequestsonlythesolutionofintervalwiselinearboundaryvalue
problemsinsteadofnonlinearones,andisquitesimplewithrespecttotheimplementation.
FInurtherwhaton,followws,ewgiveediscusssometheresultssolutionontheofptheerformancelinearinoftervthalewiseapproacbhoundaryandvpalueointoutproblems.the
limitationsanddifficultiesthatoccur.

6.1.4SolutionoftheLinearProblems–FixedPointIterationandGradient
dMetho

Investigatingthestructureofthelinearboundaryvalueproblems,theobviousandeasiest
wayofsolvingwouldbetheapplicationofamodifiedfixedpointiteration.Thebasicideaofa
fixedpointiterationonIjistoevaluateuj,zjandqjsuccessivelyinaloop:Asanexample,
insteplofthefixedpointiteration,givenacontrolqlj,wecalculatethestateuljfromthe
primalequation.Thedualstatezljcanthenbecalculatedfromthedualequationandan
updateforthecontrol,qlj+1isobtainedfromthecontrolequation.Unfortunately,numerical
experimentsshowthattheconvergencebehaviorofafixedpointiterationisextremelybad
forfullycoupledboundaryvalueproblems.Acloserinvestigationrevealsthatinthiscontext,
itisequivalenttoagradientmethodwithstepsize1/α3asweproveinthefollowing.
Remark6.8.Again,themethodsarederivedfortheundiscretizedproblemonlyandtheexten-
siontothediscretecaseisobtainedstraightforwardbyreplacingtheundiscretizedequations,
solutions,solutionoperators,functionalsandspacesthroughtheirdiscreteequivalents.

Forthepurposeofshowingtheequivalenceofafixedpointiterationandthegradientmethod,
itissufficienttoconsiderasimplifiedoptimalcontrolproblem.Theproofforthegeneral
caseofanarbitraryoptimalcontrolproblemfollowsanalogouslywiththesamebasicideas
butneedsamorecomplicatednotationalframeworkandisquitehardtofollowup.
Fortheeaseofpresentationweomittheintervalindexj.WeconsideronI=(0,T)the
problemoptimization

qmin,uJ(q,u):=α1u(t)−u¯(t)2dt+α2u(T)−u¯T2+α3q(t)2Q(6.10a)
222Itojectsub((∂tu,ϕ))+a(u,ϕ)+b(q,ϕ)+(u(0),ϕ(0))=0∀ϕ∈X.(6.10b)
Thechoiceofahomogenousinitialvalueandright-handsideissuitableduetothefact,that
forthelinearconstrainingequationthegeneralcasefollowsimmediatelybyanaffinelinear
shiftofthesolutionspace.TheLagrangianisdefinedasbeforeby

L(q,u,z)=J(q,u)−{((∂tu,z))+a(u,z)+b(q,z)+(u(0),z(0))},(6.11)

71

6SolutionTechniquesfortheMultipleShootingApproach

andthecorrespondingoptimalitysystemisgivenbythelinearboundaryvalueproblem
w.elobstatedequation:Primal((∂tu,ϕ))+a(u,ϕ)+b(q,ϕ)+(u(0),ϕ(0))=0∀ϕ∈X.
equation:Dual−((∂tz,ψ))+a(ψ,z)+(z(T)−α2(u(T)−u¯T),ψ(T))−α1((u−¯u,ψ))=0∀ψ∈X.
ation:quetrolConα3(q,χ)Q−b(χ,z)=0∀χ∈Q.
Again,weintroducethesolutionoperatorsfortheconstrainingequation(6.10b)
Q→XQ→H
S:q→uandS¯:q→u(T).
Wereformulateproblem(6.10)inthesocalledreducedformulationbyreplacinguin(6.10a)
bySqandu(T)byS¯q:
minj(q):=J(q,Sq)=α1Sq(t)−u¯(t)2dt+α2S¯q−u¯T2+α3q(t)2Q.(6.13)
222qIAmoreconcretedescriptionofthereducedapproachisgivenlateroninSection6.1.6.
Thegradientmethodinfunctionspacewithstepsize1/α3forthisproblemisdescribedby
ulaformthe(qnew,δq)Q=(qold,δq)Q−1j(qold)(δq)∀δq∈Q.
α3Weproceedwiththedevelopmentofasimplifiedrepresentationofthegradient.Fromthe
definitionofthereducedcostfunctionalin(6.13)andtheLagrangianin(6.11)weobtainfor
givenqandu=Sqtheidentity
j(q)=L(q,u,z).
Now,differentiationwithrespecttoqintothedirectionδq∈Qyieldsthefollowingrepresen-
tationofthegradientj(q)(δq):
j(q)(δq)=Lq(q,u,z)(δq)+Lu(q,u,z)(δu)+Lz(q,u,z)(δz)
whereδu=dqdu(δq)andδz=dqdz(δq).
Duetou=Sq,theprimalequationisfulfilled,andthusLz(q,u,z)(δz)=0.Choosingzsuch
thatthedualequationisfulfilled,too,weobtainadditionallyLu(q,u,z)(δu)=0andretrieve
tgradientheforj(q)(δq)=Lq(q,u,z)(δq)
(6.14)=α3(q,δq)Q−b(δq,z).
Now,wehaveeverythingathandtoinvestigatetherelationbetweenafixedpointiteration
andthegradientmethod.Wegivethepseudocodenotationforonestepofbothiterations,
andastep-by-stepcomparisonimmediatelygivesevidenceoftheirequivalence.

72

6.1SolutionTechniquesfortheIndirectMultipleShootingApproach

Algorithm6.4OnestepofthegradientmethodforalinearBVP
1:Require:Calculateqoldugivasen.
olduold=Sqold.
2:Solvedualequationforzold.
3:Evaluategradientaccordingto
j(q)(δq)=α3(qold,δq)Q−b(δq,zold).
4:Obtainnewcontrolqnewas
(qnew,δq)Q=(qold,δq)−1(α3(qold−b(δq,zold),δq)Q)
α31=α3b(δq,zold).

Algorithm6.5OnestepofthefixedpointiterationforalinearBVP
en.givqRequire:oldasuCalculate1:olduold=Sqold.
2:Solvedualequationforzold.
3:Obtainnewcontrolqnewfromthecontrolequation
1(qnew,δq)Q=α3b(δq,zold)=0∀δq∈Q.

Comparingthestepsofbothmethods,theequivalenceisquiteobvious.Thefirstthreesteps
arethesameforbothalgorithms,andstepfourandfiveofthefirstalgorithmcorrespondto
thefourthstepofthesecondone.Thebadconvergencebehaviorofthefixedpointiteration
canthusbeexplainedbythecorrespondingstepsizeofthegradientmethod.Forasmall
regularizationparameterα3>0,thestepsize1/α3becomesextremelylargeandconvergence
teed.guaranlongernois

Apromisingalternativeforthesolutionoftheboundaryvalueproblemsistheapplicationof
aspaceconjugateproblemgradiendifferstfrommethothed(CGformmethoulationd).forThefinitedimedescriptionnsionalofthismatrixmethovectordforthesystemsfasunctionthe
theirequivalendualsttoonthefunctionsusualinmatrixQ.vTheectorCGpromethoductsdisforgivtheenbysolutiontheoftheapplicationlinearofbopeoundaryratorsvalandue
gradienproblemstshallmethodnotinbefunctionsdescribedspaceindetailfortheinthissolutionsectionofb–othwelinearpresentandtheSteinonlinearhaugbconjugateoundary
valueproblemsinthecontextofthereducedapproachinSubsection6.1.6.

73

6SolutionTechniquesfortheMultipleShootingApproach

6.1.5ApplicabilityofNewton’sMethodfortheIntervalProblems
Atfirstsight,theapplicationofNewton’smethodworksquitewellforthesolutionofthe
exampleproblemsleadproblemsstopoorconsideredconvinergencethereinstroultsorduction.noconHovwever,ergenceatapplicationall,whictohwmoreeinvestigatecomplicatedin
thefollowing.Forreasonsofsimplicity,weconsiderthesolutionofanonlinearODEinitial
valueproblem,thewellknowninstableLorenzattractor,byNewton’smethodaspresented
aboveandherebypointouttheweakpointsoftheapproach.
Example6.3.(Lorenzattractor)Weconsiderthenonlinearsystemofordinarydifferential
equations∂tu1=−au1+au2,u1(0)=10,
∂tu2=−bu1−u2−u1u3,u2(0)=0,(6.15)
∂tu3=−cu3+u1u2,u3(0)=25,
parametersengivwitha=10,b=28,c=2.67.

TheefficientsolutionofthenonstiffLorenzattractorsystemisusuallyperformedbyhigher
orderexplicittimesteppingschemes.However,inourexemplarycontextitissufficientto
considerthesolutionwiththeimplicitEulertimesteppingscheme.Thesolutionof(6.15)
onatimeintervalI=(0,20)withstepsize0.002andapplicationofNewton’smethodin
everytimestepyieldsthecommonlyknownsolutionoftheLorenzattractorsystemshownin
6.5.Figure

50403020100302010-200-10-10010-2020

Figure6.5:SolutionoftheLorenzattractor.
Nosystemw,weforapplytheNewtonNewton’supmethodatedu˜btoythethesamenonlinearimplicitsystemEuleroftimeequationssteppingandsolvscehemetheasblinearizedefore.

74

6.1SolutionTechniquesfortheIndirectMultipleShootingApproach

Theresultinglinearsystemisgivenbytheequations
∂tu˜1=−au˜1+a˜u2−r1,
∂tu˜2=−bu˜1−u˜2−u˜3u3−u1u˜3−r2,
∂tu˜3=−cu˜3+u˜1u2+u1u˜2−r3,
withhomogenousinitialvaluesandresidualsasfollows:
u˜1(0)=0,r1=∂tu1−(−au1+au2),
u˜2(0)=0,r2=∂tu2−(−bu1−u2−u1u3),.
u˜3(0)=0,r3=∂tu3−(−cu3+u1u2).
WeobservethatevenonsmalltimeintervalstheNewtoniteratesconvergeveryslowlyto
thesolution.RegardingFigure6.6,weseethedependenceofthenumberofNewtonsteps
performeduntilconvergenceonthelengthofthetimeinterval.Thereasonforthisbehaviorof
35030025020015010050011.522.533.544.5
Figure6.6:ConvergencebehaviorofNewton’smethodfordifferentintervalsI=(0,T).
(x-axis:T,y-axis:stepsuntilconvergence)
Newton’smethodcanbeillustratedbyvisualizingtheNewtonresidualsfordifferentNewton
steps.ThisisdoneinFigure6.7fromwhichweclearlyseethatwithintheNewtonmethod,
theerrorisannihilatedsuccessivelyintime,butsimultaneouslytheerroratthelatertime
stepsgrowsexponentially.
Thisbehaviorcanbeunderstoodbyconsideringthefactthattheapplicationofanimplicit
EulertimesteppingschemeleadstoafastgrowingapproximationerroroftheNewton
incrementintime.Therefore,theNewtonupdatesforthesolutiononearliertimestepsare
closetotheexactupdatewhiletheupdatesatlatertimestepsareonlyapoorapproximation.
ThusthenextNewtoniterateisabetterapproximationtothesolutiononearlytimesteps
butaworseinlaterones.Now,ineachNewtonstep,furthertimestepsoftheupdatecan

75

6SolutionTechniquesfortheMultipleShootingApproach

06.000..0204
0−−00..0204
06.0−00.511.522.53
2=l(a)

00..0408
0−−00..0408
00.511.522.53
4=l(b)

04.0004.0−00.511.522.53
6=l(c)

3.5

3.5

3.5

4

4

4

4.5

4.5

4.5

0.300.150.00.15−0.30−00.511.522.533.544.5
8=l(d)Figure6.7:Newtonresidualsr1(red),r2(darkblue),r3(blue)withincreasingnumberlof
Newtonstepsfrom(a)to(d).(x-axis:time,y-axis:error)

beapproximatedquitewellandtheresidualsvanishsuccessively,butslowly,afteracertain
numberofNewtonsteps.
Ontheotherhand,thesolutionoftheproblemwiththecommonapproach,thatisthe
applicationofthetimesteppingschemetothenonlinearproblemandsolvingeachtimestep
withNewton’smethod,worksquitewell.ThereasonforthisdifferentbehaviorofNewton’s

76

6.1SolutionTechniquesfortheIndirectMultipleShootingApproach

methodcanberevealedbymeansoftheconditionnumbersofthesystemmatrices.Itiswell
knownthattheLorenzattractorsystemislocallywellconditionedandgloballyillconditioned
asexplainedinthefollowing.Forthecommonapproach,ineachNewtonstepasystem
matrixofsize3×3isinverted,whereasoneNewtonstepoftheNewtonapproachforthe
wholeproblemcanbeinterpretedasthesolutionofalinearsystemofsize3∙n×3∙nwhere
ndenotesthenumberoftimesteps.Thislargesystemmatrixisobviouslybuiltupfromthe
timestepmatricesonthediagonalandadditionalidentitymatricesontheuppersecondary
diagonal.Figure6.8pointsout,howtheconditionofthesystemmatrixchangeswiththe
enlargementofthematrix.Whilethesmallmatricesofeachtimestephaveacondition
ofapproximately1,theconditionofthelargesystemgrowsintime,andsodoestheerror.
Accordingly,forfixedterminaltimeanddecreasingtimestepsize,theconditiongrows,too,
410

310

210

11001002003004005006007008009001000
Figure6.8:ConditionnumberwithgrowingTforfixedtimestepsize.(x-axis:numberof
condition)-axis:ysteps,timeasshowninFigure6.9.Overall,wecansummarizethatNewton’smethodforthesolution
oftheinitialvalueproblemhascrucialdrawbackswhichleadtoabreakdownofthemethod.
Theproblemsdiscussedabovecandirectlybetransferredtotheapplicationontheintervalwise
boundaryvalueproblemsoftheindirectmultipleshootingapproach.Thus,wedonotfollow
thisapproachfurtheron,butpresentanalternativemethodforthesolutionofbothlinear
andnonlinearboundaryvalueproblems.

6.1.6SolutionoftheIntervalProblems–TheReducedApproach
Thesecondapproachstartswiththereformulationofthenonlinearboundaryvalueproblems
intermsofintervalwiseoptimizationproblemswhichcanbesolvedbyapplicationofthe
reducedapproach.ThistransformationisdonebycalculationoftheLagrangianbymeansof
evaluatingtheantiderivativewithrespecttothestatesandcontrol.

77

6SolutionTechniquesfortheMultipleShootingApproach

510

410

310

210

110

01000.10.20.30.40.50.60.70.80.91
Figure6.9:ConditionnumberwithdecreasingtimestepsizeforfixedT.(x-axis:timestep
condition)-axis:ysize,

First,wesetuptheoptimizationproblem
α3j(qj,uj)∈minQj×XjJj(qj,uj):=J1(uj)+2qj2Qj+(uj(τj+1),λj+1)(6.16a)
suchthat(qj,uj)fulfillstheconstrainingequation
((∂tuj,ϕ))j+a(uj)(ϕ)+b(qj)(ϕ)+(uj(τj)−sj,ϕ(τj))=((f,ϕ))j∀ϕ∈Xj.(6.16b)
ThecorrespondingLagrangianisgivenas
Lj(qj,uj,zj):=J1j(uj)+α3qj2Qj+(uj(τj+1),λj+1)
2−((∂tuj,zj))j+a(uj)(zj)+b(qj)(zj)+(uj(τj)−sj,zj(τj))−((f,zj))j,
anddifferentiationwithrespecttothecontrolandthestatesyieldstheoptimalitysystem
((∂tuj,ϕ))j+a(uj)(ϕ)+b(qj)(ϕ)+(uj(τj)−sj,ϕ(τj))=((f,ϕ))j∀ϕ∈Xj.(6.17a)
−((∂tzj,ψ))j+au(uj)(ψ,zj)+(zj(τj+1)−λj+1,ψ(τj+1))=α1J1j(uj)(ψ)∀ψ∈Xj.(6.17b)
bq(qj)(χ,zj)−α3(qj,χ)Qj=0∀χ∈Qj.(6.17c)
whichisexactlythenonlinearboundaryvalueproblem(4.8).
Remark6.9.(BoundednessofJj)Inthecasethatudependslinearlyonq,thatisu=γq,
andacostfunctionalasstatedinRemark2.4itcaneasilybeverifiedthatthecostfunctional
Jjin(6.16)isboundedfrombelow.WithJ1j(uj)+α23qj2Qjboundedfrombelowdueto
thesolvabilityoftheoriginalproblem,andthus
J1j(uj)=O(q2Qj)forqQj→∞,

78

6.1SolutionTechniquesfortheIndirectMultipleShootingApproach

ntaiobewJj(qj,uj)=J1j(uj)+α23qj2Qj+γ(qj,λj+1)
=J1j(uj)+α3qj2Qj+γ(qj,λj+1)
2=J1j(uj)+21(qj,α3qj+2γλj+1)>−∞.
Thisisduetothefactthat21(qj,α3qj+2γλj+1)=O(qQj)forqQj→∞.Inthegeneral
case,theboundednesscanbeshownanalogouslybymoresophisticatedtransformations.
Inordertosimplifythenotation,wedroptheintervalindexjinthefollowingpresentation.A
detaileddescriptionofthereducedapproachisgivenin[29].Applyingittotheoptimization
problem(6.16)weintroducethesolutionoperatorS:Q→Xfortheprimalequationand
thereducedcostfunctionalj:Q→Rwhichisdefinedas
j(q):=J(q,Sq).
Wecannowreformulate(6.16)intermsofanunconstrainedoptimizationproblem
Minimizej(q),q∈Q,
forwhichthefirstorderoptimalitycondition(6.18)isobtainedbydifferentiationwithrespect
tothecontrolq:
j(q)(δq)=0∀δq∈Q.(6.18)
Inordertosolve(6.18)byNewton’smethod,differentiationof(6.18)withrespecttoqleads
tothelinearsystem(6.19)fortheNewtonupdates:
j(q)(δq,τq)=−j(q)(τq)∀τq∈Q.(6.19)
Consequently,Newton’smethodneedstheevaluationofthefirstandsecondderivativesofj.
In[29]representationformulasforthesederivativesaredevelopedanddiscussedindetail.
Thegeneralideaistheusageoftherelation
j(q)=L(q,u,z)
whichisvalidforu=Sq,andthedifferentiationofthisexpressionwithrespecttoq.
Exploitingtherelationu=Sqweobtain
j(q)(δq)=Lu(q,u,z)(δu)+Lq(q,u,z)(δq).
KeepinginmindthatLu(q,u,z)(ϕ)=0forallϕ∈Xyieldsthedualequation,theevaluation
ofthefirstderivativeofthereducedcostfunctionalcanbebroughtdowntothefollowing
steps:owt1.Calculatez∈Xsuchthatzsolvesthedualequation(6.17b).
2.Evaluatej(q)(δq)accordingtotheidentity
j(q)(δq)=Lq(q,u,z)(δq)
=bq(q)(δq,z)−α3(q,δq)Q.

79

6SolutionTechniquesfortheMultipleShootingApproach

Forthesecondderivative,similarideascanbeappliedbutamorecomplexnotational
frameworkisneededduetotheconsiderationofthesecondderivativesofL.Thesecond
derivativeofjisobtainedbydifferentiationandapplicationofthechainrule.Weintroduce
abbreviationsthe2δu:=dqdu(δq),τu:=dqdu(τq),δτu:=ddq2u(δq,τq),
2δz:=dz(δq),τz:=dz(τq),δτz:=d2z(δq,τq)
dqdqdqandobtainbyelementarycalculustherepresentationofthesecondderivative:
j(q)(δq,τq)=Lqq(q,u,z)(δq,τq)+Lqu(q,u,z)(δq,τu)+Lqz(q,u,z)(δq,τz)
+Luq(q,u,z)(δu,τq)+Luu(q,u,z)(δu,τu)+Luz(q,u,z)(δu,τz)
+Lzq(q,u,z)(δz,τq)+Lzu(q,u,z)(δz,τu)
+Lu(q,u,z)(δτu)+Lz(q,u,z)(δτz).
Withuandzsolutionsofprimalanddualequation,thelasttwotermsofthesumvanish,
eretrievewandj(q)(δq,τq)=Lqq(q,u,z)(δq,τq)+Lqu(q,u,z)(δq,τu)+Lqz(q,u,z)(δq,τz)
+Luq(q,u,z)(δu,τq)+Luu(q,u,z)(δu,τu)+Luz(q,u,z)(δu,τz)
+Lzq(q,u,z)(δz,τq)+Lzu(q,u,z)(δz,τu).
Inanalogytothecalculationofthefirstderivative,werecommendthatδuandδzare
thathsucdetermined,Lqz(q,u,z)(δq,ϕ)+Luz(q,u,z)(δu,ϕ)=0∀ϕ∈X,(6.20)
Lqu(q,u,z)(δq,ψ)+Luu(q,u,z)(δu,ψ)+Lzu(q,u,z)(δz,ψ)=0∀ψ∈X.(6.21)
Theseadditionalequations(6.20)and(6.21)aredenotedasthetangentequationandthe
additionaladjointequation.Reattachingtheintervalindex,theseequationsread
((∂tδuj,ϕ))j+au(uj)(δu,ϕ)+bq(qj)(δqj,ϕ)+(δuj(τj),ϕ(τj))=0∀ϕ∈Xj,(6.22)
−((∂tδzj,ψ))j+au(uj)(ψ,δzj)+(δzj(τj+1),ψ(τj+1))+auu(uj)(δuj,ψ,δzj)
−α1Jj1(uj)(δujψ)=0∀ψ∈Xj.(6.23)
Thesecondderivativeofjcannowbecalculatedbythefollowingprocedure:
1.Solvethetangentequation(6.22)forδu.
2.Solvetheadditionaladjointequation(6.23)forδz.
3.Evaluatethesecondderivativesbymeansof
j(q)(δq,τq)=Lqq(q,u,z)(δq,τq)+Luq(q,u,z)(δu,τq)+Lzq(q,u,z)(δz,τq)
=bq(q)(δq,δz)+bqq(q)(δq,τq,z)−α3(δq,τq)Q.

80

6.1SolutionTechniquesfortheIndirectMultipleShootingApproach

Wedonotgofurtherintodetailconcerningthederivationoftheadditionalequations.A
thoroughinvestigationanddescriptioncanbefoundintheliteraturementionedintheintro-
ductionofthisparagraph.Instead,wefocusonthesetupofaglobalizedNewtonmethodfor
thesolutionof(6.18).Sofarinthissubsection,wehaveconsideredtheundiscretized,infinite
dimensionalproblemformulation,whileinthefollowing,whendescribingtheNewtonmethod,
wewantconsideragainthefinitedimensionalcase.Therefore,weneedthediscretesolution
operatorSkh:Qd→Xkjh,r,s,accordingtothediscretizationof(6.17a),andfurthermoreall
partialdifferentialequations(6.17b),(6.22),(6.23)havetobereplacedbytheirdiscretizations.
Consequently,fromnowon,weconsiderthereducedfunctional
jkh(qσ):=J(qσ,Skh(uσ)).
Remark6.10.Fortheeaseofpresentation,Newton’smethodforthisproblemispresentedin
functionspace.Whenevernecessary,wehintattheformulationintermsofthecoefficient
ectors.vWehavealreadydiscussed,thatonestepofNewton’smethodrequiresthesolutionofthe
linearsystem(6.19),whichisthefirstorderoptimalityconditionofthelinearquadratic
problemtrolconoptimal1δqminσm(qσ,δqσ):=jkh(qσ)+jkh(qσ)(δqσ)+2jkh(qσ)(δqσ,δqσ).(6.24)
Thus,ifδqσisasolutionof(6.24),itisasolutionof(6.19).Furthermore,ifthesecond
derivativeispositivedefinite,jkh(qσ)(δqσ,δqσ)>0∀δqσ∈Qd=0,thefirstorderoptimality
conditionisnotonlynecessarybutalsosufficient,andthereversalholds,too.Inthefollowing,
wedemandanadditionalconstraintontheNewtonupdate,δqσQ≤µ,µ∈R∪{∞},for
thisproblem,whichallowsustoapplyaconjugategradientmethodincombinationwith
hniques:tecglobalizationregiontrust1δqminσm(qσ,δqσ):=j(qσ)+j(qσ)(δqσ)+2jkh(qσ)(δqσ,δqσ)s.t.δqσQ≤µ.(6.25)
Remark6.11.(InterpretationofgradientandHessian)TheformulationofNewton’smethod
infunctionspacerequiresthesetupandcalculationofthegradientjkh(qσ)∈Qdandthe
Hessian2jkh(qσ):Qd→QdwhichhavetobeunderstoodbymeansoftheHilbertspace
tificationsiden(j(qσ),τq)Q=jkh(qσ)(τq)∀τq∈Qd,
(2jkh(qσ)(δqσ),τq)Q=(jkh(qσ)(δqσ),τq)Q∀τq∈Qd.
Remark6.12.(CalculationofdiscretegradientandHessian)Inthediscrete,finitedimensional
case,wehavetoconsiderthevectorrepresentationwithrespecttothebasisofthediscretized
controlspace.ThecoefficientvectorsrepresentinggradientandHessiancanbecalculated
explicitly.Exemplarywith{τq0,...,τqn}abasisofthediscretecontrolspaceQd,thevector
gforthegradientisobtainedasthesolutionofthelinearsystem
r=Gg

81

6SolutionTechniquesfortheMultipleShootingApproach

forwhichtheGramianmatrixGandright-handsideraredeterminedvia
Gij=(τqi,τqj)Qandri=jkh(qσ)(τqi).
Accordingly,thevectorrepresentationdofthedirectionalderivative2jkh(qσ)(δqσ)is
systemlinearthefromobtainedh=Gdsidet-handrighwithhi=jkh(qσ)(δqσ,τqi).
Bymeansoftheseconcretizations,allalgorithmspresentedinthissubsectioncaneasilybe
broughtforwardtothevectorbasedimplementationoftheapproach.
Withtheequivalenceof(6.24)and(6.19)athand,wecanformulatetheNewtonalgorithm
space.functioninAlgorithm6.6Newton’smethodforthesolutionof(6.18)
1:Startwithaninitialguessqσ,0∈Qd,µ0∈R∪{+∞},setl=0.
eatrep2:3:Solvethestateequationforuσ,l.
4:Solvethedualequationforzσ,l.
5:Calculatethegradientjkh(qσ,l)accordingtoRemark6.11.
problemtheeSolv6:δqσ∈minQdm(qσ,l,δqσ)s.t.δqσQ<µl.

7:Chooseµl+1andνlaccordingtothebehaviorofthealgorithm.
8:Calculatenextiterateqσ,l+1=qσ,l+νlδqσ.
9:Setl=l+1.
10:untiljkh(qσ,l)Q<tol

Thesolutionoftheconstrainedminimizationprobleminstep6ofAlgorithm6.6ispreferen-
tiallydonebyuseofaniterativesolverwhichrequiresmerelytheevaluationof2j(qσ,l)(δqσ).
TheevaluationprocedureforthissecondderivativecanbeperformedasdescribedinAlgorithm
6.7.Algorithm6.7Calculationof2jkh(qσ,l)(δqσ)
Require:uσ,landzσ,lhavealreadybeencomputedforgivenqσ,l.
1:Solvethediscretizedtangentequationforδuσ,l.
2:Solvethediscretizedadditionaladjointequationforδzσ,l.
3:Obtain2j(qσ)(δqσ)accordingtoRemark6.11.

Fµl,andurthermore,thetheactualregionNewtonofinstepterestisinspeacecifiedhstepbyisthechoicedeterminedofbtheythevdampingalueoftheparameterradiusνl.

82

6.1SolutionTechniquesfortheIndirectMultipleShootingApproach

BothvaluesaredeterminedanewineachstepofNewton’smethod,inordertooptimize
theconvergencebehavior.Theseglobalizationtechniquesaredescribedattheendofthis
subsection.Next,weconsiderthesolutionofthelinearproblem(6.25)fortheevaluationof
theNewtonupdateδqσwhichwasnotyetdeterminedforstep6ofAlgorithm6.6.Wetherefor
applytheSteihaugconjugategradientmethodasdescribedinthesequel.Themethodforthe
solutionoftheconstrainedoptimizationproblemwasdevelopedin[38],andthealgorithm
forourfunctionspacesettingisrecapitulatedinAlgorithm6.8.Thisalgorithmterminates,
1.Ifthecurvatureofthecalculateddirectionisnegative.Inthiscasewemovetothe
boundaryofthedomainifitsradiusisfinite,µl<∞,otherwisewetaketheprevious
iterate.2.Ifthenormoftheiterateistoolarge.Thenwetakealinearcombinationoftheprevious
iterateandthecurrentonewhichliesontheboundaryofthedomain.
3.IftheNewtonupdateisapproximatedsufficientlywell.
Algorithm6.8TheSteihaugconjugategradientmethodforthesolutionof(6.25)
1:Setp0=0,r0=−jkh(qσ),g0=r0andi=0.
oplo2:3:Computethedirectionalderivativeh=2(qσ,l)(gi)usingAlgorithm6.7.
4:Setγ=(h,gi)Q.
5:ifγ≤0then
6:ifµl<∞then
7:Computeξ>0suchthatpi+ξgiQ=µl.
8:Setδqσ=pi+ξgi.
else9:10:Setδqσ=pi−1ord=p0ifi=0.
11:break(Negativecurvaturefound.)
12:Computeα=|ri|2/γ.
13:Setpi+1=pi+αgi.
14:ifpi+1Q>µlthen
15:Computeξ>0suchthatpi+ξgiQ=µl.
16:Setδq=pi+ξgi.
17:break(Normofapproximationtoolarge.)
18:Computeri+1=ri−αh.
19:ifri+1Q/r0Q<tolthen
20:Setδqσ=pi+1
21:break(Approximationgoodenough.)
22:Computeβ=ri+12Q/ri2Q.
23:Setgi+1=ri+1+βgi.
24:Seti=i+1.

Finally,wehavetospecifythechoiceofµandνinthecontextofglobalizationtechniques
ofNewton’smethod.Wearefacedwiththechoicebetweenlinesearchmethodsandtrust
regionmethods.Inthiscontext,wehavedecidedtoapplyatrustregionglobalization.

83

6SolutionTechniquesfortheMultipleShootingApproach

Algorithm6.9Determinationofµl+1andνl
1:ifρl<0.25then
2:Setµl+1=0.25δqσQ.
3:elseifρl>0.75andδqσQ=µlthen
4:Setµl+1=min(2µl,µmax).
else5:6:Setµl+1=µl.
7:ifρl>θthen
8:Setνl=1.
else9:10:Setνl=0.

ANewtonstepiseitheracceptedorrejected,thatisνl=1orνl=0whilethediameter
µ<∞ofthecurrenttrustregionisadaptedineachNewtonstep.Accordingtothe
ltextbookofNocedalandWright[32]thetrustregionmethodisdeterminedbythechoiceof
aitsparameterparameters,θ∈[0whic,0h.25)arebythewhichmaximalweradiusdetermineµmaxthe>0,minimtheuminitialpradiuserformanceµ0∈(0needed,µmaxfor)theand
Newtonsteptobeaccepted.Therefore,wedefinetheratio
ρl:=jkmh((qqσ,l,)0)−j−khm((qqσ,l,+δδqqσ))
σ,lσ,lσwhichgivesinformationonthequalityoftheapproximationofjbythefunctionalm.The
parametersµl+1andνlarethendeterminedbyAlgorithm6.9.
Sofar,wehaveseenhowthe(nonlinear)intervalwiseboundaryvalueproblemscanbesolved
bythereducedapproachforPDEconstrainedoptimizationproblems.Wehaveshortly
solvpreseners.tedNext,thethebasicsolutionideasoftecthishniquesapproacforthehanddirectpoinmtedultipleoutshotheotingalgorithmicapproachaspareectspresenoftedthe
.analogously

6.2SolutionTechniquesfortheDirectMultipleShooting
roachApp

Thesolutiontechniquesforthedirectapproachstartanalogouslytotheindirectapproach
withtheapplicationofNewton’smethodtothesystemofmatchingconditions.Thelinearized
systemapproachisin[either14],orsolvedotherwisedirectlyacbyondensingapplicationteofchniqueaKrylovreducessubsptheaceproblemmethotod,alinearsimilartosystemthe
montheultipleconshotrolotingspace.approacThehforcondensedODEdirectconstrainedmultipleshooptimization.otingapproacTherefore,hcomeswepoinclosettoouttheat
theendofthissectionwhycertainefficientsolutiontechniquesfromtheODEapproach
canlimitationsnotbewithtransferredrespecttotothethePDEefficiencyapproac.h,andconsequentlywhythePDEapproachhas

84

6.2SolutionTechniquesfortheDirectMultipleShootingApproach

Newton’smethodrequirestheevaluationoftheresidual,andthelinearsolverneedsthe
calculationofthedirectionalderivativesformingtheleft-handsideofthelinearizedproblem.
Therefore,thelinearsolverrequiresthesolutionofintervalwiselinearinitialvalueproblems
(IVPs)whilethecalculationoftheresidualneedsthesolutionofnonlinearinitialvalue
problems.TheinterrelationofthedifferentproblemsisshowninFigure6.10.

hingMatcconditionsNewton

LinearizedSystem

IVPNonlinear

LinearIVPjQonequationsNonlinearjQonequationsLinear

ResidualesativDerivDirectional

Figure6.10:Differentsolutionapproachesfordirectmultipleshooting.

6.2.1SolutionoftheMultipleShootingSystem
ThelinearsystemdeterminingtheNewtonupdate(Δq˜σ,Δs˜h,Δp˜h)isobtainedbydifferenti-
ationof(4.35)withrespecttothemultipleshootingvariables.
Remark6.13.Inthefollowing,weconsiders0andpmasdefinedbythecorresponding
equationsin(4.35).Therefore,thesevariablesdonothavetobedeterminedbyNewton’s
method,andwecandefineΔs0:=0andΔpm:=0.
Weobtainforj=0,...,m−1,thefollowingequations:
(Δshj+1−S¯jsj(Δshj)−S¯jqj(Δqσj),ϕ)=−(ruj,v)∀ϕ∈Hhj+1,
(Δphj−Ξ¯jpj+1(Δphj+1)−Ξ¯jsj(Δshj)−Ξ¯jqj(Δqσj),ψ)=−(rzj,ψ)∀ψ∈Hhj,
α3(Δqσj,χ)Qj−bqq(qj)(Δqσj,χ,Ξj(pjh+1,shj,qσj))
−bq(qσj)(χ,Ξjsj(Δshj)+Ξjqj(Δqσj)+Ξjpj+1(Δphj+1))=−(rqj,χ)Qj∀χ∈Qdj
(6.26)

85

6SolutionTechniquesfortheMultipleShootingApproach

withtheNewtonresidualsdefinedas
(ruj,ϕ):=(shj+1−S¯j(shj,qσj),ϕ),
(rzj,ψ):=(phj−Ξ¯j(phj+1,shj,qσj),ψ),
(rqj,χ)Qj:=α3(qσj,χ)Qj−b(qσj)(χ,Ξj(phj+1,shj,qσj)).
Here,thederivativesofthesolutionoperatorsfortheprimalequationhavetobeunderstood
accordingtoLemma4.3.Forthesolutionoperatorsofthedualequationananalogous
statementholdsandcaneasilybederivedbydifferentiationofthedualequationwithrespect
topj+1,sj,andqj.Weskiptheexplicitformulationbutannotatethatthediscretesolution
operatorsasconsideredherearedefinedaccordinglyforthediscreteequations.Wecontinue
withtheintroductionofanabbreviatorynotationasdonein(6.2)byintroducingadditional
operatorsEjforj=1,...,m−1,Fjforj=0,...,m−1,andGjforj=0,...,m−2in
ulation:formeakw(Ej(Δshj),χ):=bq(qσj)(χ,Ξjsj(Δshj)),
(Fj(Δqσj),χ):=α3(Δqσj,χ)Qj−bqq(qσj)(Δqσj,χ,Ξj(phj+1,shj,qσj))−bq(qσj)(χ,Ξjqσj(Δqσj)),
(Gj(Δphj+1),χ):=bq(qσj)(χ,Ξjpj+1(Δphj+1)).
Wedefinethepseudomatricesforthedifferentblocksofthelinearizedsystem:
I−S¯0I−S¯j,qj0Fj0
A0:=0F00,q,Aj:=0Fj¯0,Am−1:=−Ξ¯m−1,qm−1I,
0−Ξj,qjI
−S¯1,s10−S¯j+1,sj+100−Em−100
B0:=−E¯10,Bj:=−E¯j+100,Bm−2:=−Ξ¯m−1,sm−100,
−Ξ1,s10−Ξj+1,sj+100
00000
C0:=000,Cj:=00−Gj,Cm−2:=0−Gm−2.
00−G000−Ξ¯jpj+10−Ξ¯m−2pm−1
Furthermore,fortheright-handsideandtheincrement,wedefineforthedifferentintervals
(r0,0,ϕ)(sh1−S¯0(sh0,qσj),ϕ)
(r0,1,χ)Qj:=α3(qσ0,χ)Qj−b(qσj)(χ,Ξj(ph1,sh0,qσ0)),
(rj,0,ϕ)(shj+1−S¯j(shj,qσj),ϕ)
(rj,1,χ)Qj:=α3(qσj,χ)Qj−b(qσj)(χ,Ξj(phj+1,shj,qσj)),
(rj,2,ψ)(phj−Ξ¯j(phj+1,shj,qσj),ψ)
(rm−1,0,χ)Qj:=α3(qσm−1,χ)Qm−1−b(qσm−1)(χ,Ξj(phm,shm−1,qσm−1)),
(rm−1,1,ψ)(phm−1−Ξ¯j(phm,shm−1,qσm−1),ψ)
x0:=(Δsh1,Δqσ0)T,
xj:=(Δshj+1,Δqσj,Δphj)T,
xm−1:=(Δqσm−1,Δphm−1)T,

86

6.2SolutionTechniquesfortheDirectMultipleShootingApproach

suchthatthelinearsystem(6.26)readsinpseudomatrixnotation
A0C0x0r0
B0A1C1x1r1
............=−....(6.28)
Bm−3Am−2Cm−2xm−2rm−2
=:K=:x=:b
Bm−2Am−1xm−1rm−1
ThealgorithmicformulationofNewton’smethodforthesolutionofthedirectmultiple
shootingapproachisequivalenttotheformulationfortheindirectapproachexceptforthe
definitionofthelinearsystem.TheexplicitformulationisgiveninAlgorithm6.10.
Algorithm6.10Newton’smethodforthedirectapproach
1:Seti=0.
2:Startwithinitialguessformultipleshootingvariables
xi=(s1h,i,qσ0,s2h,i,qσ1,ph1,...,smh,i−1,qσm,i−2,pmh,i−2,qσm,i−1,pmh,i−1)T.
eatrep3:4:Calculatetheresidualsbi=(r0,i,...,rm−1,i)Taccordingto(6.3).
5:Solvethelinearsystem(6.28)fortheNewtonupdate
KiΔxi=−bi,
Δxi=(Δs1h,i,Δqσ0,Δs2h,i,Δqσ1,Δph1,...,Δsmh,i−1,Δqσm,i−2,Δpmh,i−2,Δqσm,i−1,Δpmh,i−1)T.
ateiternextCalculate6:xi+1=xi+Δxi.
1−m7:Seti=i+1.
8:untilbi:=r0,i2Hh1×Qd0+j=0rj,i2Hhj+1×Qdj×Hhj+rm−1,i2Qdm−1×Hhm−1<tol.
Remark6.14.(Matrix-vectorbasedformulationofNewton’smethod)Theconsiderationof
Newton’smethodinthediscretecaseneedsthedeterminationofthecoefficientvectorsof
thesolutionsfromalinearsystemofequations.Therefore,weneedtheconcretizationofthe
linearsystem(6.28)forgivenbasesofthediscretizedspaces.Again,wedothisexemplarily
foronlyoneblockrow(Bj−1AjCj)ofthepseudoblockmatrixK.Wememorizethe
correspondingequationsofthelinearizedsystem(6.26).
(Δshj+1−S¯jsj(Δshj)−S¯jqj(Δqσj),ϕ)=−(ruj,ϕ)∀ϕ∈Hhj+1,
(Δphj−Ξ¯jpj+1(Δphj+1)−Ξ¯jsj(Δshj)−Ξ¯jqj(Δqσj),ψ)=−(rzj,ψ)∀ψ∈Hhj,
α3(Δqσj,χ)Qj−bqq(qj)(Δqσj,χ,Ξj(phj+1,shj,qσj))
−bq(qσj)(χ,Ξjsj(Δshj)+Ξjqj(Δqσj)+Ξjpj+1(Δpjh+1))=−(rqj,χ)Qj∀χ∈Qdj.

87

6SolutionTechniquesfortheMultipleShootingApproach

Weassumethatϕ0j,...,ϕnjjisabasisofthediscretespaceHhj,andthatχ0j,...,χνjjisabasis
ofthediscretecontrolspaceQdj.WedenotethecoefficientvectorsfortheNewtonupdates
Δshj,ΔphjandΔqσjwithrespecttothesebasesbyΔξj,s∈Rnj,Δξj,p∈Rnj,andΔξj,q∈Rνj.
Withthisnotationathand,weareabletorewritethelinearequationsas
I0Δξj+1,s−B0Δξj,s−A0Δξj,q=−d0,
I1Δξj,p−C1Δξj+1,p−B1Δξj,s−A1Δξj,q=−d1,
−C2Δξj+1,p−B2Δξj,s−A2Δξj,q=−d2.
ThematricesI0∈Rnj+1×nj+1,I1∈Rnj×nj,A0∈Rnj+1×νj,A1∈Rnj×νj,A2∈Rνj×νj,
B0∈Rnj+1×nj,B1∈Rnj×nj,B2∈Rνj×nj,C1∈Rnj×nj+1,C2∈Rνj×nj+1havetobe
understoodaccordingtothefollowingidentities:
(I0)li=(ϕij+1,ϕlj+1),
(I1)li=(ϕij,ϕlj),
(A0)li=(S¯jqj(χij),ϕlj+1),
(A1)li=(Ξ¯jqj(χij),ϕlj),
(A2)li=−α3(χij,χlj)Qj+bqq(qj)(χij,χlj,Ξj(phj+1,shj,qσj))+bq(qσj)(χlj,Ξjqj(χij)),
(B0)li=(S¯jsj(ϕij),ϕlj+1),
(B1)li=(Ξ¯jsj(ϕij),ϕlj),
(B2)li=bq(qσj)(χlj,Ξjsj(ϕij)),
(C1)li=(Ξ¯jpj+1(ϕij+1),ϕlj),
(C2)li=bq(qσj)(χlj,Ξjpj+1(ϕji+1)).
Theright-handsidevectorsd0∈Rnj+1,d1∈Rnj,andd2∈Rνjaregivenaccordingto
(d0)l=(ruj,ϕlj+1),(d1)l=(rzj,ϕlj),and(d2)l=(rqj,χlj)Qj.
Withthisinterpretationathand,thetransferofthefunctionspacemethodtoavectorbased
oneisstraightforwardbysettingupthediscretelinearsystemforthedeterminationofthe
coefficientvectorsoftheNewtonupdates.Inaddition,itisoftenconvenientandsuitableto
replacetheHilbertspacenormonHbythel2-normonthefinitedimensionalspaceRnj.
Inthefollowing,weconsiderExample(2.3)toillustratetheapplicationofNewton’smethod
case.concreteainExample6.4.(Newton’smethodforExample2.3)Wereconsiderthemultipleshooting
approachforExample2.3with2intervals(Example4.2).Weremark,thats0andp2are
determinedbytheboundaryvaluesoftheoriginalproblemandarenotpartoftheunknowns
problem:dlinearizetheof(S¯0,q0(Δqσ0)−Δsh1,ϕ)=−(S¯0(sh0,qσ0)−s1,ϕ)∀ϕ∈Hh1,
(Ξ¯1,s1(Δsh1)+Ξ¯1,q1(Δqσ1)−Δph1,ψ)=−(Ξ¯1(sh1,qσ1,ph2)−ph1,ψ)∀ψ∈Hh1,
(Δqσ0,χ)Q0+(Ξ0,q0(Δqσ0)+Ξ0,p1(Δph1),χ)=−((qσ0,χ)Q0+(Ξ0(sh0,qσ0,ph1),χ))∀χ∈Qd0,
(Δqσ1,χ)Q1+(Ξ1,s1(Δsh1)+Ξ1,q1(Δqσ1),χ)=−((qσ1,χ)Q1+(Ξ1(sh1,qσ1,ph2),χ))∀χ∈Qd1.

88

6.2SolutionTechniquesfortheDirectMultipleShootingApproach

Inpseudomatrixnotationthisspecialsystemiswrittenas
−IS¯0,q000Δsh1S¯0(sh0,qσ0)−sh1
0I+Ξ0,q00Ξ0,p1Δqσ0=−qσ0+Ξ0(sh0,qσ0,ph1).
Ξ1,s10I+Ξ1,q10Δqσ1qσ1+Ξ1(sh1,qσ1,ph2)
00Ξ¯1,q1−IΔph1Ξ¯1(s1h,qσ1,ph2)−ph1
Differentiationofthedualequationyieldsthedeterminingequationforthederivativesofthe
solutionoperators,andaccordinglyfortheirdiscretizationsbyconsideringthecorresponding
discreteequations.However,wepresentforthesakeofbrevityoftheequationsonlythe
case.ousutinconWedeterminex,y,w∈Xj,suchthatforallϕ∈Xj,j=0,1,thefollowingPDEsarefulfilled:
−((∂txj,ϕ))j+((xj,ϕ))j+((3(uj)2xj,ϕ))j+(xj(τj+1),ϕ(τj+1))
=((Sj,sj(ξ),ϕ))j−((6ujSj,sj(ξ)xj,ϕ))j,
−((∂tyj,ϕ))j+((yj,ϕ))j+((3(uj)2yj,ϕ))j+(yj(τj+1),ϕ(τj+1))
=((Sj,qj(ρ),ϕ))j−((6ujSj,qj(ρ)xj,ϕ))j,
−((∂twj,ϕ))j+((wj,ϕ))j+((3(uj)2wj,ϕ))j+(wj(τj+1)−ζ,ϕ(τj+1))=0.
Thederivativesofthesolutionoperatorswithrespecttosj,qj,andpj+1aregivenasthe
solutionoperatorsoftheequationsabovebythefollowingmappings:
Ξj,sj(ξ)=x,Ξj,qj(ρ)=y,Ξj,pj+1(ξ)=w,
Ξ¯j,sj(ξ)=x(τj),Ξ¯j,qj(ρ)=y(τj),Ξ¯j,pj+1(ξ)=w(τj).

6.2.2TheGMRESMethodfortheSolutionoftheLinearizedSystem

Asintheindirectmultipleshootingapproach,thepseudomatrixnotationofproblem(6.28)
hasasparsetridiagonalblockstructure,butwithmorecomplicatedblocksonthediagonal.
cNevhoice,ertheless,andthetheapapplicationplicationofaofthepreconditionergeneralizeisdabsolutelyminimumrnecessaryesidual.methoThisdfactisisanoboutlinedvious
bandyadifferenconsiderationtnumofberofExampleinterv6.2alswhicasshhohaswnbineenFiguresolved6.11.forAsdifferenbeforetlyafineforwardbdiscretizationsackward
blockGaussSeidelpreconditionerisduetothestructureofthepseudosystemmatrixKan
appropriatechoice.Thisassumptionisadditionallysupportedbyitsoutstandingperformance
inmatrixcomparisondecompwithositiondiinfferentotalowerpreconditionerstriangularaspsshoeuwndoinblockFigurematrix6.12.L,anAgain,uppweerintrtroianducegulara
pseudoblockmatrixU,andadiagonalpseudoblockmatrixD:
A0C0
...−UB0A1C1
D:≡..........
−L...Bm−3Am−2Cm−2
Bm−2Am−1

89

6SolutionTechniquesfortheMultipleShootingApproach

105InIntervtervalsals
alstervIn15

700600500400105InIntervtervalsals
alstervIn15300200100002004006008001000
Figure6.11:ConvergencebehavioroftheGMRESmethodfordifferentlyfinediscretizations
calculatedfordifferentnumbersofshootingintervalsinExample6.2.(x-axis:
numberofcells,y-axis:numberofGMRESsteps)
1000

100

10

GSFW-BWGSFWGSBWID

1510152025303540
Figure6.12:ResultsfortheforwardbackwardGaussSeidelpreconditioner(red),the
forwardGaussSeidelpreconditioner(darkblue),thebackwardGaussSeidel
preconditioner(blue)andwithoutpreconditioner(magenta)calculatedfor
Example6.2.(x-axis:numberofintervals,y-axis:numberofiterations)
ThepreconditioningpseudomatrixPfortheforwardbackwardblockGaussSeidelprecondi-
tioningwasalreadyintroducedintheprevioussectioninequation(6.5)as
P:=(D−L)D−1(D−U),

90

6.2SolutionTechniquesfortheDirectMultipleShootingApproach

andthepreconditionedproblemisgivenasinequation(6.6)bythelinearsystem

P−1Kx=P−1b.

bloWhileckthestructuregeneralofthesetupofdiagonalthebloGMREScksresultsmethodinisathemoresameassophisticatedinAlgorithmblock6.in2,vtheersionindifferenthet
caseofdirectmultipleshooting.Therearedifferentpossiblewaystoinvertadiagonalblock
ofthepseudomatrixD.Ontheonehand,adiagonalblockofDcanbeinterpretedasan
optimizationproblem.Inthiscase,standardtechniquesforthesolutionofoptimization
ofanproblemsiterativcanebesolvapplied.er.TheOnintheterpretationotherhandinthetermsinvofersionancanbeoptimizationachievedbyproblemdirectisthoroughlyapplication
discussedin[20],wheredifferentexamplesandnumericalresultsinthecontextoflinear
solverquadraticfortheinoptimizationversionofproblemsthearediagonalgiven.blocWk.efolloButwthefirst,otherletuswaydescribandeapplyforanthesakiterativeofe
completenessthemultiplicationwiththepreconditioningpseudoblockmatrixbysuccessive
substitution.Asmentioned,theprocedureofpreconditioningismainlyperformedasbefore
exceptfortheadditionalmultiplicationwithD.First,theinversionofthelowerdiagonal
pseudoblockmatrixD−Lisdonebyforwardsubstitution,asshowninequation(6.29).
ofAfterwtheuppards,erwemtriangularultiplypseudwithotheblockdiagonalmatrixpDseudo−UisblockobtainedmatrixbyDbacandkwardfinallythesubstitutioninversionas
(6.30).equationinengiv

(6.29)

1−x0=A0r0
−1
.x1=A1(r1−B0x0)
(D−L)x=r⇔..(6.29)
−1
xm−2=Am−2(rm−2−Bm−3xm−3)
xm−1=Am−1(rm−1−Bm−2xm−2)
−1
1−xm−1=Am−1rm−1
−1
xm−2=Am−2(rm−2−Cm−2xm−1)
.(D−U)x=r⇔..(6.30)
−1
x1=A1(r1−C1x2)
x0=A0(r0−C0x1)
−1
Theevaluationofthedirectionalderivativeshasalreadybeendiscussedintheprequel.We
nowdescribetheblockinversionofAjbymeansofaniterativesolver.Wedecidedtoapply
astandardGMRESmethodasalreadystatedinAlgorithm6.2forthesolutionofthelinear
blockproblem,wherethepseudomatrixvectorproducty=Ajxisobtainedbysolutionof
additionalproblemsasgiveninAlgorithm6.11.

91

6SolutionTechniquesfortheMultipleShootingApproach

Algorithm6.11EvaluationofthepseudomatrixvectorproductforadiagonalblockAj.
1:ifj=0then
2:Calculatev0∈Hh1from(v0,ϕ)=(S¯0,q0(x1),ϕ)∀ϕ∈Hh1.
3:Calculatev1∈Qd1from(v1,χ)=α3(x1,χ)Qj−bqq(qσ0)(x1,χ,zσ0)∀χ∈Qd1.
4:Sety0=−x0+v0.
5:Sety1=v1.
.yreturn6:7:elseifj>0andj<m−1then
8:Calculatev0∈Hhj+1from(v0,ϕ)=S¯j(x1)∀ϕ∈Hhj+1.
,qj9:Calculatev2∈Hhjfrom(v2,ψ)=(Ξ¯jqj(x2),ψ)∀ψ∈Hhj.
10:Calculatev1∈Qdjfrom(v1,χ)=α3(x1,χ)Qj−bqq(qσj)(x1,χ,zσj)∀χ∈Qdj.
11:Sety0=−x0+v0.
12:Sety1=v1.
13:Sety2=−x2+v2.
.yreturn14:else15:16:Calculatev1∈Hhm−1from(v1,ψ)=(Ξ¯jqm−1(x0),ψ)∀ψ∈Qdm−1.
17:Calculatev0∈Qdm−1from(v0,χ)=α3(x0,χ)Qj−bqq(qm−1)(x0,χ,zj)∀χ∈Qdm−1.
18:Sety0=v0.
19:Sety1=−x1+v1.
.yreturn20:

Remark6.15.Theconversionofthealgorithmtoavectorbasedoneisstraightforward,and
canbeachievedbyconsiderationofRemark6.14fortheinterpretationofthesolutionsand
solutionoperatorsintermsofvectorsandmatrices.

ofWethewanttosystem.giveanTherefore,impressionweofassumetheadthebditionalestcase,effortwhicwhichhisisthatneededweforaretheableprtoinveconditioninerttheg
diagonalblockAjinonestepoftheinnerGMRESmethod.Theanalysisofthecostsyields
ofthe2∙follomwinginitialvaluecoherence:problems,1stepofwhereasthe1plainstepofGMRESthemethopreconditioneddneedstheGMRESnumericalmethodsolutionneeds
thesolutionofatleast6∙minitialvalueproblems.Thus,acompensationoftheadditional
effortisachievedonlyifthenumberofGMRESiterationsisatleastreducedto1/3ofthose
intheunpreconditionedcase.Thebestcaseusuallydoesnotholdforourapplications,such
thatanenormouslylargernumberofadditionalinitialvalueproblemshastobesolvedthan
predictedbythissimpleformula.Nevertheless,thereductionobtainedbypreconditioning
stillcouldbreducesetheobtainedoverallwithouteffortprecandonenablesditioning.ustoWesolvhaevethosesolvedproblemsExamplefor6.2whichwithnoandconvwithoutergence
initialvpreconditioningalueproblemsondiffereforntdifferennumtbnersumbofersinoftervmualsltipandleshohaveotingplottedintervthealsninumbFigureerof6.13.solved

Anotherapproach.Wepromisinggiveaapproacdetailedhisthedescriptionapplicationofthisofideaincondensingthenexttechniquesubsection.ssimilartotheODE

92

2.0∙104
1.5∙104
1.0∙104
5.0∙103

6.2SolutionTechniquesfortheDirectMultipleShootingApproach

preconditionedpreconditionednot

0510152025303540
Figure6.13:Numberofsolvedinitialvalueproblemsfordifferentnumbersofshooting
intervals,calculatedwithandwithoutpreconditioning.(x-axis:numberof
intervals,y-axis:numberofsolvedproblems)

6.2.3CondensingTechniquesfortheSolutionoftheLinearizedSystem
Werecalltheformulationofthelinearizedsystem(6.26)andaimatthereductionofthis
systemontothespaceofthecontrolvariablesbysuccessivesubstitution.Forthispurpose,
wemakeuseofthefollowingrelations.
(Δsh1,ϕ)=(S¯jq0(Δqσ0),ϕ)−(ru0,ϕ),
(Δsh2,ϕ)=(S¯js1(Δsh1)+S¯jq1(Δqσ1),ϕ)−(ru1,ϕ),
...(Δsm−1,ϕ)=(S¯m−2(Δsm−2)+S¯m−2(Δqσm−2),ϕ)−(rum−2,ϕ),
hm−2shm−2qσ
(Δphm−1,ψ)=(Ξ¯jsm−1(Δshm−1)+Ξ¯jqm−1(Δqσm−1),ψ)−(rzm−1,ψ),
(Δphm−2,ψ)=(Ξ¯jpm−1(Δphm−1)+Ξ¯jsm−2(Δshm−2)−Ξ¯m−2(Δqσm−2),ψ)−(rzm−2,ψ),
qjσ...(Δph1,ψ)=(Ξ¯jp2(Δph2)+Ξ¯js1(Δsh1)+Ξ¯jq1(Δqσ1),ψ)−(rz1,ψ).
WecannowwritetheunknownNewtonincrementsforprimalanddualshootingvariablesas
afunctionofthecontrolvariables.Introducing
jfj:Q˜d→Hh,
jgj:Q˜d→Hh

93

6SolutionTechniquesfortheMultipleShootingApproach

thathsucΔshj=fj(Δqσ0,...,Δqσm−1)
Δphj=gj(Δqσ0,...,Δqσm−1),
wereducethelinearmultipleshootingsystem(6.26)tothedeterminationofthecontrol
incrementsΔqσ0,...,Δqσm−1fromthecondensedmultipleshootingsystem(6.31).
Letforj=0,...,m−1thefollowingequationsbefulfilled:
α3(Δqσj,χ)Qj−bqq(qσj)(Δqσj,χ,Ξj(phj+1,shj,qσj))
−bq(qσj)(χ,Ξjsj(fj(Δqσ0,...,Δqσm−1))+Ξjqj(Δqσj))
+bq(qσj)(χ,Ξjpj+1(gj(Δqσ0,...,Δqσm−1)))=−(rqj,χ)Qj∀χ∈Qdj.(6.31)
ThewholesolutionprocedureofeachNewtonstepisthusbroughtdowntothesolutionof
(6.31)byapplicationofaGMRESmethod.ThealgorithmicperformanceofthisGMRES
methodneedstheevaluationofthedirectionalderivativesgivenontheleft-handsideof
(6.31),whichisdoneasstatedinAlgorithm6.12.
Algorithm6.12CalculationofthedirectionalderivativesvjintheGMRESmethod
Require:IteratesΔqσ0,...,qσm−1.
1:forj=1,...,m−1do
2:EvaluateΔshj=fj(Δqσ0,...,Δqσm−1).
3:forj=m−1,...,1do
4:EvaluateΔphj=gj(Δqσ0,...,Δqσm−1).
5:forj=0,...m−1doj
6:Calculatedirectionalderivativevj∈Qdfrom
(vj,χ)=α3(Δqσj,χ)Qj−bqq(qσj)(Δqσj,χ,Ξj(phj+1,shj,qhj))
−bq(qσj)(χ,Ξjsj(Δshj)+Ξjqj(Δqσj)+Ξjpj+1(Δphj+1))∀χ∈Qdj.
7:returnvj∈Q˜d.
Remark6.16.(Algorithm6.12intermsofthecoefficientvectors)Thealgorithmcanexplicitely
bewritteninmatrixvectornotationbyapplyingthenotationintroducedinRemark6.14.
ForexamplewiththevectorsdefinedasinRemark6.14werewritestep6as
ζj=−C2Δξj+1,p−B2Δξj,s−A2Δξj,q
wherethecoefficientvectorofthedirectionalderivativeisdenotedasζj∈Rνj.
ThereductionofthenumberofGMRESstepsbycondensingthesystemisquiteremarkable
andtheadditionaleffortislimitedtothecondensinginthebeginningandtheexpansion
afterconvergence.Theseproceduresmakeatotalamountof4∙madditionalinitialvalue
problems.Figure6.14givesanimpressionoftheimprovementobtainedbyapplicationofthe

94

6.2SolutionTechniquesfortheDirectMultipleShootingApproach

condensingtechniquesincomparisontothepreconditioneddirectmultipleshootingapproach.
Example6.2hasbeensolvedon1024cellsbybothtechniques,andthenumberofsolved
initialvalueproblemsgivesevidenceofthecomputationaleffortsavedbycondensing.
1.2∙104
1.0∙104direct(preconditioned)
condensed8.0∙103
6.0∙103
4.0∙103
2.0∙103
0510152025303540

Figure6.14:Numberofsolvedinitialvalueproblemsfordifferentnumbersofshooting
(inxterv-axis:als,numbcalculatederofinfortervals,ypreconditioned-axis:numandberofcondensedsolveddirectproblems)multipleshooting.
Ascomesmenticloseonedtointhetheideaintroofmductionultipleofshothisotingsection,fortheODEcondensedconstrainedmultipleoptimizationshootingapproacproblems.h
Ashotthisotingpoint,approacweharefromableODEstotounderlinePDEs.theThisisrestrictionsdoneinofthethefollotransfwingerofthesubsection.directmultiple

6.2.4FromODEstoPDEs–Limitations
IndirectChaptermultiple3wsehohavotingegivenapproacanohvforerviewODEofconstrainedstate-of-the-arttecoptimization.hniquesTheforthemainadvsolutionantageoftheof
multipleshootingforODEsisthehighefficiencywhichisnotonlyobtainedbyapplicationof
condensingtechniques,butdependscruciallyontheefficiencyofthetimesteppingschemes
andthesimultaneouscalculationofalldirectionalderivativesfortheexplicitassemblingand
inversionofthelargematrixofalldirectionalderivatives.
InfurthertheconresearctexthofisPDEsnecessarynot.allFirstofoftheall,latterwediscrefeaturestizeincanbtimeebyapplieddisconfortincuousertainorconreasons,tinuousand
whicGalerkinhcanmebethinods,terpretedandareasathusGalerkinlimitedtomethothed.applicationNumericalofthosecomputationstimeinthissteppingthesisschemeshave
areonlybneededeenptotesterformedhigherfortheorderdG(0)timefirststeppingordersctimehemessteppingandthescpheme.ossibleFurtherimprovinvementestigationsofthe
multipleshootingapproach.

95

6SolutionTechniquesfortheMultipleShootingApproach

Second,thesimultaneouscalculationofalldirectionalderivativesfortheassemblingofthe
largesystemmatrixresultsinanunacceptablehighcomputationaleffortwheneverthecontrol
isdiscretizedsufficientlyfine.Considerforexamplethecaseofadistributedcontrolona
spatialmeshof5000nodes,andthestatesdiscretizedaccordingly.OneachintervalIj,the
directionalderivativeswithrespecttoqj,sjandλjhavetobecalculatedforeverydirection,
thatiswithrespecttoeverybasisvectorofthefiniteelementdiscretization.Assuming
bilinearelements,thismakesatotalof5000forwardand5000backwardinitialvalueproblems
perinterval.Andforatotalof10intervals,weobtainanamountof100000initialvalue
problemsfortheassemblingofthematrix.
Third,thestoragerequirementsareextremelylarge.Thesystemmatrixforthecondensed
systemintheabovementionedexampleisofthesizeof10∙5000×10∙5000=50000×50000
and(duetothecondensing)usuallynotofsparsestructure.Ifthematrixisonceassembled,
theexplicitinversionisquiteexpensivetoo,andconsequently,thisprocedureisnotsuitable
inthecontextofPDEswithfinespatialdiscretizationsofthecontrolspace.
Finally,wewanttodevotetherestofthischaptertoanumericalcomparisonoftheefficiency
ofthepreviouslypresentedmultipleshootingapproachesforPDEconstrainedoptimization
problems.

6.3NumericalComparisonoftheDirectandIndirectMultiple
ShootingApproach

Inthisfinalsection,anumericalcomparisonbetweentheindirectmultipleshootingapproach,
thedirectmultipleshootingapproachandthecondensedmultipleshootingapproachisgiven.
WeconsideranonlinearoptimizationproblemwithNeumannboundarycontrol.
Example6.5.(NeumannboundarycontrolonaT-shapeddomain)Wesearchtominimize
functionalcostdistributedthe12α2
(q,u)∈minQ×XJ(q,u)=2Iu−u¯Γodt+2IqQdt
subjecttotheconstrainingnonlinearequation
∂tu−Δu−u3−u=0onI×ΩT,
u(0)=u0onΩT,
∂nu=qonI×Γc,
∂nu=0onI×ΩT\Γc.
ΩTisaT-shapeddomainwithcontrolboundaryΓcandobservationboundaryΓoassketched
inFigure6.15.Intheconcretecaseofinterest,wepicktheparametersu0=0.9,u¯≡1,
α=0.01andthetimedomainI=(0,2).Furthermore,thetimesteppingschemeisinitialized
withatimestepsizeof0.01.

96

6.3NumericalComparisonoftheDirectandIndirectMultipleShootingApproach

Γo

ΓcFigure6.15:ΩTwithobservationandcontrolboundary(ΓoandΓc).
Thenumericalcomparisonoftheefficiencyofthethreeapproachesismainlybasedontime
measurements–firstofallthetotalcomputationtimeoftheapproachesgivesevidenceof
theefficencyofeachapproach,butadditionallywepointout,whatthetimeisspentfor
indetail.Throughoutthissection,weconsiderthesolutionfor5,10,15and20multiple
shootingintervals.Thecontrolsareintervalwiseconstantintimebutdistributedinspace.
Thevaluesofalltimemeasurementsaregiveninseconds.
Remark6.17.Thefollowingcomputationsareperformedononlyonefixedspatialmesh.
Nevertheless,numericalcomputationsonmeshesofdifferentrefinementlevelssuggestthe
assumptionthattheresultspresentedinthissectionareindependentofthemeshsize.
ThetotaltimeneededforthesolutionofExample6.5bythedifferentapproachesisdepicted
againstthenumberofintervalsinFigure6.16.Whilethetimeneededforthesolutionofthe
100000condenseddirectindirect

10000

1000

10068101214161820
Figure6.16:TotaltimefordifferentnumbersofintervalsforExample6.5,solvedbydifferent
multipleshootingapproaches.(x-axis:numberofintervals,y-axis:timein
seconds)initialvalueproblemsisinverseproportionaltothenumberofintervals,thetotalnumber
ofinitialvalueproblemsgrowsproportionaltothenumberofintervals.Furthermore,the

97

6SolutionTechniquesfortheMultipleShootingApproach

numberofstepsoftheouterNewtonmethodandtheouterGMRESmethodarenearly
independentfromthenumberofintervals,suchthattheapproximatelyconstanttotaltime
fordirectandcondensedapproachisthelogicalconsequence.Ontheotherhand,forthe
indirectapproach,thenumberofiterativestepsneededforthesolutionoftheintervalwise
boundaryvalueproblemsdecreaseswiththeintervallength,whiletherelationsabovestill
hold.Therefore,thetotaltimeneededforthesolutionoftheproblembytheindirectapproach
decreaseswithagrowingnumberofintervals.
Thedistributionofthetotaltimeontothesolutionofthelinearandnonlinearinterval
problemsfordifferentmultipleshootingparameterizationsisshowninFigure6.17.The

100000

10000

1000

linearcondensedlineardirectindirectlinearnonlinearcondensednonlineardirectnonlinearindirect

10068101214161820

Figure6.17:Timespentforthesolutionofthelinearandthenonlinearintervalproblems
forExample6.5,solvedbydifferentmultipleshootingapproaches.(x-axis:
numberofintervals,y-axis:timeinseconds)

mostplotsofconfirmthetimewhatiswspeendhaveforthealreadypsolutionostulatedoftheinthelinearizedprevioussystemsections:ofmatcForhingallapproacconditions.hes,
Precisely,Figures6.18and6.19pointoutthatfordirectandindirectmultipleshooting
onlywithoutapproxicondensingmately2ab%outpass97%ofduringthethecomputationcalculationtimeofthareeneededresidual.forThethisrestproofcedure,thetimewhileis
spentfortheinitialization,theoutput,andotherprocedureswhichdonotdirectlybelongto
themequilibrated,ultipleshothoughotingstillmethoaboutd.F82or%ofthectheontimedensedisapneededproacforhthethetimesolutionofdistributiontheislinearizedmore
problemwhileonly17%ofthetimearespentforthecalculationoftheresidual.
Sofar,wehaveseen,thatthecondensedapproachisthemostpromisingoneoutofthethree
differentapproachesconsideredinthischapter.Inthenexttwochapters,wewanttocombine
themultipleshootingapproachwithmeshadaptationgainedbyamodificationofthedual
weightedresidualmethod.Thisideaallowsustotheincreasetheefficencyofthemethodsby

98

6.3NumericalComparisonoftheDirectandIndirectMultipleShootingApproach

condensednonlinear
nonlineardirectnonlinearindirect

18161412condensednonlinear
nonlineardirect10nonlinearindirect8642068101214161820
Figure6.18:Percentageofthetimeforthesolutionofthenonlinearintervalproblems
forExample6.5,solvedbydifferentmultipleshootingapproaches.(x-axis:
numberofintervals,y-axis:percentageoftime)
1009896949290linearcondensed88lineardirect86linearindirect84828068101214161820
Figure6.19:Percentageofthetimeforthesolutionofthelinearintervalproblemsfor
Example6.5,solvedbydifferentmultipleshootingapproaches.(x-axis:number
ofintervals,y-axis:percentageoftime)

linearcondensedlineardirectindirectlinear

solvingthePDEsunderconsiderationonmeshesthatareascoarseaspossibleforobtaining
acertaindiscretizationerrorinthecostfunctional.

99

7APosterioriErrorEstimation

Thischapterisdevotedtothepresentationofaposteriorierrorestimation.Startingwiththe
repetitionoftheoriginalideasofdualweightedresidualaposteriorierrorestimation(DWR
method)inSection7.1,weproceedwiththeapplicationoftheclassicalerrorestimatorto
themultipleshootingapproach.Wepointout,thattheclassicalerrorestimatorislimitedto
thecaseofcoincidingadjacentmeshesatthemultipleshootingnodes.Next,anewerror
estimator,especiallysuitedfortheapplicationtothemultipleshootingsolution,isderived
inSection7.2.Thiserrorestimatorisdesignedtoincorporatetheprojectionerrorsonthe
multipleshootingnodesduetodifferentadjacentmeshes.Itisbasedontheconsideration
oftheconvergedmultipleshootingsystemintermsofaGalerkinapproach.Whilethe
errorestimatorsaredevelopedfortheerrorbetweenthefullydiscretesolutionandthe
continuoussolutioninfunctionspace,forthenumericalconsiderationwelimitourselves
tothefurtherinvestigationofthespatialpartoftheerrorfortheremainingpartofthe
chapter.Thedevelopederrorestimatorscontaintheunknownsolutionoftheoptimization
problem.Therfore,theevaluationoftheerrorestimatorsneedsreliableapproximationsof
theaforementionedsolutions.Consequently,thepracticalevaluationoftheerrorisdiscussed
inSection7.3.Finally,thischapterclosesbypresentingsomenumericalexamplesinSection
7.4whichillustratetheperformanceofthedevelopederrorestimators.
Remark7.1.Weshouldremarkthattheapplicationoftheconsiderederrorestimatorsto
themultipleshootingsolutionisindependentofthemultipleshootingapproach.Theerror
estimatorasanadditionalfeatureisappliedtothesolutionafterconvergenceandneedsno
informationontheapproachitself.

7.1TheClassicalErrorEstimatorfortheCostFunctional

Westartthissectionbyreviewingamodificationofafundamentalresultfrom[7].Thismodi-
ficationisforexamplederivedin[29]andisanessentialingredientforalllaterdevelopments.
Lemma7.1.LetYbeafunctionspaceandLathreetimesGâteauxdifferentiablefunctional
onY.Weseekastationarypointy∗ofLonY1⊂Y,thatis
L(y1∗)(y1)=0∀y1∈Y1.
Thisequationis∗approximatedbyaGalerkinmethod,usingaspaceY2⊂Y.Theapproximative
problemseeksy2∈Y2suchthat
L(y2∗)(y2)=0∀y2∈Y2.(7.1)

101

7APosterioriErrorEstimation

(7.2)

Ifthecontinuoussolutiony1∗fulfillsadditionally
L(y1∗)(y2)=0∀y2∈Y2,(7.2)
andthefollowingerrorrepresentationholdsforarbitraryy2∈Y2.
L(y1∗)−L(y2∗)=1L(y2∗)(y1∗−y2)+R.
2TheremaindertermRisgivenby
1R=21L(y2∗+se)(e,e,e)∙s∙(s−1)ds
0wheretheerroreisdefinedase=y1∗−y2∗.
Proof.First,fromthemaintheoremofcalculus,thefollowingidentityholdsfory1∈Y1,
y2∈Y2:1
L(y1∗)−L(y2∗)=0L(y2∗+se)(e)ds.
Applicationofthetrapezoidalruletotheintegralyields:
11L(y2∗+se)(e)ds=21L(y2∗)(e)+21L(y∗1)(e)+21L(y2∗+se)(e,e,e)∙s∙(s−1)ds.
00=:R
Thesecondaddendvanishesaccordingto(7.2),anddueto(7.1)wehaveforarbitraryy2∈Y2
1L(y2∗)(e)=1L(y2∗)(y1∗−y2).
22

of.prothecompletesThisRemark7.2.(Notation)ThroughoutthischapterdifferentLagrangians,costfunctionals,and
residualsaredefined.Inordertokeepthenotationeasilyunderstandableweusethesame
notationwithinthedifferentsubsectionswhereasthecertainmeaningbecomesclearinthe
context.TheLagrangiansinfunctionspacearedenotedbyL,Lagrangiansofthediscretized
problemsbyL˜.Furthermore,allresidualsarenamedρ˜with,ifnecessary,additionalindices
tospecifythecorrespondingintervalnumberandequation.
Wewanttorecallsomeresultsontheestimationofthediscretizationerrorinthecost
functionalbythedualweightedresidualmethod.Theideaofthisapproachwasfirst
developedbyBecker,Kapp,andRannacherin[4],[5],[3],[7]andextensionstooptimization
problemshavebeendevelopedbyMeidnerandVexlerin[6],[30].WhereasMeidnerandVexler
areinterestedinsplittingthediscretizationerrorofthefullydiscretizedsolutionintodifferent
partsduetothespatialdiscretization,thetemporaldiscretization,andthediscretizationof
thecontrolspace,wefirstderivetheerrorestimatorsfortheerrorJ(q,u)−J(qσ,uσ)andlater
inthenumericalapplicationsrestrictourselvestothecaseofestimatingJ(qk,uk)−J(qσ,uσ).
WestartbyrecallingaresultfortheDWRerrorrepresentationofthecostfunctionalJ
onthewholetimeinterval.Therefore,weneedsomepreparatoryresultspresentedinthe

102

7.1TheClassicalErrorEstimatorfortheCostFunctional

following.WerecallthedefinitionoftheLagrangianL:Q×X×X→Rfromequation
(4.3),whichwasgivenby
L(q,u,z):=J(q,u)−{((∂tu,z))+a(u)(z)+b(q)(z)+(u(0),z(0))−((f,z))−(u0,z(0))}.
ItsdiscretecounterpartL˜:Qd×X˜kr,s×X˜kr,s→RreadsintermsofthechosendG(r)cG(s)
discretizationnL˜(qσ,uσ,zσ):=J(qσ,uσ)−((∂tuσ,zσ))l+a(uσ)(zσ)+b(qσ)(zσ)
l=1n−1
+([uσ]l,zk+,l)+(uσ−,0−u0,zσ−,0)−((f,zσ)).(7.3)
=0lForthederivativesofthediscreteLagrangianthefollowingrelationsholdtrue:
Lemma7.2.Letfortheoptimizationproblemofinterest,lastlystatedin(4.2),thesolutions
ofthecontinuous,resp.discretized,problembegivenas
(q,u,z)∈Q×X×Xand(qσ,uσ,zσ)∈Qd×X˜kr,hs×X˜krh,s.
Thenthefollowingequationsholdtrue:
L˜(qσ,uσ,zσ)(qˆσ,uˆσ,zˆσ)=0∀(qˆσ,uˆσ,zˆσ)∈Qd×X˜kr,hs×X˜krh,s,(7.4)
L˜(q,u,z)(qˆσ,uˆσ,zˆσ)=0∀(qˆσ,uˆσ,zˆσ)∈Qd×X˜kr,hs×X˜krh,s.(7.5)
Proof.Equation(7.4)isequivalenttothefirstorderoptimalityconditionofthediscretized
optimizationproblemandthusfulfilledforitssolution(qσ,uσ,zσ)∈Qd×X˜krh,s×X˜kr,hs.The
proofofequation(7.5)ismoresophisticated.Inordertoclarifythebasicideasofthisproof,
weshowhowtheverificationisperformedfortheprimalequation
L˜zj(q,u,z)(zˆσ)=0∀zˆσ∈X˜krh,s.(7.6)
Theprooffortheremainingequations
L˜ju(q,u,z)(uˆσ)=0∀uˆσ∈X˜krh,s,
L˜qj(q,u,z)(qˆσ)=0∀qˆσ∈Qd.

.analogouslywsfolloRegardingthedefinitionoftheLagrangianin(7.3),thejumpsandtheinitialconditioncancel
outforthecontinuoussolution.Weremainwiththereformulationof(7.6)asfollows:
nj((∂tu,zˆσ))l+a(u)(zˆσ)+b(q)(zˆσ)=0∀zˆσ∈X˜krh,s.(7.7)
=1lNor,w,sby2densityofXinL2(I,V)withrespecttothenormonL2(I,V),andbytheinclusion
X˜kh⊂L(I,V),weareabletoapproximatezˆσbyasequenceoffunctionsinX.Considering
thelimit,weobtainthat(7.7)istrueandconsequentlythatequation(7.6)holdstrue.

103

7APosterioriErrorEstimation

Now,wehaveeverythingathandtoconsidertheerrorrepresentationofthefunctionalerror
inthecostfunctionalJ:
J(q,u)−J(qσ,uσ).
Weareabletoprovethefollowingwellknownresultforthediscretizationerror:
Theorem7.3.Let(q,u,z)beastationarypointoftheLagrangianL,thatis,itfulfillsthe
firstorderoptimalityc˜ondition(4.4).Furthermore,let(qσ,uσ,zσ)beastationarypointof
thediscreteLagrangianL,thatis,itsolvesthediscretizationof(4.4).Fortheerrorofthe
costfunctionalthefollowingrepresentationholds:
J(q,u)−J(qσ,uσ)=21L˜(qσ,uσ,zσ)(q−qˆσ,u−uˆσ,z−zˆσ)+Rσ(7.8)
where(qˆσ,uˆσ,ˆzσ)∈Qd×Xkrh,s×Xkrh,sarbitraryandtheremaindertermsRσhasthesame
formasinLemma7.1withL=L˜j.
Prtheoof.fullyFromdiscreteprevioussolutionthoughfulfillstswitseknow,discretization.thattheconTherefore,tinuousthepsolutionrimalfulfillsequation(4.4)in,theand
formulationoftheLagrangianvanishesandwedirectlyobtain
J(q,u)−J(qσ,uσ)=L˜(q,u,z)−L˜(qσ,uσ,zσ).
TheremainingpartoftheproofisobtainedfromLemma7.1with
Y1:=Q×X×X,
Y2:=Qd×X˜krh,s×X˜kr,hs,
Y:=Y1+Y2.
DuetoLemma7.2,condition(7.2)inLemma7.1holdsforL=L˜,andweretrievethe
yequalitL˜(q,u,z)−L˜(qσ,uσ,zσ)=21L˜(qσ,uσ,zσ)(q−qˆσ,u−uˆσ,z−zˆσ)+Rσ
forarbitrary(qˆσ,uˆσ,zˆσ)∈Qd×Xkrh,s×Xkrh,s.Altogetherthisyieldsthestatedcondition
J(q,u)−J(qσ,uσ)=21L˜(qσ,uσ,zσ)(q−qˆσ,u−uˆσ,z−zˆσ)+Rσ
of.prothecompletesand

WecanfurthermorewritetheerroridentitiesofTheorem7.3bymeansoftheresiduals
˜ρu(q,u)(ϕ),ρ˜z(q,u,z)(ψ),ρ˜q(q,u,z)(χ)ofthediscretizedprimal,dual,andcontrolequation:
ρ˜u(qσ,uσ)(ϕ)=L˜z(qσ,uσ,zσ)(ϕ),
ρ˜z(qσ,uσ,zσ)(ψ)=L˜u(qσ,uσ,zσ)(ψ),
ρ˜q(qσ,uσ,zσ)(χ)=L˜q(qσ,uσ,zσ)(χ).
Introducingthisnotationandneglectingtheremainderterms,system(7.8)turnsinto
J(q,u)−J(qσ,uσ)
1≈{ρ˜u(qσ,uσ)(z−zˆσ)+ρ˜z(qσ,uσ,zσ)(u−uˆσ)+ρ˜q(qσ,uσ,zσ)(q−qˆσ)}.(7.9)
2

104

7.1TheClassicalErrorEstimatorfortheCostFunctional

Example7.1.(ResidualsforthecG(s)dG(0)discretization)Tomakethesetheoretical
resultsmoresubstantial,theresidualsforthecG(s)dG(0)discretizationarestatedinthe
following.Thenotationisaccordingtoequation(5.9).
ρ˜u(qσ,uσ)(ψ)=(Uσ,0−u0,Ψ0)
n+{(Uσ,l−Uσ,l−1,Ψl)
=1l+kla¯(Uσ,l)(Ψl)+klb¯(Qσ,l)(Ψl)−kl(f(tl),Ψl)
ρ˜z(qσ,uσ,zσ)(ψ)=(Zσ,0−Zσ,1,Ψ0)
1−n+(Zσ,l−Zσ,l+1,Ψl)+kla¯u(Uσ,l)(Ψl,Zσ,l)
=1l−klF(Uσ,l)(Ψl)+(Zσ,n,Ψn)−Ju(Qσ,n,Uσ,n)(Ψn)
nρ˜q(qσ,uσ,zσ)(ψ)=b¯q(Qσ,l)(Ψl,Zσ,l)−(Qσ,l,Ψl)R
=1l

Wecontinuebyembeddingtheerrorestimatorintothecontextofmultipleshooting.Therefore,
letusfirststatetwopreliminaryremarks.
Remark7.3.(Discretizationofthemultipleshootingvariables)Inthefollowing,weassume
thatadjacentmeshesatthemultipleshootingnodesareequivalentforreasonsofconsistency.
weThehamveVultiplej,nj,ssho=Votingj+1,v0,s.ariablesarediscretizedaccordingly,andforthespatialdiscretization
hhRemark7.4.Themultipleshootingsolutiononthedifferentstagesofdiscretizationfulfills
thematchingconditions.Therefore,accordingtoRemark7.3,wehavethefollowingidentities
forthefullydiscretemultipleshootingsolution:
uσj−,nj=uσ(,j0+1)−=shj+1andzσ(,j0+1)−=zσj,n−j=λhj+1.

WeprovethefollowingtheoremwhichdisplaystheclassicalDWRerrorrepresentationofthe
functionalerrorintermsoftheintervalwiseresiduals:
Theorem7.4.Let(q,u,z),(qσ,uσ,zσ)besolutionsofproblem(4.2)onthedifferentstages
ofdiscretization,andletusdenotetheirintervalwiserestrictionsasbefore.Letρ˜u,ρ˜z,ρ˜q
denotethediscreteprimal,dual,andcontrolresidualoftheequationsin(5.7).Forthe
functionalerrorthereholds

J(q,u)−J(qσ,uσ)
1={ρ˜u(qσ,uσ)(z−zˆσ)+ρ˜z(qσ,uσ,zσ)(u−uˆσ)+ρ˜q(qσ,uσ,zσ)(q−qˆσ)}(7.11a)
2

105

7APosterioriErrorEstimation

wheretheresidualsareobtainedbysummationoftheintervalwiseresiduals:
1−mρ˜u(qσ,uσ)(z−zˆσ)=ρ˜uj(qσj,uσj)(zj−zˆσj),
=0j1−mρ˜z(qσ,uσ,zσ)(u−uˆσ)=ρ˜zj(qσj,uσj,zσj)(uj−uˆσj),(7.11b)
=0j1−mρ˜q(qσ,uσ,zσ)(q−qˆσ)=ρ˜qj(qσj,uσj,zσj)(qj−qˆσj).
=0jHere,thetermsρ˜uj(qσj,uσj)(ϕ),ρ˜zj(qσj,uσj,zσj)(ψ),andρ˜qj(qσj,uσj,zσj)(χ)denotetheresidualsof
thediscretizedintervalwiseoptimalitysystem(5.6).

Proof.ThefirststatementhasalreadybeenproveninLemma7.3.Therefore,onlytheproof
ofthelaststatementremainsundone.Weprovetheidentityonlyfortheprimalresidual,the
resultfordualandcontrolresidualfollowsanalogously.Fromequation(5.2a)weretrieveby
summationoverthemultipleshootingintervals
1−mρ˜uj(qσj,uσj)(zj−zˆσj)=
=0jm−1nj
((∂tuσj,zj−zˆσj))j,l+a(uσj)(zj−zˆσj)+b(qσj)(zj−zˆσj)
=1l=0jnj−1
+([uσj]l,zlj+−zˆσj,l+)+(uσj,−0−shj,z0j−−zˆσj,−0)−((f,zj−ˆzσj))j.
=0lDuetoRemark7.4theterms(uσj,−0,z0j−−zˆσj,−0)and(shj,z0j−−zˆσj,−0)canceloutforj=1,...,m−1,
andonlyj+thej−termforj=0remains.Furthermore,wecansubstituteforcontinuouszjthe
termszl=zl=z(tj,l)andobtain
1−mρ˜uj(qσj,uσj)(zj−zˆσj)=
=0jm−1nj
((∂tuσj,zj−zˆσj))j,l+a(uσj)(zj−zˆσj)+b(qσj)(zj−zˆσj)
=1l=0jnj−1m−1
+([uσj]l,z(tj,l)−zˆσj,l+)+(uσ0,−0−u0,z(τ0)−zˆσ0,−0)−((f,zj−zˆσj))j.
=0j=0lFromRemark7.4weseefurtherthatforj=0,...,m−2thejumptermsforl=0canbe
aswritten([uσj+1]0,z(τj+1)−zˆσ(,j0+1)+)=(uσ(,j0+1)+−uσj−,nj,z(τj+1)−zˆσ(,j0+1)+).

106

7.2APosterioriErrorEstimationfortheMultipleShootingSystem

Weinsertthisintotheequationandobtain
1−mρ˜ju(qσj,uσj)(zj−zˆσj)=
=0jm−1nj
((∂tuσj,zj−zˆσj))j,l+a(uσj)(zj−zˆσj)+b(qσj)(zj−zˆσj)
=1l=0jnj−1m−2
+([uσj]l,z(tj,l)−zˆσj,l+)+([uσ0]0,z(τ0)−zˆσ,0+0)+(uσ(,j0+1)+−uσj−,nj,z(τj+1)−zˆσ(,j0+1)+)
=0j=1l1−m+(uσ0,−0−u0,z(τ0)−zˆσ0,−0)−((f,zj−zˆσj))j.
=0jTheright-handsideofthisequationisequivalenttotheresidualofthediscreteprimal
(5.7):ofequation1−mρ˜uj(qσj,uσj)(zj−zˆσj)=ρ˜u(qσ,uσ,zσ)(z−zˆσ).(7.12)
=0jof.prothecompletesThis

SofarwehaveseenhowtheclassicalDWRerrorestimatorcanbecomposedfromintervalwise
residualsundertheassumptionofidenticaladjacentmeshesonthemultipleshootingnodes.
Thisassumptionisduetothefact,thatthesolutionobtainedbymultipleshootingmust
coincidewiththesolutionfromtheclassicalapproach.Inthenextsubsection,wedevelopan
approachthatallowsustohandletheerrorestimatorforsolutionswithdifferentlyrefined
adjacentmeshes.Thiserrorestimatorincorporatesadditionalprojectionerrorsonthe
multipleshootingnodes.

7.2APosterioriErrorEstimationfortheMultipleShooting
System

Thefollowingapproachtoaposteriorierrorestimationforthefunctionalerrorisbasedon
embeddingthewholemultipleshootingapproachafterconvergenceintothecontextofGalerkin
methods.Thisprocedureenablesustotreateachintervalseparatelywithoutanyassumptions
ontheadjacentgridsfromdifferentintervals.Theresultingerrorestimatorinheritspartsof
theclassicalDWRerrorestimatorbutadditionallyincorporatesprojectionerrorsforboth
primalanddualshootingvariablesonthemultipleshootingnodes.
Remark7.5.(Errorestimationfordirectandindirectmultipleshooting)Wederivethis
estimatorbymeansoftheindirectmultipleshootingapproachonly,butremarkthatthe
obtainedsolutionsfordirectandindirectmultipleshootingareidentical.Therefore,the
resultsholdaccordinglyforthesolutionofthedirectmultipleshootingapproach.

107

7APosterioriErrorEstimation

Letusconsiderthefollowingoptimizationproblemforwhichweshowthatitsoptimality
systemisequivalenttotheindirectmultipleshootingformulation(4.8)withmatching
conditions(4.9).Usingthenotationasintroducedin(4.10),wesetuptheminimization
problemm−1
s,˜minu,˜q˜J(q˜,˜u,s˜):=α1J1j(uj)+α3qj2Qj+α2J2(sm)(7.13a)
2=0jthathsuc1.forallϕ∈Xj,j=0,...,m−1,
((∂tuj,ϕ))j+a(uj)(ϕ)+b(qj)(ϕ)+(uj(τj)−sj,ϕ(τj))−((f,ϕ))j=0,(7.13b)
2.forallv∈H,j=0,...,m−1
(s0−u0,v)=0,
(7.13c)(sj+1−uj(τj+1),v)=0.
TheLagrangianof(7.13),mappingfromQ˜×X˜×H˜×X˜×H˜→Rreads
1−mL(q˜,˜u,s,˜z˜,˜λ)=α1J1j(uj)+α3qj2Qj+α2J2(sm)
2=0j1−m−((∂tuj,zj))j+a(uj)(zj)+b(qj)(zj)+(uj(τj)−sj,zj(τj))−((f,zj))j
=0j1−m+(sj+1−uj(τj+1),λj+1)+(s0−u0,λ0).(7.14)
=0jExplicitcalculationofthefirstorderoptimalitycondition
L(q˜,˜u,s,˜z˜,λ˜)(χ,˜ϕ,˜ξ˜,ψ˜,η˜)=0∀(χ,˜ϕ,˜ξ˜,ψ˜,η˜)∈Q˜×X˜×H˜×X˜×H˜
yieldsbacktheintervalwiseoptimalitysystems(4.8)oftheindirectapproachtogetherwith
thematchingconditions(4.9):
Differentiationof(7.14)withrespecttotheprimalvariables(˜u,s˜)yieldsforall(ϕ,˜ξ˜)∈X˜×H˜
conditionthe1−mα1J1j(uj)(ϕj)+α2J2(sm)(ξm)
=0j1−m−((∂tϕj,zj))j+au(uj)(ϕj,zj)+(ϕj(τj)−ξj,zj(τj))
=0j+(ξj+1−ϕj(τj+1),λj+1)+(ξ0,λ0)=0
m−1
=0j

108

7.2APosterioriErrorEstimationfortheMultipleShootingSystem

whichbypartialintegrationtransformsinto
1−mα1J1j(uj)(ϕj)+α2J2(sm)(ξm)
=0j1−m−−((∂tzj,ϕj))j+au(uj)(ϕj,zj)+(zj(τj+1),ϕj(τj+1))+(zj(τj),ξj)
=0j1−m+(ξj+1−ϕj(τj+1),λj+1)+(ξ0,λ0)=0.
=0jNow,summingupthetermsappropriately,weendupwiththeequation
1−mα1J1j(uj)(ϕj)+α2J2(sm)(ξm)−(λm,ξm)
=0j1−m−−((∂tzj,ϕj))j+au(uj)(ϕj,zj)+(zj(τj+1)−λj+1,ϕj(τj+1))
=0j1−m+(λj−zj(τj),ξj)=0.
=0jWeseparatetheequationsandobtainforj=0,...,m−1thefollowingPDE:
Determinezj∈Xjsuchthatforallϕj∈Xj
α1J1j(uj)(ϕj)−−((∂tzj,ϕj))j+au(uj)(ϕj,zj)+(λj+1−zj(τj+1),ϕj(τj+1))=0.
Thisequationisequivalenttothedualequation(4.8b).Furthermoreweobtainasetof
conditionswhichareequivalenttothematchingconditionsforthedualvariablein(4.9):
(zj(τj)−λj,ξj)=0∀ξj∈H,
α2J2(sm)(ξm)−(λm,ξm)=0∀ξm∈H.
Analogouslyweobtainbydifferentiationof(7.14)withrespecttoq˜theintervalwisecontrol
(4.8c).equationsNow,asbefore,weareinneedoftheLagrangianofthediscretizedprobleminorderto
developtheerrorestimatorforthecostfunctional.WedenotetheLagrangianforthefully
discreteproblembyL˜:Q˜d×X˜kh×H˜h×X˜kh×H˜h→R.ThediscreteLagrangianisdefined
bythefollowingidentity:
1−mL˜(q˜σ,u˜σ,s˜h,z˜σ,λ˜h):=α1J1j(uσj)+α3qσj2Qj+α2J2(shm)
2=0jm−1njnj−1
−((∂tuσj,zσj))j,l+a(uσj)(zσj)+b(qσj)(zσj)+([uσj]l,zσj,l+)+(uσj,−0−sσj,zσj,−0)−((f,zσj))j
j=0l=1l=0
+(shj+1−uσj(τj+1),λhj+1)+(sh0−u0,λh0).
m−1
=0j

109

7APosterioriErrorEstimation

WeobtainbydifferentiationoftheLagrangianwithrespecttoq˜σ,u˜σ,s˜hthefullydiscrete
intervalwiseoptimalitysystemasin(5.6)andthediscretematchingconditions(5.8).Thiscan
eitherbeverifiedasdoneforthecontinuouscasebyelementarycalculusorbyargumentingthat
discretizationanddualizationinterchangeforGalerkindiscretizations.Now,thederivation
oftheerrorestimatorproceedsasalreadyseenfortheclassicalDWRerrorestimatorin
Subsection7.1.Letusstatethefollowingtheorem:
Theorem7.5.(Errorestimatorforthemultipleshootingsolution)Let(q˜σ,u˜σ,s˜h)bea
stationarypointofthefullydiscretizedproblem(5.6)withmatchingconditions(5.8).Then,
forthefunctionalerrorthefollowingidentityholds:
J(q˜,˜u,s˜)−J(q˜σ,u˜σ,s˜h)
1=2L˜(q˜σ,u˜σ,s˜h,z˜σ,λ˜h)(q˜−q˜ˆσ,u˜−u˜ˆσ,s˜−s˜ˆh,z˜−z˜ˆσ,λ˜−λ˜ˆh)+Rσ
1−m=1ρ˜uj(qσj,uσj,shj)(zj−zˆσj)+ρ˜zj(qσj,uσj,zσj,λhj)(uj−uˆσj)+ρ˜qj(qσj,uσj,zσj)(qj−qˆσj)
2=0j1+2ρ˜sj(uσj,shj)(λj−λˆhj)+ρ˜λj(zσj,λhj)(sj−sˆhj)+Rσ(7.15)
forarbitraryuˆσj,zˆσj∈X˜kjh,r,s,qˆσj∈Qdj,sˆhj,λˆhj∈HhjandRσaremaindertermofthesame
formasinLemma7.1withL=L˜.Theresidualsaredefinedasfollows:
njρ˜uj(qσj,uσj,shj)(zj−zˆσj):=((∂tuσj,zσj−zˆσj))j,l+a(uσj)(zσj−zˆσj)+b(qσj)(zσj−zˆσj)
=1l1−nj+([uσj]l,zσj,l+−zˆσj,l+)+(uσj,−0−shj,zσj,−0−zˆσj,−0)−((f,zσj−zˆσj))j,(7.16a)
=0lρ˜jz(qσj,uσj,zσj,λhj)(uj−uˆσj):=−α1J1j(uσj)(uσj−uˆσj)
nj+−((∂tzσj,uσj−uˆσj))j,l+au(uσj)(uσj−uˆσj,zσj)
l=1nj−1
−([zσj]l,uσ−,l−uˆσ−,l)(zσj,n−j−λhj+1,uσ−,nj−uˆσ−,nj),(7.16b)
=0lρ˜qj(qσj,uσj,zσj)(qj−qˆσj):=bq(qσj)(zσj,qj−qˆσj)−α3(qσj,qj−qˆσj)Q,
(7.16c)

jjjjˆj(shj−u0,λj−λˆhj)j=0,
hσhρ˜s(uσ,sh)(λ−λh):=(sj−uj−1(τj),λj−λˆj)j=1,...,m,(7.17a)
jjjj
ρ˜j(zj,λj)(sj−sˆj):=(λ−zσ(τj),s−sˆh)j=0,...,m−1,(7.17b)
σλhh(λj,sj−sˆhj)−α2J2(sj)(sj−ˆshj)j=m.

110

7.2APosterioriErrorEstimationfortheMultipleShootingSystem

Proof.Fromthevalidityofthediscretizedformulationofcondition(7.13b)and(7.13c)for
bothcontinuousandfullydiscretestationarypointsoftheproblem,weobtainasforthe
estimatorerrorclassicalJ(q˜,˜u,s˜)−J(q˜σ,u˜σ,s˜h)=L˜(q˜,˜u,s,˜z˜,λ˜)−L˜(q˜σ,u˜σ,s˜h,z˜σ,λ˜h).(7.18)
WiththesameargumentationasinLemma(7.2)wecanconcludethatthefollowingequalities
holdforall(χ˜σ,ϕ˜σ,ξ˜h,ψ˜σ,η˜h)∈Q˜d×X˜kh×H˜h×X˜kh×H˜h:
L˜(q˜,˜u,s,˜z˜,λ˜)(χ˜σ,ϕ˜σ,ξ˜h,ψ˜σ,η˜h)=0,
L˜(q˜σ,u˜σ,s˜h,z˜σ,λ˜h)(χ˜σ,ϕ˜σ,ξ˜h,ψ˜σ,˜ηh)=0.
Wechoosethespaces
Y1:=Q˜×X˜×H˜×X˜×H˜,
Y2:=Q˜d×X˜kh×H˜h×X˜kh×H˜h,
Y:=Y1+Y2
andapplyLemma(7.1)totherighthandsideof(7.18).Thisyieldstherelation
J(q˜,˜u,s˜)−J(q˜σ,u˜σ,s˜h)=21L˜(q˜σ,u˜σ,s˜h,z˜σ,λ˜h)(q˜−q˜ˆσ,u˜−u˜ˆσ,s˜−s˜ˆh,z˜−z˜ˆσ,λ˜−λ˜ˆh)+Rσ.
ExplicitcalculationofL˜directlyyieldsthestatedconditionintermsoftheresiduals.

Acloserinvestigationoftheerrorrepresentation(7.15)revealsthecompositionoftheerror.
First,theintervalwisediscretizationerrorsreplayintheresidualsρ˜uj,ρ˜zj,andρ˜qjofthe
equations.Second,thecontributionoftheprojectionerrorsonthenodesisgivenbythe
residualsρ˜sandρ˜λofthematchingconditions.
Remark7.6.(Simplifications)LetusnoteatthispointthatspecialchoicesofthespacesHhj
fortheshootingvariablesyieldsimplificationsoftheerrorestimator.First,ifwechooseHhj
asthesupersetofthetracespacesfrombothadjacentintervals(equivalently,themeshat
τjisthecommonrefinementofbothadjacentmeshes),thentheshootingresiduals(7.17)
vanish,andtheclassicalerrorestimatorpresentedintheprevioussectionremains.
Alternatively,wecanshifttheerrorsduetoinconsistenciesattheshootingnodescompletely
totheshootingresidualsbychoosingHhjastheintersectionofthetwoadjacenttracespaces.
Inthiscase,thetermscorrespondingtotheboundaryvaluesofprimalanddualresidualin
(7.16)vanish.Whilethisapproachwouldyieldthesmallestshootingsystem,wedidnotuse
ithere:itcanbefearedthatitmayyieldanuncontrollablelossofaccuracyatthenodesif
theshootingresidualsarenotproperlytakenintoaccountfortherefinementoftheadjacent
meshes.Ontheotherhand,ifHhjisequaltothetracespacefromtheleftorright,thentheterms
accordingtotherightandtheleftboundaryvalueinthesecondandfirstresidualin(7.16)
vanish,respectively.Additionally,eitherρ˜sjforj=1,...,morρ˜λjforj=0,...,m−1in
(7.17)vanishaswell.

111

7APosterioriErrorEstimation

Wefollowthislastapproachinthenumericalexperiments,wherewechooseHhjtobethe
tracespacefromtherighthandside,yieldingthesimplifiedresiduals
njρ˜uj(qσj,uσj,shj)(zj−zˆσj)=((∂tuσj,zσj−zˆσj))j,l+a(uσj)(zσj−zˆσj)+b(qσj)(zσj−zˆσj)
l=1nj−1
+([uσj]l,zσj,l+−zˆσj,l+)−((f,zσj−zˆσj))j,(7.19a)
=0ljjjjj0j=0,...,m−1,
ρ˜λ(zσ,λh)(s−sˆh)=(λm,sm−sˆhm)−α2J2(sm)(sm−sˆhm)j=m.(7.19b)
Allotherresidualsremainunchanged.

Wehavedevelopedanerrorestimatorforbothcasesofmeshhandling,dynamicallychanging
mesheswithidenticaladjacentmeshesonthenodesandintervalwiseconstantmesheswith
differentlyrefinedadjacentmeshes.Consequently,wehavetodiscusshowtheerrorestimators
canbeevaluatedinpractice,thatishowtheexactsolutionscanbeapproximatedproperly
fortheevaluationoftheerrorestimators.
Remark7.7.Intheintroductiontothischapterwealreadymentioned,thatweareonly
interestedintheestimationofthespatialdiscretizationerror.Nevertheless,fromthetheo-
reticalpointofview,thewholediscretizationerrorcanbetreatedanalogously.However,the
practicalimplementationoftime-spacefiniteelementsishighlysophisticatedandisnotdone
inthisthesis.Thus,inthefollowing,weconsider(qk,uk,zk)and(qˆk,uˆk,zˆk)insteadofthe
continuoussolution.Thisisallowedbecauseallderivationsdonesofarinthischapterhold
forthetimediscreteformulation,too.

7.3EvaluationoftheErrorEstimators

Inwhicthishwseedevction,elopweedinpresenthetpossiblepreviousevaluationsections.tecOurhniquesmainforgoaltheisathepevosteriorialuationerroroftheestimatorserror
estimatorspresentedinSubsections7.2and7.1.Wewanttoevaluatetheestimatorforthe
discretizationerrorsuchthattheresultingerrorindicatorsreflecttheerrordistributionin
spaceproperly,andtheerrorestimatorconvergestothediscretizationerrorforh0.
Remarkdimensional7.8.caseofThroughoutquadrilateraltheremainingmeshes.partAofgenerthiscalizationhapter,towetherestrictthreeourselvdimensionalestothecasetwofo
hexalateralmeshesisstraightforward.

Inthesequel,wewanttoconsidertheevaluationofthediscretizationerrorestimators
presentedbefore.First,forbotherrorestimators,theevaluationprocedureoftheintervalwise
residualsincludesanapproximationoftheinterpolationerrors
(zkj−zˆσj),(ukj−uˆσj),(qkj−qˆσj),

112

7.3EvaluationoftheErrorEstimators

wherezˆσj=Ih(s)zkj∈X˜kjh,r,s,uˆσj=I(hs)ukj∈X˜kjh,r,sandqˆσj=Idqkj∈Qdjarethecorrespond-
inginterpolantsofthecurrentsolutions.Furthermore,fortheevaluationof(7.15),the
errorsolationterpin(sj−sˆhj),(λj−λˆhj)
withsˆhj=Ih(s)sj∈Hhj,λˆjh=Ih(s)λj∈Hhjhavetobeapproximated.
Weintroducelinearoperatorsthatmapthecomputedsolutionstoapproximationsofthese
interpolationerrorsasfollows:
zkj−Ih(s)zkj≈Phzjσ,
ukj−Ih(s)ukj≈Phuσj,
qkj−Idqk≈Pdqσj,
sj−Ih(s)sj≈Pˆhλhj,
λj−Ih(s)λj≈Pˆhshj.
ForthespecialcaseofquadrilateralmeshesandacG(s)spacediscretization,aconcretization
oftheoperatorsPhandPˆhisgivenasfollows:
Ph=I2(2hs)−idwithI2(2hs):X˜kjh,r,s→X˜kj,r(2,h2)s,
Pˆh=Iˆ2(2hs)−idwithIˆ2(2hs):Hhj→Hj(2h).
Hereby,thepiecewisebiquadraticinterpolationI2(2hs)caneasilybecomputedduetothe
requestedpatchstructureoftherefinedmeshintroducedinSection5.2.Thesecondoperator
PdhastobedefinedaccordingtothechoiceofQd.AccordingtoSection5.4werestrict
ourselvestoacG(p)dG(r),p≤s,discretizationofthecontrolspaceQ.Therefore,an
appropriatechoiceofPdisgivenby
Ph=I2(2hp)−idwithI2(2hp):X˜kjh,r,p→X˜kj,r(2,h2)p.
Remark7.9.Wheneverthediscretizationofthestatesandcontrolcoincide,thatisp=s,
theerrorduetothecontrolequationvanishes:
ρ˜qj(qσj,uσj,zσj)(qj−qˆσj)=0.

Wediscussthepracticalevaluationoftheerrortermsinthefollowing.Weeasilyobtain
afullycomputableversionofbotherrorestimators,(7.11)and(7.15),asfollows:Forthe
intervalwiseformulationoftheclassicalDWRerrorestimator,weobtain

J(qk,uk)−J(qσ,uσ)
=1ρ˜uj(qσj,uσj,zσj)(Phzσj)+ρ˜zj(qσj,uσj,zσj)(Phuσj)+ρ˜qj(qσj,uσj,zσj)(Pdqσj)+Rσ.
m−1m−1m−1
2j=0j=0j=0

113

7APosterioriErrorEstimation

Accordingly,thecomputableformulationoftheerrorestimatorforthecompletemultiple
shootingsystemisgivenby
J(q˜k,u˜k,s˜)−J(q˜σ,u˜σ,s˜h)
1−m=1ρ˜uj(qσj,uσj,shj)(Phzσj)+ρ˜zj(qσj,uσj,zσj,λhj)(Phuσj)+ρ˜qj(qσj,uσj,zσj)(Pdqσj)
2=0j+1ρ˜sj(uσj,shj)(Pˆhλhj)+ρ˜λj(zσj,λhj)(Pˆhshj)+Rσ.
2Finally,toconcludethischapter,letuspresentsomenumericalexamplesinordertoverify
thevalidityoftheerrorestimators.

ExamplesNumerical7.4

bWeefore,presenthetnerrorumericalestimatorresultsforofthethewholeerrormultipleestimatorsshoforotingsExampleystem2.1withandinter2.4.vAsalwisemenconstantionedt
meshesisconsideredtobemorepromisingfromthecomputationalpointofview.Therefore,
wemainlyfocusonthisapproachhenceforth,butneverthelessgivecomparativeresultson
theexampleclassicalDconstrainedWRerrorbytheestimatorheatforequationdynamicallywithconctrolhangingasamesourceshes,termtoo.onWethestartrighwitht-handan
side.Thecostfunctionalmeasurestheterminaltimedeviationfromadesiredstated.
Example7.2.(Distributedcontroloftheheatequation)ConsideringExample2.1with
parametersα2=1,α3=10−3,T=2.5,anddesiredterminaltimestateu¯=0.5,weapply
mtimeultiplestepsshoeacotingh,thatwithisaatotalconstannumtbertimeof10stepinsizetervofals.0.01Theforintervthealstimearesubdiscretization.dividedintoThe25
spatialdomainischosenasasquareΩ=(−1,1)×(−1,1).Furthermore,theinitialcondition
isgivenbythefunctionu(0)=cos(2πx)cos(π2x),andtheboundaryconditionsaregivenas
conditions.hletDirichomogenous

Wtheemconsiderultipleshoglobalotingrefinemenmethotdafterontheconvnewergencemesh.oftheThismyieultipleldsshotheotingconvmethoergencedandresultsrestartfor
theillustratemultiplethatshowithotingincreasingerrorestimatorrefinemenastsholevel,wnintheTerrorable7.1.estimatorThenconumericvergesaltothecomputationscorrect
valueofthefunctionalerror.Inthiscaseofidentical,globallyrefinedmeshesonallintervals,
theprojectionerrorisduetotheprojectionofu0intothediscretizedspace.Wealsoconsider
theerrorestimatorforthecaseoflocallyrefinedmeshes.Therefore,weapplythemesh
intervrefinemenalwisettecconstanhniquestwhicmesheshwillthebeinresultstroducedareinaccordinglyChapter,8.asAsexpoutlinedected,inTforablelocally7.2.Inrefined,the
following,Nisthetotalnumberofcellsfromallmeshes(ofalltimesteps),ehdenotesthe
correcterrorvalue,ηdenotestheestimateofthediscretizationerror,ηistheestimate
oftheprojectionerror,discandηistheestimateforthewholefunctionalerrorproejh.Thesocalled
ηonefficiencythedevindexelopmenIefft:=ofhthe/ehapprogivesximation.informationIeffonshouldtheconvaccuracyergetoof1theforNerror→∞ifestimatortheerrorand
ell.workswestimator

114

ExamplesNumerical7.4

Table7.1:ErrorsExampleand7.2withestimatesforfunctionalglobalvalueJrefinemen(q,u)t=with0.in0553066terv.alwiseconstantmeshesfor

NehηdiscηprojηIeff
1080−1.669∙10−01−6.090∙10−02−6.167∙10−11−6.090∙10−020.365
4320−8.395∙10−02−2.006∙10−02−1.384∙10−11−2.006∙10−020.239
17280−2.824∙10−02−9.728∙10−03−1.152∙10−12−9.728∙10−030.344
69120−7.162∙10−03−5.519∙10−036.500∙10−12−5.519∙10−030.770
276480−1.718∙10−03−1.933∙10−031.396∙10−13−1.933∙10−031.125
1105920−4.225∙10−04−4.522∙10−048.020∙10−14−4.522∙10−041.070
4423680−1.050∙10−04−1.073∙10−041.981∙10−15−1.073∙10−041.022
Table7.2:Errorsandestimatesforlocalrefinementwithintervalwiseconstantmeshesfor
Example7.2withfunctionalvalueJ(q,u)=0.0553066.

NehηdiscηprojηIeff
1080−1.669∙10−01−6.090∙10−026.166∙10−11−6.090∙10−020.365
2052−8.395∙10−02−2.006∙10−02−1.020∙10−07−2.006∙10−020.239
4644−2.824∙10−02−9.729∙10−03−1.714∙10−07−9.729∙10−030.344
8532−7.165∙10−03−5.521∙10−03−1.912∙10−07−5.521∙10−030.771
17604−1.721∙10−03−1.939∙10−03−1.916∙10−07−1.940∙10−031.127
37044−4.161∙10−04−4.323∙10−04−1.917∙10−07−4.325∙10−041.039
77220−1.809∙10−04−1.631∙10−04−1.917∙10−07−1.633∙10−040.903
139428−1.079∙10−04−1.092∙10−04−1.974∙10−07−1.094∙10−041.013
291060−5.213∙10−05−5.095∙10−05−1.974∙10−07−5.115∙10−050.981

Localrefinementwithdynamicallychangingmeshesyieldsdifferentmesheswithintheprocess
ofmeshadaptation.However,theerrorestimatorηinthiscaseconvergestothecorrect
valueehasgiveninTable7.3.

Table7.3:ErrorsExampleand7.2withestimatesforfunctionallocalvaluerefinemenJ(q,ut)=with0.0553066dynamically.changingmeshesfor

NehηIeff
1080−1.669∙10−01−6.090∙10−020.365
1956−8.395∙10−02−2.006∙10−020.239
4152−2.824∙10−02−9.730∙10−030.345
8592−7.168∙10−03−5.524∙10−030.771
17412−1.724∙10−03−1.938∙10−031.124
35796−4.282∙10−04−4.575∙10−041.068
75612−1.094∙10−04−1.114∙10−041.018
165804−2.966∙10−05−2.981∙10−051.005

115

7APosterioriErrorEstimation

Asexpected,byrefinementwithdynamicallychangingmeshesweobtainbetterapproxima-
ationssufficienofthetlyfularnctigeonnalumbateraofinsmallertervnals,umwbeerofcancells.compWeensatehintthisatthedrafact,wbackthatforbinyctervhoalwiseosing
meshes.tconstanTocompletethissectionofnumericalexamples,weconsidernowanexampleconstrained
byanonlinearPDEwithNeumannboundarycontrol.Weminimizeatimedistributedcost
functionalandtherebyapproximateatimedistributedfunctionu¯(t).
Example7.3.(Neumannboundarycontrol)WeconsiderExample2.4onaspatialdomain
Ω=(−0.5,0.5)×(−0.5,0.5)withinitialconditionu0=1anddesiredtimedistributedstate
u¯(t)=t∙cos(4πx)cos(4πy).Fortheregularizationparameterwechoseα3=10−4andforthe
timedomainI=(0,2).Thenumberofmultipleshootingintervalsischosenasm=10,and
thetheintimetervalwisesteppingconsctrolshemeareisrequestedinitializedtowithbeaconstanconstanttintimetime,stepthatsizeisoffor0j.01=.0,F...,10urthermore,
Qj=v∈L2(Ij,R)v(t)=cj∈R⊂L2(Ij,R).
Asinthepreviousexample,theresultsforExample7.3showanappropriateconvergence
behavioroftheerrorestimatorforthedistributedcostfunctional.
Inthisexample,weconsideronlythecaseofintervalwiseconstantmeshes.InTable7.4the
obtainederrorandthecorrespondingerrorestimatorforthedifferentstepsofglobalmesh
refinementarepresented.
ForthesakeofcompletenessTable7.5replaystheanalogousresultsforlocalrefinementwith
intervalwiseconstantmeshes.Inbothcasestheerrorestimatorηconvergestothecorrect
valueehwithproceedingmeshrefinement,andtheefficiencyindexIeffconvergesaccordingly
.1toTable7.4:ErrorsandestimatesforglobalrefinementforExample7.3withfunctionalvalue
J(q,u)=0.295083.

NehηdiscηprojηIeff
33602.299∙10−022.020∙10−020−2.020∙10−02−0.879
134408.932∙10−−0303−4.174∙10−−040304.174∙10−−04030.047
2150405376052..709314∙∙1010−04−−52..787184∙∙1010−040052..787184∙∙1010−0410..014944
8601601.420∙10−04−1.427∙10−0401.427∙10−041.005
34406403.534∙10−05−3.548∙10−0503.548∙10−051.004
Wehaveseensofarthattheerrorestimatorsyieldreliableestimatesforthetotalfunctional
error.adaptivInemeshadditiontorefinementhet.correctTheresultsappropresenximationtedofabtheoveerror,andbweelowareinalreadyterestedindicateinthatappropriatelocal
meshrefinementfortheseexamplesyieldsbetterapproximationsofthefunctionalerrorat
fewercellsthanglobalmeshrefinement.Ingeneral,wewanttorefinethosecellswithlarge
errorcontributions.Therefore,weneedknowledgeonthecellwiseerrordistribution.

116

ExamplesNumerical7.4

Table7.5:ErrorsExampleand7.3withestimatesforfunctionallocalvaluerefinemenJ(q,ut)=with0.in295083terv.alwiseconstantmeshesfor

NehηdiscηprojηIeff
33602.299∙10−022.020∙10−02−0.000∙10−00−2.020∙10−02−0.879
76441.154∙10−023.663∙10−03−1.735∙10−05−3.661∙10−03−0.317
184806.665∙10−03−1.179∙10−033.009∙10−051.176∙10−030.176
431762.339∙10−03−1.637∙10−031.965∙10−071.637∙10−030.700
981128.231∙10−04−6.540∙10−04−4.217∙10−086.541∙10−040.795
2253723.260∙10−04−3.163∙10−04−5.213∙10−093.163∙10−040.970
5184481.385∙10−04−1.395∙10−04−1.876∙10−091.395∙10−041.007
10637765.960∙10−05−5.963∙10−05−4.521∙10−105.963∙10−051.000
21161283.095∙10−05−3.120∙10−05−5.949∙10−113.120∙10−051.008

theThepronextcesscofhaptermeshisfirstadaptationaboutthewithinlothecalizationmultipleoftheshootingdiscretizationmethod.errorandsecondabout

117

8MultipleShootingandMeshAdaptation

Inthischapter,wepresentlocalizationtechniquesfortheerrorestimatorsdevelopedin
Chapter7andanadaptivealgorithmforthepurposeofmeshadaptation.Meshadaptation
theplaysdeanimpterminationortantofroleaninoptimaltheconmeshtextinoftimeefficienandtspacesolutionwithtecresphniquesecttoforthePDEsdesiredandaimsaccuracyat.
Thatis,wesearchforapreferablycoarsemeshonwhichthesolutionisapproximatedsuch
thatwereachabeforehanddefinederrorboundforthecostfunctional.
Wesubdividethischapterintotwomainparts.Section8.1isonthemeshadaptationfor
theclassicalDWRerrorestimatoringeneral.ItissubdividedintoSubsection8.1.1on
thedetailsofthelocalizationprocedurefortheerrorandSubsection8.1.2onthemesh
adaptationalgorithm.Section8.2isengagedwiththecorrespondingproceduresforthenewly
developederrorestimatorforthemultipleshootingsystemfromChapter7.Itisanalogously
partitionedintoSubsections8.2.1and8.2.2.Thechapterisclosedbysubstantiatingthe
obtainedalgorithmsbynumericalexamplesinSection8.3.

8.1MeshAdaptationbytheClassicalDWRErrorEstimator

8.1.1LocalizationoftheErrorEstimator

Wedenotedstartbythisηwithsectionbyappropriateintroducingindicesforsomethespenotation.cificationTheinertherorparestimatorsticularconaretext:generally
ηh:=1{ρ˜u(qσ,uσ)(Phzσ)+ρ˜z(qσ,uσ,zσ)(Phuσ)+ρ˜q(qσ,uσ,zσ)(Pdqσ)},
2ηhj:=1ρ˜uj(qσj,uσj)(Phzσj)+ρ˜zj(qσj,uσj,zσj)(Phuσj)+ρ˜qj(qσj,uσj,zσj)(Pdqσj),
2andthusm−1
ηh=ηhj.
=0jFurthermoretheintervalwiseerrorestimatorηhjissplitintoitssubintervalwisecontributions,
njηhj=ηjh,l,
=1l

with

ηjh,l=1ρ˜uj,l(qσj,uσj)(Phzσj)+ρ˜zj,l(qσj,uσj,zσj)(Phuσj)+ρ˜qj,l(qσj,uσj,zσj)(Pdqσj).
2

119

8MultipleShootingandMeshAdaptation

Hereρ˜uj,l,ρ˜zj,landρ˜qj,ldenotethosepartsoftheresidualsρ˜uj,ρ˜zjandρ˜qjwhichbelongtothe
timesubintervalIj,l.
Withthisnotationathandweproceedwiththemeshwiselocalizationprocedure.Wewant
tobreakuptheerrorcontributionsηjh,lintocellwiseerrorindicatorswhichcorrespondtothe
cellwisecontributiontotheerrorofprimal,dual,andcontrolequation.Forthelatter,the
splittingcanbeperformedrathersimplebywritingthescalarproductswithintheresidual
˜ρqj,lasasumoverthecellsofthetriangulationThj,l.Concerningthesplittingoftheprimal
anddualresiduals,ourapproachfollowstheonepresentedin[7].Wetransformtheresiduals
ρ˜uj,landρ˜zj,lbycellwisepartialintegrationof(uσj,Phzσj)and(zσj,Phuσj)suchthatwe
obtain(uσj,Phzσj)=(−Δuσj,Phzσj)K+(∂nuσj,Phzσj)∂K,
,lj∈TKh(zσj,Phuσj)=(−Δzσj,Phuσj)+(∂nzσj,Phuσj)∂K.
,lj∈TKhWefollowtheargumentsofBeckerandRannacherin[7]thatPhuσjandPhzσjhavecontinuous
tracesalongtheboundary∂KofeachcellK.Eachinnerfaceoccurstwicewithinthesum
withchangingsignofthenormalderivative(c.f.Figure8.1).Therefore,weobtain
(uσj,Phzσj)=(−Δuσj,Phzjσ)K+21([∂nuσj],Phzσj)∂K,
,lj1K∈Th
(zσj,Phuσj)=j,l(−Δzσj,Phuσj)+2([∂nzσj],Phuσj)∂K.
∈TKhHere,wedefine[∂nuσj]oninnerfacesasthejumpoverthefaceΓfromKtotheneighbor
cellKandonouterfacesastheouternormalderivative:
[∂nuσj]Γ:=∂nuσjK+∂nuσjKand[∂nuσj]Γ:=2∂nuσj.

nnKK

Figure8.1:Adjacentcellsandouternormalvector.

WiththeresidualstransformedasdescribedaboveandwrittenasasumoverthecellsofThj,l,
wearenowabletoconsiderthecellwisecontributionηjh,l,KofthecellKtotheresiduals.
writeewTherefore,jjm−1njj
ηh,l=j,lηh,l,Kandηh=j=0l=1j,lηh,l,K.
K∈ThK∈Th

120

8.1MeshAdaptationbytheClassicalDWRErrorEstimator

Next,wewanttodiscusstheprocessofmeshadaptationaccordingtothecellwiseerror
indicators.

8.1.2TheProcessofMeshAdaptation

Wheneverwedecidetorefinethespatialmeshesaftertheevaluationoftheerrorestimators,
wehavetoassignacriterionthatallowsustodistinguishbetweenthosecells,whichwillbe
errorrefinedconandtributionsthosecellsfromthatdifferenwillbtekcellseptarethesameequilibrated,.Wewishtorefinethemesh,suchthatthe
ηjh,l,K≈const,
forallintervalsIjforj=0,1,...,m−1,allsubintervalsIj,lforl=1,2,...,nj,andallcells
,ljKtotal∈Tnhum.bFeroroftheMsakcellseofK1,.simplicit..,KyMwithwithrespcorrespecttotheondingnotation,errorletindicatorsusassumeηKn,nthat=w1,e..ha.,veMa.
Theprocessofequilibrationisperformedasfollows.Wesortthecellindicatorsindescending
order,ηKπ1≥ηKπ2≥...≥ηKπM,
where(π1,...πM)denotesapermutationof(1,...,M).Thenwechooseafixednumber
nthisrefwhicfractionalhdenotesamounthetoffractionalcellswhichamounhatveofthecellslargestwhichwerrorecowishntoretributions.fine.InFinally,wealgorithmicalrefine
formulationthisprocedurereads:

Algorithm8.1SpatialmeshadaptationfortheclassicalDWRerrorestimator
Require:SetofinitialmeshesThj,l.
1:Choosenref.
2:Evaluatesolutionsqσ,uσ,zσ.
3:EvaluateerrorestimatorsandretrievecellwiseindicatorsηKi.
4:Sorterrorindicators:ηKπ1≥ηKπ2≥...≥ηKπM.
5:forn=1toMdo
6:ifn<M∙nrefthen
7:SetrefinementindicatoronKπn.
8:Refinemeshesaccordingtothecellwiserefinementindicators.
9:Replaceadjacentmeshesonthenodesbycommonrefinement.

Themeshproadapceduretationpreissepntederformedaboveafterdescribeachesonlsolutionyonecyccycleleofofthespatialoptimalitmeshysystem.adaptation.IntheUsuallycase,
ofmultipleshooting,thismeanstheevaluationoftheerrorestimatorandtherefinementof
thecellsafterconvergenceoftheouterNewtonmethod.Furthermore,asalreadyseenin
thepreviouschapter,theapproachoftheclassicalDWRerrorestimatorislimitedbythe
assumptionofequivalentadjacentmeshesonthenodes.Therefore,adifferentrefinement
strategyisappliedinthecaseoftheerrorestimatordevelopedinSection7.2.

121

8MultipleShootingandMeshAdaptation

8.2MeshAdaptationbytheErrorEstimatorfortheMultiple
SystemngiotSho

Beforewebeginwiththedescriptionoftheadaptationstrategy,letusstateapreliminary
assumption,namely,accordingtoChapter5,weassumeconstantmeshesoneachofthe
multipleshootingintervals.

8.2.1LocalizationoftheErrorEstimator

Thelocalizationprocedurefortheerrorestimatorisquitesimilartotheonepresentedin
theprevioussection.Theerrorestimatorforthemultipleshootingsystemconsistsoftwo
differenttypesofintervalwiseresiduals.First,wehavetheintervalwiseresidualtermsdueto
thediscretizationerrorontheintervalsρ˜uj,ρ˜zj,andρ˜qj.Second,wehavetheresidualsdueto
theprojectionerroronthenodesρ˜sjandρ˜λj.Weusethesameabbreviationsηhjandηjh,las
beforeforthediscretizationerrorestimatesandadditionallyintroduceanabbreviationfor
errorsjectionprotheηpju:=21ρ˜sj(uσj,shj)(Pˆhλhj),
ηpjz:=1ρ˜λj(zσj,λhj)(Pˆhshj).
2jInThelotegratingcalizationbackbproyparts,cedureforandηshummingdirectlyupfolloovwserallthecells,approacwehforobtaintheasDbWeforeRerrorestimator.
jnjj
ηh=l=1K∈Tjηh,l,K.
hTheprojectionerrorsareeasilylocalizedtocellwisecontributionsηpju,Kandηpjz,Konthe
nodemeshThτjby
ηpju=τjηpju,Kandηpjz=τjηpjz,K.
K∈ThK∈Th
Overall,weobtain
m−1njm−1
ηh=ηjh,l,K+ηpju,K+ηpjz,K.
j=0l=1K∈Thjj=0K∈Thτj
Henceforth,consideringtheadaptationstrategy,wehavetodefinethechoiceofTτj.As
mentionedinChapter7,wefollowtheapproachwhichisusedinthenumericalexpheriments.
WehaveThτj=Thj,thatis,thenodemeshonτjisequivalenttothetracemeshfromthe
right.Thus,accordingto(7.19),ρ˜λj(zσj,λhj)(Pˆhshj)vanishes,andonlytheprojectionerrorfor
remains.ariablevdualthe

122

8.2.2TheProcessofMeshAdaptation

8.3ExamplesNumerical

Incontrasttodynamicallychangingmeshes,intervalwiseconstantmeshesneedthecalculation
ofappropriatecellwiseerrorindicatorsnotonlyoutofthecellwiseindicatorsineachtime
stepbutalsooutoftheadditionalprojectionerrors.Ourbasicideaistocalculatethecell
indicatorforacertaincellasthemaximumabsolutevalueoveralljtimestepsofthejinterval
andtheprojectionerror.LetthecellindicatorforacellKofThbedenotedbyηK,then
ηjK:=max|ηpjz,K|,l=1max,...,nj|ηjh,l,K|.(8.1)
AsbeforeweassumethatwehaveatotalnumberofNcellsfromtheintervalwisemeshes,
K1,...,KNforwhichwecalculatethefinalerrorindicatorsηKiaccordingto(8.1).Inthis
notation,theadaptivealgorithmreads:
Algorithm8.2Spatialmeshadaptationforthemultipleshootingerrorestimator
Require:SetofinitialmeshesThj.
.nChose1:ref2:Evaluatesolutionsqσ,uσ,zσ.
3:Evaluateerrorestimatorsandcellwiseindicatorsforalltimestepsandprojectionerrors.
4:CalculatecellwiseindicatorsηKi.
5:Sorttheseindicators:ηKπ1≥ηKπ2≥...≥ηKπM.
6:forn=1toMdo
7:ifn<M∙nrefthen
8:SetrefinementindicatoronKπn.
9:Refinemeshesaccordingtothecellwiserefinementindicators.

Weclosethischapterbyaconsiderationoftwodifferentexampleproblemsinthenextsection.

ExamplesNumerical8.3

Inthissection,numericalexampleswithdifferenttypesofcostfunctionalsandcontrolsare
estigated.vinRemark8.1.Fromtheviewpointofmultipleshootinginthecontextofpartialdifferential
equations,itisadvisabletoallowdifferentmeshesforstatesandcontrol.Thisapproach
impconsequenortancetlyforallothewstheefficiencyreductionoftheofmtheultipledishomensiotionngofmethothedconastroloutlinedspace,inwhichChapterisof6.crucialDue
toimplementationalaspects,wearenotyetabletoconsiderthiscaseofmeshrefinement,
suchthatinthefollowingexamplesthemeshesforstatesandcontrolareequivalent.
Firstofall,werecalltheformulationfromExample2.2andconsidertwodifferenttimedomain
localdecompmeshositionsrefinemeninthetduefollotowing.thecThishosenexampleterminalservtimeeswcostelltofunctional.demonstrateForthethepropresolutionertiesofof
thesingularityontheboundaryattheterminaltimepointwecanexpectathinboundary
layeroflocallyrefinedcellsatthecorrespondingspatialmesh.

123

8MultipleShootingandMeshAdaptation

Example8.1.(Distributedcontroloftheheatequation)Wechoosethespatialdomain
Ω=(−1,1)×(−1,1),thetimedomainI=(0,2.5),andtheparameterα=10−3.The
optimalcontrolproblemofinterestisgivenby
α1(q,u)∈minQ×XJ(q,u):=2u(T)−0.5L22(Ω)+2Iq(t)L22(Ω)dt(8.2a)
subjecttothelinearheatequation
∂tu−Δu=qinΩ×I,
u=0on∂Ω×I,(8.2b)
u=cosπxcosπxinΩ×{0}.
22Wewanttominimizetheterminaltimefunctionalconstrainedbythelinearheatequationfor
whichthecontrolactsasasourcetermontheright-handside.Theneedformeshadaptation
resultsfromthechoiceofthegoal,J1(q)=21Iu(T)−0.52dtwithinhomogeneousboundary
valueswhilethesolutionissubjecttoahomogeneousDirichletboundarycondition.We
chooseaconstanttimestepsizeof0.01andextrapolatethecorrectvalueofthefunctionalas
0.0553066.Theresultsofthecomputationswithlocalmeshrefinementfor10intervalsare
8.2.Figureinwnsho

1−102−10

−310

4−10

loglobalcalrefinemenrefinementt
refinemendynamicalcallot

5−10104105106107
Figure8.2:Functionalerrorfordifferentlyfinemeshes,gainedbydifferentstrategiesof
meshrefinementforExample8.1.(x-axis:log(N),y-axis:log(|eh|))
Additionally,Figure8.3illustratestherelationbetweentheprojectionerrorestimateand
theerrorestimateforthediscretizationerrorontheintervals.Meshrefinementisperformed
appropriatelysuchthattheprojectionerrorestimatedoesnotbecomedominant.Itstays
belowthediscretizationerroranddecreasesifnecessary.

124

ExamplesNumerical8.3

errordiscretizationerrorjectionpro

1−1010−2prodiscretizationjectionerrorerror
3−104−105−106−107−108−109−1010−1011−10104105106
Figure8.3:Discretizationerrorestimateandprojectionerrorestimatewithproceedingmesh
refinementforExample8.1.(x-axis:log(N),y-axis:log(|ηproj|)andlog(|ηdisc|))
Weseethatlocalrefinementduetobothdynamicallychangingandintervalwiseconstant
meshesisbyfarbetterthanglobalrefinement.Weneedalargernumberofcellsforintervalwise
constantmeshestoobtainacertainerrorboundthanfordynamicallychangingones.This
canbeexpectedduetothelessflexiblespatialmeshes,butnevertheless,thisdrawbackcan
easilybeovercomebyincreasingthenumberofintervals.
Todemonstratethisproperty,werecalculateExample8.1for50intervals.Theresults
presentedinFigure8.4pointoutthatalargernumberofintervalsyieldssignificantlybetter
results,inthiscaseevenbetterthanwithdynamicallychangingmeshes.However,weremark
thatforfurtherrefinement,thecurveswillcrossandrefinementwithdynamicallychanging
meshesyieldsbetterresultsthanwithintervalwiseconstantmeshes.Theobtainedsequenceof
locallyrefinedmeshes(seeFigure8.8onpage128)underlinestheassumptionthatrefinement
isparticularlyimportantfortheresolutionofthesingularityontheboundaryattheterminal
t.oinptimeInthefollowing,wedonotpursuetheideaofdynamicallychangingmeshesfurtheron,
butrestrictourselvesontheapplicationofintervalwiseconstantmeshes.Inthefollowing
example,weconsideranoptimizationproblemonaT-shapeddomain(seeFigure8.5).We
choseNeumannboundarycontrolandacostfunctionaldistributedintimeandspace.In
thisparticularcontextofmeshadaptation,thechoiceofthedomain,thecontrolandthe
costfunctionalsuggeststheassumptionthatrefinementshouldoccur,firstatthecontrol
boundary,andsecondattheinwardspointingcorners.Thisisduetothecontrollocalizedon
theboundaryofthelowerpartoftheT-shapeddomainwhichmustaffectthesolutionon
thewholedomain.Therefore,informationmustpassalongtheinwardspointingcornersto
controlthesolutionintheupperpartoftheT-Shapeddomain.

125

8MultipleShootingandMeshAdaptation

1−102−10

3−10

4−10

loglobalcalrefinemenrefinementt
trefinemendynamicalcallo

5−10104105106
Figure8.4:Functionalerrorfordifferentlyfinemeshes,gainedbydifferentstrategiesof
meshrefinement(calculatedon50intervalsforExample8.1).(x-axis:log(N),
y-axis:log(|eh|))
Example8.2.(NeumanncontrolonaT-shapeddomain)Weconsidertheproblem
12α2
(q,u)∈minQ×XJ(q,u)=2Iu(t)−u¯(t)dt+2Iq(t)Qdt
subjecttotheconstraint
∂tu−Δu+u3−u=0inI×ΩT,
u(0)=0inΩT,
qonI×Γc,
∂nu=0onI×ΩT\Γc
withΩTaT-shapeddomain(seeFigure8.5),u0=0,u¯=t,α=10−3,andI=(0,1).

ΓcFigure8.5:ΩTwithcontrolboundaryΓc.

Weconsiderthecaseof10intervals,theimplicitEulerwithstepsize0.01,andobtainby
extrapolationthecorrectvalueofthefunctionalas0.0052103.InFigure8.6wecompare
globalandlocalrefinementforintervalwiseconstantmeshes.

126

3−104−10

5−10

ExamplesNumerical8.3

loglobalcalrefinemenrefinementt

errordiscretizationerrorjectionpro

6−10104105106
Figure8.6:FmeshunctionalrefinemenerrortforforExampledifferently8.2.fine(x-axis:meshes,log(Ngained),yby-axis:logdifferen(|eth|))strategiesof
Theoutstandingperformanceoflocalrefinementisillustratedbythesteepdescentofthe
errorvalueforthecorrespondingcurve.Thedevelopmentofthedifferentpartsoftheerror
thatestimatortheisestimategivenforintheFigurepro8.7.jectionItshoerrorwsevendecreasesmoreclaccordinglyearlythanwithinthetheestimateexamplebforefore,the
error.discretization4−10errordiscretization10−5projectionerror
6−107−108−109−1010−1011−1012−10104105106
Figure8.7:Discretizationerrorestimateandprojectionerrorestimatewithproceedingmesh
refinementforExample8.2.(x-axis:log(N),y-axis:log(|ηproj|)andlog(|ηdisc|))

127

8MultipleShootingandMeshAdaptation

Thevisualizationofthemeshesobtainedbytheprocedureoflocalrefinementattimest=0,
t=0.2,t=0.4,t=0.6,t=0.8andt=1isgiveninFigure8.9attheendofthechapteron
page129.Itillustratesthereasonforthegoodperformanceoflocalrefinementincomparison
toglobalrefinement:thesingularitiesatthecornersareresolvedbylocallyrefinedmeshes.
Now,wehavethemeshadaptivemultipleshootingmethodathandandproceedinthenext
sectionwiththeconsiderationofthesolidfuelignitionmodelinthecontextofmultiple
oting.sho

0=t(a)

2=t(c)

(b)t=1.5

(d)t=2.5

Figure8.8:Intervalwiseconstantmeshesatt=0,t=1.5,t=2,t=2.5forExample8.1.

128

0=t(a)

(c)t=0.4

(e)t=0.8

ExamplesNumerical8.3

(b)t=0.2

(d)t=0.6

1=t)(f

Figure8.9:Intervalwiseconstantmeshesatt=0,t=0.2,t=0.4,t=0.6,t=0.8and
t=1forExample8.2.

129

9ApplicationtotheSolidFuelIgnition
delMo

Thischapterfocusesontheapplicationofmeshadaptivemultipleshootingtotheoptimal
controlofthesolidfuelignitionmodel.WeintroducethesolidfuelignitionmodelinSection
9.1andbrieflyreviewthetheoreticalbackgroundinSection9.2.Finally,inSection9.3we
considertheoptimalcontrolofthemodelfordifferentconfigurationsanddifferenttypesof
cfuncost.tionals

9.1TheSolidFuelIgnitionModel

Thesolidfuelignitionmodeloriginatesfromcombustiontheoryforrapidexothermicchemical
reactions.Weconsidertheone-stepirreversiblereaction
νFF+νOO→νPP
whichdescribestheoxidationofafuelFwiththeoxidantOtotheproductPofthis
combustionprocess.Inthisconnection,thestoichiometricconstantsνF,νO,andνPdescribe
theabsolutepartsinwhichthereactantsandproductsinteract,andweadditionallyintroduce
thecorrespondingmassfractionsofallspecies,yF,yO,andyP.IfbothreactantsFandO
areavailableincorrectproportionsandtheinitialstockyF0andyO0isofthesameorderof
magnitudethenbothspeciesareconsumedentirelyduringtheprocess.Therefore,thereaction
ratedependsstronglyonthemassfractionyF0andyO0whichmakesthemathematical
descriptionoftheprocessunfortunatelyrathercomplicated.However,ifweassumethat
thefuelFisavailableinexcess,thatisyF0yO0,themassfractionofFdoesnotchange
significantlyduringthechemicalprocessandcanbeassumedtobeconstant.Withthis
simplificationsathand,theprocessreducestoasinglespeciesreactiondeterminedbythe
developmentofyOduringtheprocessofcombustion.Thederivationofthemathematical
formulationofthereactionprocessismoreelaboratedandisperformedbyastepbystep
applicationofdifferentconservationlaws(thatistheconservationofmass,theconservation
ofspecies,theconservationofmomentumandtheconservationofenergy).Wedonotintend
togivetheexplicitderivationhere,butrefertothetextbookofBebernesandEberly[2].
Thesoobtainedequationsdescribingthesystemareofacomplexstructureandincorporate
notonlythemassfractions,butalsothedensity,thespeciesvelocity,thepressure,andthe
temperature.Sinceweareinterestedintheconsiderationofasolidfuel,whichweassumeto
beanondeformablematerialofconstantdensity,theequationssimplify.First,thespecies
velocityvanishestozero,secondthedensitycanbeassumedtobeconstantequalto1.With

131

9ApplicationtotheSolidFuelIgnitionModel
someadditionalminorsimplificationsweobtainthemathematicaldescriptionofthesolid
fuelcombustionwithtemperatureT,timeintervalI=(0,∞),andfuelmassfractionyas
∂tT−ΔT=εδymeTεT−1inI×Ω,
∂ty−βΔy=−εδΓymeTεT−1inI×Ω
withinitialvalueandboundaryconditiongivenby
T(0,x)=1,y(0,x)=1inΩ,
T(t,x)=1,∂ny(t,x)=0onI×∂Ω.
Here,β≥0,Γ>0,andδ>0isthesocalledFrank-Kamenetskiparameter.Forthefuelsof
interest,εissmall.Simplifyingthemodelbytheasymptoticfirstorderapproximationof
temperatureandmassfractionthrough
T=1+εθandy=1−εc
delmoreducedtheyields∂tθ−Δθ=δ(1−εc)me1+θεθinI×Ω,
∂tc−βΔc=−δΓ(1−εc)me1+θεθinI×Ω
withinitialvalueandboundarycondition
θ(0,x)=0,c(0,x)=0inΩ,
θ(t,x)=0,∂nc(t,x)=0onI×∂Ω.
Forε1theequationsdecoupleandweareleftwiththeconsiderationofthetemperature
θonly.Thissimplifiedsystem(9.3)isfinallydenotedasthesolidfuelignitionmodeland
describesundertheseassumptionsthethermalreactionofarigidmaterialduringtheignition
periodoftheprocess.
∂tθ−Δθ=δeθinI×Ω(9.3a)
withinitialvalueandboundarycondition
θ(0,x)=0inΩ,
θ(t,x)=0onI×∂Ω.(9.3b)
Thereactivethermalproscessingleandsptheeciesequilibratingreactiondevelopsconductiondueoftotheenergy.energyWhenevwhicerhtheissetheatfreedissipationduringtheis
sufficientlylargecomparedtothereleasedenergy,anenergeticalequilibriumofthesystemcan
bsettleecauseduringenoughtheproenergycessisofcomconductedbustion.fromInthethissystemcasetheduetosystemcodooling.esHonotwevheater,uifpscoolingignificandotlyes
nottheproreacvidetionanisaccelerateaccordinglydhighnoticeablyheatindissipation,thisregion.aloAscalizedatempresult,ineraturethisriseostronglyccursloandcalizedand
hightemperatureregion,thefuelisburnedrapidlyinanexplosivereaction.Consequently
thesolidfuelcombustionallowsadistinctionoftwocases:
132

kgroundBacTheoretical9.21.Ifthereactionsettlesintoenergeticalequilibriumitisclassifiedassubcriticalorfizzle
t.enev2.Ifthereactionresultsinanexplosiveburstofpowergenerationwespeakofasupercritical
t.enevexplosiveorThisclassificationissupportedbytheoreticalresultswhichprovideinformationonthe
ignitionwhereabmooutsdeloftwhichehappweearanceneedinofthethenextbloswup.ection:Wefinallyintroducethesteadystatesolidfuel
−Δψ=δeψinΩ,
ψ(x)=0on∂Ω.(9.4)
Wtheegivetheoryasonhortexistencesummaryanofdtheuniquenesstheoreticalofsolresultsutionsmenfromtionedtheaboveliteratureandreviewinthefolloadditionallywing
section.BackgroundreticalTheo9.2Wesolutionhaveoverexplainedthesowholefartimethatorthesolidotherwisefuelthesignitionolutionmobdelecomesgiveninunbequationoundedat(9.3)atimehasTeither<∞a.
Thisblowuptimeismainlydeterminedbyacriticalparameterδc,thatisifδ<δcthesolution
isboundedfort∈(0,∞),otherwiseforδ>δcthesolutionexperiencesablowupatt=T.
Westartthissectionwiththediscussionoftheexistenceanduniquenessofthesolutions,
beforeweproceedwithsomeresultsprovidinganupperboundoftheblowuptimeTin
dependenceonδc.Thetheoreticalinvestigationissophisticatedandneedsanextensive
theoreticalpreparation.Therefore,wecitetheessentialresultsfrom[2]withoutproving
them.Letδcdenotethe(reactiondependant)criticalparameter.Thenthefollowingtwo
theorems(Theorem3.6and3.7in[2])hold:
2,10≤θTheorem(t,x)≤9.1.ϕ(xF),orδwher<eδcϕ,(xpr)isoblemthemi(9.3)nimalhasasolutionuniqueofthesolutionsteθady(t,xstate)∈prCoblem(I×Ω(9.4)).with
Remark9.1.WiththeassumptionsofTheorem9.1weobtainadditionalinformationonthe
convergencebehaviorofthesolutionintimewhichis
tlim→∞θ(t,x)=ϕ(x).
Theorem1,29.2.Foreachδ>δc,thereisaT∈[1/δ,∞)suchthat(9.3)hasauniquesolution
θ(t,x)∈C((0,T)×Ω).Moreover,
tlimTmaxθ(t,x)x∈Ω¯=∞.
Insolutionotherwofords,thethesteadysolutionstateθ(t,problemx)of(9.4)(9.3)ifisthegloballyparameterexistenδtstaandysbconelovwergesthetothecriticalbminimalound
δc.Otherwise,thesolutionbecomesunboundedintheL∞-senseastT.
133

9ApplicationtotheSolidFuelIgnitionModel

Furtherresearchallowsnotonlytodecidewhetherasystemexhibitsablowup,butalsoto
determineforgivenδcthemaximumtimeintervalforwhichthesolutionexists.Wecite
withoutprooffrom[2]thefollowingtwotheoremsandacorollaryansweringthisquestion.
ThetimeandbasicisaidealoiswertobfindoundanforODEtheinitialsolutionvaluetotheproblemsolidfuelwhoseignitionsolutionmodel:exhibitsablowupin
Theorem9.3.(Theorem3.9from[2])Letu(t)bethesolutionoftheinitialvalueproblem
u=δeu−λ1u,t∈(0,T)andu(0)=0
whereλ1isthefirsteigenvalueof−Δϕ=λϕ,x∈Ωandu(0)=0,x∈∂Ω.Letθ(t,x)be
thesolutionof(9.3)on[0,T)×Ω¯,then
u(t)≤supθ(t,x)x∈Ω¯
fort∈[0,T).
Itiseasytoverifythatthesolutionuonlyexistsonthetimeinterval[0,T0)whereu(t)→∞
fortT0.ThemaximumtimeT0isdeterminedbytheidentity
∞dzT0=0δez−λ1z.
Consequently,T0<∞ifδ>δ∗:=λ1/e,andinthiscasetheblowupoccursinfinitetimeas
statedinthecorollarybelow.
Corollary9.4.(Corollary3.10from[2])Ifδ>δ∗=λ1/e,thenT0<∞and
tlimT0supθ(t,x)x∈Ω¯=∞
whereT<T0.Thatis,theblowupoccursinfinitetime.

Wesubstantiatethisresultbymeansofanumericalexample.Consideringπ2Ω=(−1,1)×(−1,1)
δw∗e=1obtain.815.forFtheurthermore,smallesttheeigenvcriticalalueoftheparameterLaplacianisgivenλ1b=yδ2c=and1.734for.theWecanparameternow
investigatefordifferentparametersδ>δ∗thequalityoftheupperboundT0fortheblowup
timeT.Figure9.1showsthebehaviorofpredictedandnumericallycomputedblowuptime
forexhibitsdifferenatblowupreactioninfiniteparameterstimeandδ.fuTherthercorollarmoreygivesstatesanuppthaterforbouδnd>δfor∗>thδecblothewupsolutiontime.
∗FtheorthetimeninumericaltervalI=example(0,T)succonsideredhthatin(0,theT0)⊂nextI.Fsection,orthewesakeensureofthatδcompleteness,>δandwechofinallyose
presentresultsfortheblowuptimeforthoseremainingparametersδ∈(δc,δ∗).Bebernes
andEberlyderiveforthiscasethefollowingtheorem:
Theorem9.5.(Theorem3.13from[2])Ifδcisinthespectrumof(9.4)andifδ>δc,then
theuniquesolutionθ(t,x)of(9.3)blowsupinfinitetimeTwhere
2π2T<δc(δ−δc).

134

100

10

1

1.0

9.3OptimalControloftheSolidFuelIgnitionModel

timewupbloprediction

10∗Figure9.1:andPredictedδc=1.and734.(xcalculated-axis:blolog(δwup),ytime-axis:forlog(tdifferen))tparametersδwithδ=1.815

Withthetheoreticalresultsathand,weareabletoconstructanexamplewhosesolutionblows
upsysteminbfiniteycontime.trollingAgainsttheheatthisbacdissipation.kground,ouThatrapis,plicwechoationsosetenthedtoconctrolontrolasathissourceexplosivterme
ontherighthandsideofequation(9.3a).

9.3OptimalControloftheSolidFuelIgnitionModel

Optimalcontrolofthesolidfuelignitionmodelisofpracticalrelevanceinchemicalengineering.
Weconsiderherethe2-dimensionalcaseonasquareddomainwithdistributedcontrolfor
differentcostfunctionals.
Example9.1.(Distributedcontrolofthe2-dimensionalsolidfuelignitionmodel)Let
Ω=(−1,1)×(−1,1)⊂R2begiven,andchoseδ=2.5forthefuelignitionmodel.Bymeans
ofTheorem9.3andCorollary9.4weobtainthatinthisconfigurationthesolutionexhibitsa
blowupatT<T0=0.929.Wedefinethetimedependentdiscontinuousfunction
ππ
θ¯(t,x,y):=t∙cos2x∙cos2yfort<1,
t∙cos32πx∙cos32πyfort≥1,
thetimedependentcontinuousfunction
θˆ(t,x,y):=t∙cos3πx∙cos3πy,
22

135

9ApplicationtotheSolidFuelIgnitionModel

(9.5)(9.6)(9.7)

andthetimeindependentfunction
θ˜(x,y):=0.5.
Inthefollowing,weconsiderthreedifferentcostfunctionals:
1.Atimedistributedcostfunctionalofthestructure
J(q,θ):=1θ(t)−θ¯(t)2dt+10−3q(t)2Q.(9.5)
22I2.Aterminaltimecostfunctional
3−J(q,θ):=21θ(T)−θ˜2+102q(t)2Q.(9.6)
3.Anotherdistributedcostfunctionalgivenby
J(q,θ):=1θ(t)−θ¯(t)2dt+1eθ(t)−eθ¯(t)2dt+10−3q(t)2Q.(9.7)
222IIWeobtaintheoptimalcontrolproblemofinterestas
q∈minQJ(q,θ)
suchthat∂tθ−Δθ−δeθ=q,inI×Ω,
θ(0,x)=0inΩ,
θ(t,x)=0onI×∂Ω.
Firstproblemofall,(9.3)letonustheconsgividenerthedomainforwΩardforqsim=0.ulationTheofthecalculationproblem,breaksthatdoiswntheatt=solution0.715of,
andthedevelopmentofthesolutiontowardsthispointisshowninFigure9.2.Thesolution

(a)t=0.706(b)t=0.709(c)t=0.712(d)t=0.715
Figure9.2:SolutionfortheforwardsimulationofExample9.1.
blowsupatapproximatelyt=0.715.Aforwardsimulationwithq=0andseveralother

136

9.3OptimalControloftheSolidFuelIgnitionModel

testedinitialcontrolsonthewholetimeintervalisnotpossible,andthereforecommon
solutiontechniqueswhichrequirethesolutionoftheproblemonthewholetimeintervalI
forthecalculationofthecontrolupdatecannotbeapplied.Asoutlinedintheintroduction,
multipleshootingovercomesthisdifficultyandallowsthecalculationofpiecewisesolutions.
Forthecostfunctional(9.5)weapplymultipleshootingonthetimeintervalI=(0,2)with
40multipleshootingintervalseachconsistingof5timestepsoflength0.01.Thesolution
atdifferenttimepointsisgiveninFigure9.3.Allmeshesarerefinedlocally,suchthatthe
solutionsondifferentintervalsarecomputedondifferentlyfinediscretizations.Additionally,
weshowthecontrolatthesametimepointsandtherebythecorrespondingmeshesinFigure
9.4.

(a)t=0.5

(c)t=0.95

(b)t=0.9

(d)t=1.0

(e)t=1.05(f)t=2.00
Figure9.3:SolutionforExample9.1withcostfunctional(9.5).
Extrapefficiencyolatingoflocaltheandexactglobalminimalrefinemenvaluet.ofThethechoicefunctionaloflocalas0re.fi775194neme,ntweiscanforthiscompareexamplethe
oflessimportancethaninthoseexamplesconsideredinthepreviouschapter.Infact,no
singularitiesortravelingfrontshavetoberesolvedsuchthatlocalrefinementyieldsonlya
slightimprovementforthisexampleasillustratedinFigure9.5.
Thisisdifferentforthecaseoftheterminaltimecostfunctional(9.6).Weconsiderthe
timeintervalI=(0,1)forwhichamultipleshootingtimedomaindecompositionof10
subintervalseachconsistingof10timestepsoflength0.01ischosen.Asexpected,dueto
theDirichletboundarydata,localrefinementtakesplacemainlyonthelasttimeinterval.

137

9ApplicationtotheSolidFuelIgnitionModel

(a)t=0.5

(c)t=0.95

(e)t=1.05

(b)t=0.9

(d)t=1.0

(f)t=2.00

Figure9.4:ControlandmeshesforExample9.1withcostfunctional(9.5).

Itresultsinathinboundarylayerofrefinedcells.Theexactfunctionalvalueisobtained
fromextrapolationas0.10116413forwhichwecomparetheperformanceforlocalandglobal
refinementinFigure9.6.Thesolutionatdifferenttimepointsforthisoptimizationproblem
isshowninFigure9.7,andasbeforethecontrolandmesharegivenafterwardsinFigure9.8.
Finally,weconsidertheproblemforthethirdcostfunctional(9.7).Inthiscase,wedecompose
I=(0,1)into10multipleshootingintervalsandapplyasbeforetheimplicitEulertime
steppingschemewithstepsize0.01.Weobtainanapproximationoftheexactfunctionalby
extrapolationas0.09053600.Asinthefirstconfiguration,localrefinementyieldsonlyslight
improvementsoftheefficiency.ThisresultisshowninFigure9.9.Theobtainedsolutionand
controlatdifferenttimepointsareshownattheendofthischapterinFigures9.10and9.11
inwhichthestructureoftherefinedmeshesisoutlined.

138

1−10

2−10

3−10

9.3OptimalControloftheSolidFuelIgnitionModel

loglobalcalrefinemenrefinementt

4−10104105106107
Figure9.5:Functionalerrorfordifferentlyfinemeshes,gainedbylocalandglobalmesh
refinementforExample9.1withcostfunctional(9.5).(x-axis:log(N),y-axis:
log(|eh|))

1−102−10

3−10

loglobalcalrefinemenrefinementt

4−10104105106
Figure9.6:Functionalerrorfordifferentlyfinemeshes,gainedbylocalandglobalmesh
log(|refinemene|))tforExample9.1withcostfunctional(9.6).(x-axis:log(N),y-axis:
h

139

9ApplicationtotheSolidFuelIgnitionModel

140

(a)t=0.25

(c)t=0.75

(b)t=0.5

(d)t=1.00

Figure9.7:SolutionforExample9.1withcostfunctional(9.6).

(a)t=0.25

(c)t=0.75

(b)t=0.5

(d)t=1.00

Figure9.8:ControlandmeshesforExample9.1withcostfunctional(9.6).

2−103−10

4−10

−510

9.3OptimalControloftheSolidFuelIgnitionModel

loglobalcalrefinemenrefinementt

104105106
Figure9.9:Functionalerrorfordifferentlyfinemeshes,gainedbylocalandglobalmesh
refinementforExample9.1withcostfunctional(9.7).(x-axis:log(N),y-axis:
log(|eh|))

(a)t=0.25

(b)t=0.5

(c)t=0.75(d)t=1.00
Figure9.10:SolutionforExample9.1withcostfunctional(9.7).

141

9

toApplicationthe

142

Figure

9.11:

SolidFuelIgnitionModel

(a)t=0.25

(c)t=0.75

ControlandmeshesforExample

(b)t=0.5

(d)t=1.00

costwith9.1

functional

(9.7).

okOutloandConclusion10

Inthisthesiswedevelopedmultipleshootingapproachesforoptimizationproblemsconstrained
bypartialdifferentialequations.Additionally,weinvestigatedtheapplicationofaposteriori
errorestimationandspatialmeshadaptationinthecontextofmultipleshooting.
Startingfromthemultipleshootingapproachforordinarydifferentialequations,weextended
thedirectandindirectmultipleshootingapproachtoproblemsconstrainedbypartial
differentialequations.Inthiscontext,wealsooutlinedtherelationbetweendirectand
indirectmultipleshooting.Consequently,thedevelopmentoftheapproachesalsoinvolved
discussingthegeneraldifferencesbetweentheODEandthePDEconstrainedcase.We
concludedthatthePDEcasehasseverallimitationswithrespecttotheefficiency.While
theODEcaseexploitsefficientderivativegenerationtechniquestogetherwithfastexplicit
assemblingofthesystemmatrices,thePDEcasedoesnotallowthisprocedure.Thereason
forthisrestrictionistheextremelyhighdimensionofthesystemmatricesduetofinespatial
discretizations.Inthiscontext,possibleextensionsmightleadintothedirectionofreduction
techniques.Ontheonehand,exploitingaprioriinformationontheproblemmightyielda
reductionofthecontrolspaceand,ontheotherhand,acoarserproblemsuiteddiscretization
ofthecontrolmightreducethedimensionofthecontrolspaceevenfurther.Theseideas
mightthereforeallowtoacceleratethecondensedapproachfurther.
WediscussedthatoptimizationproblemswithhighlyunstablePDEconstraintsdonotpermit
thesolutionoverthewholetimeinterval.Theseproblemscannotbesolvedbystandard
solutiontechniques.Multipleshootingasatimedomaindecompositiontechniqueisasuitable
solutionapproachforthesolutionofsuchproblemsaswesawfortheexplosivesystemgiven
bythesolidfuelignitionmodel.
Weconsidereddifferentsolutiontechniquesappliedtodirectandindirectmultipleshooting.
WestartedfromNewton’smethodforthesolutionofthesystemofmatchingconditions,
proceededwithapreconditionedGMRESmethodforthelinearizedsystem,andfinally
concludedwiththeinvestigationofdifferentsolversfortheintervalwiseproblems.These
problemscomprisedlinearandnonlinearinitialvalueproblemsaswellaslinearandnonlinear
boundaryvalueproblems.Wediscovered,thatmostofthecomputationtimeisspentonthe
generationofthedirectionalderivativesneededfortheiterativesolutionofthelinearized
system.Numericaltestsindicatedthatpreconditioningisinevitablebutincreasesthe
numericaleffortnoticeably.Condensingtechniqueswereintroducedasapossiblealternative
tosavecomputationaleffortandtime.Promisingtopicsforfurtherresearchseemto
bepreconditioningtechniquesforthecondensedapproachandadditionalpreconditioners,
preferablyparallelizable,forthedirectapproach.
Furthermore,westudiedthecombinationofmultipleshootingwithaposteriorierroresti-
mation:firsttheapplicationofthestandardapproach,andsecondthedevelopmentofan

143

Conclusion10okOutloand

errorestimatorsuitedfortheneedsofmultipleshootingwithdifferentadjacentmesheson
themultipleshootingnodes.Furtherfuturedevelopmentsinthisareaofresearchmightlead
intothedirectionoftimeadaptivestrategies.Ontheonehand,aposteriorierrorestimation
canbeappliedforthetemporalerror,andmeshadaptationintimefollowsstraightforward.
Ontheotherhand,anappropriatesplittingofthemultipleshootingintervalswouldnot
onlyallowthereductionofthetemporalerror,butalsothereductionofthespatialerrorin
thecaseofintervalwiseconstantmeshes.Theideaoffindingadiagonalsequenceofspatial
mesheswithintheouterNewtonmethodshouldalsobeinvestigated.Thisprocedureperforms
meshrefinementwhenevertheerrorduetotheNewtonresidualbecomessmallerthanthe
discretizationerror.Thus,thenumberofNewtonstepsneededduringthesolutionprocedure
reduced.ebtmigh

Finally,afterhavingunderstoodthegeneralfeaturesofmultipleshootingwithmeshadaptation
forPDEconstrainedoptimizationproblems,theapplicationofthedevelopedmultipleshooting
approachtosophisticatedapplicationproblemsisadvisable.

144

wledgmentsnockA

ThistionalwGrorkaduwasiertenkolsupportedlegby“ComplextheGermanprocesses:ResearcMohFdeling,oundationSim(DFulationG)andthroughtheOptimization”Interna-.I
mamyveryresearchgratefulinaninthattheterdisciplinaryGermangraduateResearchscFhooloundationwhichofferedofferednotmetheonlyoppgenerousortunitytofundingdo
andparticipateprovidedinvanariousinsighwtinorkshopstoadifferenbroadtareasandoftostascienyintifictheresearUnictedh,butStatesalsoforalloatwwoedmonmetoth
visit.hresearcIwanttoexpressmygratitudetomysupervisorsProf.Dr.Dr.h.c.HansGeorgBock,
InterdisziplinäresZentrumfürWissenschaftlichesRechnen(IWR),UniversityofHeidelberg,
forandproProf.vidingDr.thisRolfintereRannacstingher,subjectInstitutandfürforAtheirngewandtesupportandMathematikman,yUnivfruitfulersityofdiscussionsHeidelboeverrg,
ears.ythreepasttheDuringthistime,manyotherpeoplehavecontributedtothiswork.Firstandforemost,I
wforaninttotrothankducingmeProf.inDr.totheGuidofiniteKanscelemehat,ntDepsoftwareartmentpacofkagedealMathematics.II,for,ThisexascontinA&MuousUnivsuppersitorty,
andinnovativeideas,forthemanyinterestingandencouragingconversationsandlastbut
InotwouldleastlikforetotheexpwrearmssmwyelcomegratitudeduringtoDr.mystayJohannesatTScexashlöder,A&MInterUniversitdisziplinäry.Fesurthermore,Zentrum
fürearlystageofWissenschaftlichesthiswork.ReIamchnenindebted(IWR),toUnivProf.ersitDr.yofMatthiasHeidelberg,Heinkforenschishloss,supportDepduringartmentthofe
ComputationalandAppliedMathematics,RiceUniversity,whokindlyofferedmetostaywith
whisorkongrouptheforsubtwjeoct,wbeeksutalsoandtoprookvidedthetimemetonotonlycriticallyanintediscussrestingandinsighimprotveinmtoyidhiseasonpreviousthe
topic.IalsomatterswanandttomythankofficemymatecolleagueBärbelDr.JanssenDominikfortheMeidnerwarmforhisandadvispleasaneintmanyatmospherecomputationalandthe
interestingdiscussionsonthedeal.iisoftwarepackage.
ofFinally,IMathematicswant,toUnivthankersitmyyoffriendsSussex,andmforyprofamily:ofreadingmysister,andmDr.yKeparenrstintsforHesse,theDepwonderfulartment
ort.suppencouraging

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