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

Helke Karen Hesse

Description

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.

Sujets

• Mathematics

Informations

• Publié par: ruprecht-karls-universitat_heidelberg
• Publié le: 01 janvier 2008
• Nombre de lectures: 12
• Langue: English
• Poids de l'ouvrage: 2 Mo

Extrait

Inaugural-Dissertation

zur

DoktorwürdederErlangung

der

Naturwissenschaftlich-MathematischenGesamtfakultät

der

ersitätt-Karls-UnivhuprecR

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vorgelegt

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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

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Prof.

Prof.

Dr.

Dr.

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Mesh

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Optimization

Rolf

Dr.

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Hans

Georg

Problems

Bo

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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

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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,