Adaptive frame based regularization methods for linear ill-posed inverse problems [Elektronische Ressource] / von Mariya Zhariy

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Adaptive Frame Based Regularization Methodsfor Linear Ill-Posed Inverse Problemsvon Mariya ZhariyDissertationzur Erlangung des akademischen GradesDoktorin der Naturwissenschaften– Dr. rer. nat. –Vorgelegt im Fachbereich III der Universit¨ at Bremenim Oktober 2008Betreuer: Prof. Dr. Gerd TeschkeProf. Dr. Ronny RamlauAbstractThis thesis is concerned with the development and analysis of adaptive regularization methods for solvinglinear inverse ill-posed problems. Based on nonlinear approximation theory, the adaptivity concept hasbecome popular in the field of well-posed problems, especially in the solution of elliptic PDE’s. Undercertain conditions on the smoothness of the solution and the compressibility of the operator it has beenshown that the nonlinear approximation guarantees a more efficient approximation with respect to thesparsity of the solution and to the computational effort.In the area of inverse problems, the sparse approximation approach has been applied in solving de-noising and de-blurring problems as well as in general regularization. An essential cost reduction hasbeen achieved by newly developed strategies like domain decomposition and specific projection meth-ods. However, the option of adaptive application, leading simultaneously to cost reduction and sparseapproximation, has not been taken into account yet.

Sujets

• Mathematics

Informations

• Publié par: universitat_bremen
• Publié le: 01 janvier 2008
• Nombre de lectures: 20
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

AdaptiveFrameBasedRegularizationMethods
forLinearIll-PosedInverseProblems

vonMariyaZhariy

Dissertionat

zurErlangungdesakademischenGrades
haftenNaturwissenscderDoktorin–Dr.rer.nat.–

VorgelegtimFachbereichIIIderUniversit¨atBremen
2008erOktobim

Betreuer:Prof.Dr.GerdTeschke
Prof.Dr.RonnyRamlau

Abstract

Thisthesisisconcernedwiththedevelopmentandanalysisofadaptiveregularizationmethodsforsolving
linearinverseill-posedproblems.Basedonnonlinearapproximationtheory,theadaptivityconcepthas
becomepopularinthefieldofwell-posedproblems,especiallyinthesolutionofellipticPDE’s.Under
certainconditionsonthesmoothnessofthesolutionandthecompressibilityoftheoperatorithasbeen
shownthatthenonlinearapproximationguaranteesamoreefficientapproximationwithrespecttothe
sparsityofthesolutionandtothecomputationaleffort.
Intheareaofinverseproblems,thesparseapproximationapproachhasbeenappliedinsolvingde-
noisingandde-blurringproblemsaswellasingeneralregularization.Anessentialcostreductionhas
beenachievedbynewlydevelopedstrategieslikedomaindecompositionandspecificprojectionmeth-
ods.However,theoptionofadaptiveapplication,leadingsimultaneouslytocostreductionandsparse
approximation,hasnotbeentakenintoaccountyet.
Inourresearchwecombinetheadvantagesofthenonlinearapproximationwiththeclassicalregular-
izationandparameteridentificationstrategies,likeTikhonovandLandwebermethods.Wemodifythe
classicalapproachforsolvingtheill-posedinverseproblemsinordertoobtainasparseapproximation
withessentiallyreducednumericaleffort.Themainnoveltyofthedevelopedregularizationmethodsis
theadaptiveoperatorapproximation.
Ingeneral,wehaveshownthatitispossibletoconstructadaptiveregularizationmethods,whichyield
thesameconvergenceratesastheconventionalregularizationmethods,butaremuchmoreefficientwith
respecttothenumericalcosts.
Theanalyticresultsofthisworkhavebeenconfirmedandcomplementedbynumericalexperiments,that
illustratethesparsityoftheobtainedsolutionanddesiredconvergencerates.

Zusammenfassung

DasHauptvorhabendieserArbeitistdieEntwicklungadaptiverRegularisierungsmethodenf¨urdieL¨osung
linearerschlechtgestellterinverserProbleme.BasierendaufdernichtlinearenApproximationhabenadap-
tiveVerfahrenindenletztenJahreninsbesondereimBereichdergutgestelltenProblemeanPopularit¨at
gewonnen.UnterbestimmtenVoraussetzungenandieGlattheitderL¨osungunddieKompressibilit¨atdes
OperatorsistesgelungendieL¨osungeinesgutgestelltenProblemsdurcheined¨unnbesetzteDarstellung
mitlinearerKomplexit¨atzuapproximieren.
InderletztenZeitsinddieaufderd¨unnenDarstellungbasiertenAns¨atzeaufdemGebietderschlecht-
gestelltenProblememehrmalserfolgreichverwendetworden,etwaimBereichderVerbesserungder
Bildqualit¨atdurchEntrauschenundSch¨arfen.DiedarauffolgendentwickeltennumerischenVerfahren,
wiez.B.GebietzerlegungsalgorithmenoderbestimmteProjektionsstrategien,erm¨oglicheneineSenkung
derIterationsschritte,ber¨ucksichtigenabernichtdieM¨oglichkeitderadaptivenAnwendung.
DieindieserArbeitvorgestelltenadaptivenRegularisierungsverfahrenbasierenaufdenklassischenMeth-
odenzurL¨osungschlechtgestellterProbleme,etwaLandweber-IterationundTikhonov-Regularisierung
sowieentsprechenderRegelnzurWahlderoptimalenRegularisierungsparameter.DieModifikationder
klassischenVerfahrenerfolgtmitdemZiel,dieL¨osungderschlechtgestelltenProblemeaufeineadaptive
WeisezuapproximierenundgleichzeitigdenBerechnungsaufwandzureduzieren.Dabeientstehenauf
nat¨urlicheWeised¨unnbesetzteRekonstruktionen.
DietheoretischenResultatederArbeitwurdenanhandeinesnumerischenBeispiels,derRekonstruktion
vonTomographie-Daten,verifiziert.Esistunsgelungen,denRechenaufwandzuverringernunddabei
dieKonvergenzratendernichtadaptivenVerfahrenzuerhalten.

tsentCon

IWaveletBasesAndFramesForOperatorEquations
1WaveletBases
1.1MultiresolutionandWaveletRieszBases............................
1.2RieszBasisPropertyandBiorthogonality...........................
1.3CharacterizationofFunctionSpaces..............................
1.3.1FunctionSpaces.....................................
1.3.2DirectandInverseEstimates..............................
1.4WaveletsonBoundedDomains.................................
2Frames
2.1BasicFrameTheory.......................................
2.2GelfandFrames..........................................
2.3LocalizationofFrames......................................
2.4FrameBasedOperatorDiscretization..............................

IIInverseProblems
B3Definitionsasic3.1Ill-PosednessandGeneralizedInverse..............................
3.2RegularizationandOrderOptimality..............................
3.3CompactLinearOperators...................................
4FilterBasedFormulation
4.1RegularizationPropertyandParameterChoiceRules.....................
4.1.1A-prioriParameterChoice................................
4.1.2Morozov’sDiscrepancyPrinciple............................
4.1.3BalancingPrinciple...................................
4.2RegularizationinHilbertScales.................................
4.2.1WaveletBasedRegularization..............................
4.3TikhonovRegularization.....................................
4.3.1RegularizationResult..................................
4.3.2TikhonovRegularizationwithanA-PrioriParameterChoice............
4.3.3TikhonovRegularizationwithMorozov’sDiscrepancyPrinciple...........
4.3.4TikhonovRegularizationwithBalancingPrinciple..................
4.3.5TikhonovRegularizationinHilbertScales.......................
4.4LandweberIteration.......................................
4.4.1RegularizationResult..................................

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4.4.2A-prioriTruncationRule................................
4.4.3Morozov’sDiscrepancyPrinciple............................
4.4.4LandweberIterationinHilbertscales.........................

IIIAdaptivityIssues
5NonlinearApproximationandAdaptivity
5.1NonlinearApproximation....................................
5.2BestN-termApproximationin2................................
6AdaptiveIterativeSchemeforWell-PosedProblems
6.1FrameBasedSetting.......................................
6.2AdaptiveDiscretizationoftheOperatorEquation.......................
6.3FrameBasedAdaptiveSolver..................................

IVAdaptiveRegularizationMethods
7ApproximateFilterBasedMethods
7.1ApproximateFilterBasedRegularization...........................
7.1.1ApproximateWaveletBasedRegularization......................
7.2ApproximateMorozov’sDiscrepancyPrinciple........................
7.3OrderOptimalitybyApproximateBalancingPrinciple....................
8AdaptiveTikhonovRegularization
8.1AdaptiveTikhonovRegularization:Formulation.......................
8.1.1RegularizationResult..................................
8.2OrderOptimalitybyA-prioriParameterChoice........................
8.2.1StandardFilterBasedMethod.............................
8.2.2A-prioriErrorEstimateinHilbertscales........................
8.3ComplexityofAdaptiveTikhonovRegularization.......................
8.4ApproximateApplicationofBalancingPrinciple.......................
9AdaptiveLandweberIteration
9.1ModifiedLandweberIteration..................................
9.2AdaptiveFormulationofLandweberIteration.........................
9.3OrderOptimalitybyA-PrioriParameterChoice.......................
9.4RegularizationbyA-PosterioriParameterChoice.......................
9.4.1AdaptiveResidualDiscrepancy.............................
9.4.2AMonotonicityofAdaptiveIteration.........................
9.4.3ConvergenceinExactDataCase............................
9.4.4RegularizationResult..................................
roblemPExample1010.1LinearRadonTransform.....................................
10.2RegularityofSolution......................................
10.3RequirementsontheSystem..................................
10.3.1RequirementsImposedbytheRegularityoftheSolution...............
10.3.2RequirementsImposedbytheOperatorDiscretization................

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Numerics

10.4.1

Summary

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Adaptive

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Discretization

Outlo

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ductiontroIn

Adaptivityisapromisingconceptinsignalrepresentationwheneverastructureorasignalundercon-
siderationisinsomesensesparse,i.e.itcanbeapproximatedbyonlyafewcomponentsintermsof
somebasisorframe.Themaintwoquestionsinsparseapproximationarehowtofindanappropriate
decomposingsystemandwhichcomponentstouse.
Inthisthesisweapplytheconceptofnonlinearapproximation[28]intermsofwavelets,whichhasits
origininthemultiresolutionanalysis[20,40].Themultiscaledecompositionbenefitsfromthesimultane-
ousrepresentationofthesignalcomponentsondifferentresolutionlevels,whichinmanyrelevantcases
resultsinamoreefficientapproximationthantherepresentationonthefinestresolutionlevel.However,
givenamultileveldecomposition,therearedifferentwaystoapproximatethedecomposedsignalfrom
itsbuildingcomponents.Theefficiencyofapproximationcanbeconsiderablyimprovedifoneconsiders
significantcomponentsofthesignalinsteadofworkingwithlinearsubspaces,builtbytruncationin
scale.Unlikethelevel-by-levelapproximati