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Computational methods for inverse elliptic problems

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Computational methods for inverse elliptic problems
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Hamad, Abdalkaleg
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Denver, CO
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University of Colorado Denver
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English

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Doctorate ( Doctor of philosophy)
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University of Colorado Denver
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College of Engineering and Applied Sciences, CU Denver
Degree Disciplines:
Engineering and applied science
Committee Chair:
Premnath, Kannan
Committee Members:
Tadi, Mosehn
Vega, Rafael Sanchez
Carey, Varis
Butler, Troy

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ComputationalMethodsforInverseEllipticProblems by ABDALKALEGHAMAD B.S.,BrightStartUniversityofTechnology,Brega-Libya,1998 M.S.inMechanicalEngineering,UniversityofColoradoBoulder,2010 M.S.inAppliedMathematics,UniversityofColoradoDenver2018 Athesissubmittedtothe FacultyoftheGraduateSchoolofthe UniversityofColoradoinpartialfulllment oftherequirementsforthedegreeof DoctorofPhilosophy EngineeringandAppliedScienceProgram 2018

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ii ThisthesisfortheDoctorofPhilosophydegreeby AbdalkalegHamad hasbeenapprovedforthe EngineeringandAppliedScienceProgram KannanPremnath,Chair MohsenTadi,Advisor RafaelSanchezVega VarisCarey TroyButler Date:December15,2018

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iii Hamad,AbdalkalegPhD,EngineeringandAppliedScienceProgram ComputationalMethodsforInverseEllipticProblems ThesisdirectedbyAssociateProfessorMohsenTadi ABSTRACT Thisdissertationisconcernedwithtwonewcomputationalmethodsforinverseelliptic problems.Thetwomethodsareiterativealgorithms.Thealgorithmsstartwithaninitialguess fortheunknownfunction,obtainabackgroundeld,andobtaintheworkingequationsforthe erroreld.Intherstmethodtheunknownfunctioniscomposedofaknownuniform backgroundandanunknownseparablepart.Itformulatestwowell-posedproblemsfortheerror eldwhichmakesitpossibletoobtainthecorrectionterm.Weapplythismethodtoinverse sourceproblemsforPoissonandHelmholtzequations.Inthesecondmethodtheunknown functionisageneralpositivefunctionwhosevalueisawayfromzero.Thismethodgeneratesa setoffunctionsthatsatisfy some oftherequiredboundaryconditions.Werefertothisspaceas propersolutionspace .Thecorrectiontotheassumedvaluecanthenbeobtainedbyimposingthe remainingboundaryconditions.Weapplythismethodtoaninverseevaluationofanunknown wavenumberinaHelmholtzequation,andaninversescatteringproblem.Weconsiderthe evaluationofrealandcomplexwavenumbers.Wealsoapplythepropersolutionspacemethod toreconstructtheexuralrigidityofaKirchho-Loveplate.Themethodcanbeappliedto problemsinvariousgeometries.Numericalresultsindicatethatbothalgorithmscanrecoverclose estimatesoftheunknownfunctionsbasedonmeasurementscollectedattheboundary. Theformandcontentofthisabstractareapproved.Irecommenditspublication. Approved:MohsenTadi

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iv DEDICATION Iwouldliketodedicatemyworktomyparentsandmywife.

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v ACKNOWLEDGEMENTS IwouldliketothankmysupervisorAssociateProfessorMohsenTadiforhissupportandhis guidance.Also,Iwouldliketothankmydissertationcommittee,RafaelSanchezVega,Kannan Premnath,VarisCarey,andTroyButler.

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vi TABLEOFCONTENTS CHAPTER I INTRODUCTION1 I.1BackgroundforInverseProblems.............................1 I.2ClassicationofInverseProblemsBasedonTheUnknownFunctions.........2 I.3ExamplesforInverseProblems..............................4 I.3.1TheGravimetryInverseProblem.........................4 I.3.2TheAdmittivityInverseProblem.........................5 I.3.3InverseScatteringProblem............................5 I.3.4IdenticationofFlexuralRigidityInKirchho-LovePlates..........6 I.4PracticalExamplesofNon-InvasiveImagingTechniques................7 I.4.1Vibrothermography................................8 I.4.2ElectricImpedanceTomography.........................8 I.4.3MicrowaveImaging.................................9 I.4.4Acousto-ElectromagneticTomography......................11 I.4.5BioluminescenceTomography...........................13 I.5Well-PosedandIll-PosedProblems............................15 I.6ExamplesofIll-PosedProblems..............................17 I.6.1SystemsofLinearEquations...........................17 I.6.2CauchyProblemforALaplaceEquation.....................18 I.6.3InverseEvaluationofACoecientInAPartialDierentialEquation....18 I.7TechniquesforRegularizationofIll-PosedProblems..................19 I.7.1TruncatedSVDRegularization..........................19 I.7.2TikhonovRegularization..............................20 I.8SomeExistingMethodsforInverseEllipticProblems..................21

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vii I.8.1OptimizationMethod...............................22 I.8.2Themethodoffundamentalsolutions......................23 I.8.3Layer-strippingmethod..............................25 I.8.4Levelsetmethod..................................26 I.8.5D-barmethod....................................28 I.8.6Aniterativemethodbasedonmultipleforwardsolutions............30 I.9Organizationofthedissertation..............................31 II INVERSESOURCEPROBLEMFORPOISSONANDHELMHOLTZEQUATIONS33 II.1Introduction.........................................33 II.2Inversesourceproblemin2-D...............................36 II.2.1Updateinydirection...............................37 II.2.2Updateinxdirections..............................41 II.2.3Recoverywithpartialdata............................41 II.3Inversesourceproblemin3-D...............................43 II.3.1Updateinzdirection...............................44 II.3.2Updatein x and y directions...........................46 II.4Helmholtzinversesourceproblem.............................48 II.5Numericalexperiments...................................49 II.5.1Example1......................................49 II.5.2Example2......................................50 II.5.3Example3......................................55 II.5.4Example4......................................59 II.5.5Example5......................................59 II.6Summary..........................................61 III ONIDENTIFICATIONOFAWAVENUMBERFORHELMHOLTZEQUATION62 III.1Introduction.........................................62

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viii III.2Problemstatementandtheidenticationalgorithm...................63 III.3Propersolutionspacefortheidenticationofawavenumber.............65 III.4Complexwavenumber...................................67 III.5Numericalimplementation.................................67 III.6Numericalexamplesfortheidenticationofawavenumbermethod.........70 III.6.1Example1......................................70 III.6.2Example2......................................73 III.6.3Example3......................................75 III.6.4Example4......................................77 III.6.5Example5......................................80 III.6.6Example6......................................82 III.6.7Example7......................................82 III.6.8Example8......................................85 III.7Summary..........................................90 IV INVERSESCATTERINGPROBLEMSBASEDONPROPERSOLUTIONSPACE91 IV.1Introduction.........................................91 IV.2Problemstatementandtheidenticationalgorithm...................92 IV.3Propersolutionspaceforthescatteringinverseproblem................93 IV.4Numericalimplementationfortheinversescatteringproblemmethod.........95 IV.5Numericalexamplesfortheinversescatteringproblem.................98 IV.5.1Example1.....................................98 IV.5.2Example2.....................................101 IV.5.3Example3.....................................105 IV.6Summary..........................................105 V IDENTIFICATIONOFFLEXURALRIGIDITYINKIRCHHOFF-LOVEPLATES109 V.1Introduction.........................................109

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ix V.2Problemstatementandtheidenticationmethod....................109 V.3Propersolutionspaceforthereconstructionoftheexuralrigidity..........111 V.4Numericalimplementationfortheidenticationofexuralrigiditycoecient....113 V.5Numericalexamplesfortheidenticationofexuralrigiditycoecient.......116 V.5.1Example1......................................116 V.5.2Example2......................................119 V.5.3Example3.....................................119 V.6Summary..........................................122 VI CONCLUSION 125 BIBLIOGRAPHY 127

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x LISTOFTABLES TABLE I.1Afewexamplesforill-posedandwell-posedproblems..................16

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xi LISTOFFIGURES FIGURE I.1Directproblemvs.inverseproblem............................2 I.2Inversescatteringproblem.................................6 I.3Geometryoftheproblem:aburiedheatsourceofareacontainedintheplane x =0,perpendiculartothesamplesurface z =0...................9 I.4Schematicoftheconstantcurrentinjectionandtheboundarydatacollectionin electricalimpedancetomographyEITforacirculardomain.............10 I.5SchematicofEITimagingofahumanchest.......................10 I.6Schematicofbreastscreeningbymicrowaveimaging..................11 I.7ChestMicrowaveimaging.................................12 I.8PrincipleofMagneto-AcousticTomographywithMagneticInduction.........13 I.9Acousto-electromagneticimagingdevice.........................14 I.10Thesetoffunctionsforunknownshape.........................27 I.11Therstellipticproblemfortheerroreld........................32 I.12Thesecondellipticproblemfortheerroreld.......................32 II.1Erroreldwithover-speciedboundaryconditions....................36 II.2Erroreldwithover-speciedboundaryconditions....................42 II.3Erroreldwithpartialdataattheboundary.......................42 II.4Therstwell-posedproblemforthecomponentsoftheerroreld...........47 II.5Thesecondwell-posedproblemforthecomponentsoftheerroreld..........47 II.6Reductionintheerrorfortheexample1.........................50 II.7Convergenceoftherecoveredsourcefunctioninthe x directionforexample1.....51 II.8Convergenceoftherecoveredsourcefunctioninthe y directionforexample1.....51

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xii II.9Reductionintheerrorfortheexample1withpartialdata...............51 II.10Convergenceoftherecoveredsourcefunctioninthe x directionforexample1with partialdata..........................................52 II.11Convergenceoftherecoveredsourcefunctioninthe y directionforexample1with partialdata..........................................52 II.12Reductionintheerrorfortheexample2.........................53 II.13Actualsourcefunctionfortheexample2.Thebackgroundeldis 0 : 0025......54 II.14Recoveredsourcefunctionfortheexample2after140iterations............54 II.15Reductionintheerrorfortheexample2withhighernoiselevel............54 II.16Recoveredsourcefunctionfortheexample2after120iterations............55 II.17Actualtargetwithina3-Ddomainfortheexample3..................56 II.18Recoveredtargetwithina3-Ddomainfortheexample3after60iterations......56 II.19Comparisonoftheactualfunctionandtherecoveredsourcefunctionin x forthe example3...........................................57 II.20Comparisonoftheactualfunctionandtherecoveredsourcefunctionin y forthe example3...........................................57 II.21Comparisonoftheactualfunctionandtherecoveredsourcefunctionin y forthe example3...........................................58 II.22Actualsourcefunctionwith4targetsfortheexample4................59 II.23Recoveredsourcefunctionafter60iterationsfortheexample4............60 II.24Recoveredsourcefunctionafter80iterationsfortheHelmholtzoperatorinexample5.............................................61 III.1TheactualwavenumbergiveninIII.19toberecoveredforexample1........71 III.2Thereductioninerrorasafunctionofthenumberofiterationsforexample1....71 III.3Therecoveredfunctionafter400iterationsforexample1................71

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xiii III.4Comparisonoftherecoveredfunctionwiththeactualfunctionattwocross-sections forexample1.........................................72 III.5TheactualwavenumbergiveninIII.27toberecoveredforexample2.......73 III.6Thereductioninerrorasafunctionofthenumberofiterationsforexample2....74 III.7Therecoveredfunctionafter350iterationsforexample2................74 III.8Comparisonoftherecoveredfunctionwiththeactualfunctionatadiagonalcrosssectionsforexample2....................................74 III.9TheactualwavenumbergiveninIII.28toberecoveredforexample3........75 III.10Therecoveredfunctionwith =1 : 2after200iterationsforexample3.........76 III.11Therecoveredfunctionwith =0 : 7after200iterationsforexample3.........76 III.12Therecoveredfunctionwith =0 : 2after200iterationsforexample3.........76 III.13Thereductionintheerrorasafunctionofthenumberofiterationsforthethree dierent 'sforexample3.................................77 III.14TheactualfunctiongiveninIII.29toberecoveredforexample4..........78 III.15Therecoveredfunctionwith =0 : 005after600iterationsforexample4.......78 III.16Comparisonoftherecoveredfunctionwiththeactualfunctionatadiagonalcrosssectionsforexample4....................................79 III.17Convergenceoftherecoveredwavefunctionalongthediagonalinexample4.....79 III.18TheactualfunctiongiveninIII.31toberecoveredforexample5..........80 III.19Therecoveredfunctionwith =0 : 005after600iterationsforexample5.......81 III.20Therecoveringofawavenumberforexample6.....................83 III.21Therecoveredfunctionafter2500iterationsforexample6...............83 III.22Comparisonoftherecoveredfunctionwiththeactualfunctionatadiagonalcrosssectionsforexample6...................................84 III.23Convergenceoftherecoveredwavefunctionalongthediagonalinexample6.....84 III.24TheactualfunctiongiveninIII.35toberecoveredforexample7..........85 III.25Therecoveredfunctionwith =0 : 05after2500iterationsforexample7.......86

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xiv III.26Comparisonoftherecoveredfunctionwiththeactualfunctionatadiagonalcrosssectionsforexample7....................................86 III.27Convergenceoftherecoveredwavefunctionalongthediagonalinexample7.....87 III.28TheactualfunctiongiveninIII.36toberecoveredforexample6..........87 III.29Therecoveredfunctionwith =0 : 05after2500iterationsforexample6.......88 III.30Comparisonoftherecoveredfunctionwiththeactualfunctionatadiagonalcrosssectionsforexample8....................................88 III.31Convergenceoftherecoveredwavefunctionalongthediagonalinexample8.....89 IV.1TheactualfunctiongiveninEq.toberecoveredinexample1..........99 IV.2Therecoveredfunctionforexample1after20iterations =0 : 2 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10........99 IV.3Therecoveredfunctionforexample1after20iterations =0 : 1 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(11........99 IV.4Comparisonoftheactualfunctionwiththerecoveredfunctionsat y =0 : 3for threedierentvaluesof .................................100 IV.5Thereductioninerrorforexample1asfunctionofthenumberofiterations = 0 : 1 e 11............................................100 IV.6Therecoveredfunctionforexample1after50iterationswithDiricheltdataand =0 : 1 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10........................................100 IV.7Thereductioninerrorforexample1asfunctionofthenumberofiterationswith Dirichletdata........................................101 IV.8TheactualfunctiongiveninEq.IV.24toberecoveredinexample2........102 IV.9Therecoveredfunctionforexample2after75iterationswithNeumanndata = 0 : 2 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10..........................................102 IV.10Therecoveredfunctionforexample2after75iterationswithDirichletdata = 0 : 2 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10..........................................103 IV.11Comparisonoftheactualfunctionwiththerecoveredfunctionsalongthediagonal y = x forNeumannandDirichletdata.........................103

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xv IV.12Thereductioninerrorforexample2asfunctionofthenumberofiterations.....103 IV.13Thereductioninerrorforexample2asfunctionofthenumberofiterationsfor threedierentlevelsofnoise................................104 IV.14Therecoveredfunctionforexample2after75iterationswithNeumanndata = 0 : 2 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10with8%noise..................................104 IV.15TheactualfunctiongiveninEq.IV.25regionsoflargegradientstoberecovered inexample3.........................................106 IV.16TherecoveredfunctionforExample3after30iterationswithNeumanndata = 0 : 8 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10with4%noise..................................106 IV.17TheactualfunctiongiveninEq.IV.26toberecoveredinexample3.........107 IV.18Therecoveredfunctionforexample3Eq.IV.26after30iterationswithNeumanndata =0 : 8 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10with4%noise.........................107 IV.19Thereductioninerrorforexample3asfunctionofthenumberofiterationsfor thetwodierentfunctions.................................107 V.1Theexactfunctionofexuralrigiditycoecientforexample1.............117 V.2Thenumericallyrecoveredfunctionofexuralrigiditycoecientexample1......117 V.3Thecomparisonoftheactualfunctionwiththerecoveredfunctionalongthediagonal.forexample1.....................................118 V.4Convergenceoftherecoveredexuralrigiditycoecientalongthediagonalinexample1...........................................118 V.5Theexactfunctionofexuralrigiditycoecientforexample2.............119 V.6Thenumericallyrecoveredfunctionofexuralrigiditycoecientexample2......120 V.7Thecomparisonoftheactualfunctionwiththerecoveredfunctionalongthediagonalforexample2.....................................120 V.8Convergenceoftherecoveredexuralrigiditycoecientalongthediagonalinexample2...........................................121

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xvi V.9Theexactfunctionofexuralrigiditycoecientforexample3.............122 V.10Thenumericallyrecoveredfunctionofexuralrigiditycoecientexample3......123 V.11Thecomparisonoftheactualfunctionwiththerecoveredfunctionalongthediagonal.forexample3.....................................123 V.12Convergenceoftherecoveredexuralrigiditycoecientalongthediagonalinexample3...........................................124

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1 CHAPTERI INTRODUCTION ThisdissertationtitledComputationalMethodsforInverseEllipticProblemsresulted inthreepapersthathavebeenpublished. A.HamadandM.Tadi, "Anumericalmethodforinversesourceproblemsfor poissonandhelmholtzequations" ,PhysicsLettersA ,vol.380,no.44,pp. 3707-3716,2016. A.Hamad,Y.Wang,andM.Tadi, "Onidenticationofawavenumberfor Helmholtzequation" JournalofSoundandVibration ,vol.437,pp.447-456,2018. A.HamadandM.Tadi," Inversescatteringbasedonpropersolutionspace ", JournalofTheoreticalandComputationalAcoustics ,p.1850033,2018. I.1BackgroundforInverseProblems Historically,theproblemofthedeterminationoftheunknownfunctionsinvolvedin mathematicalmodelsdescribingvariousphysicalsystemsisreferredto inverseproblems .This identicationisbasedonexcitingthephysicalsystembyexternalinputandrecordingthe output.Initialworksinthiseldincludesinverseproblemsofquantumscattering[1],inverse problemsinseismology[2],andastronomy[3,4].FigureI.1showsacomparisonofatypical directforwardproblemwithatypicalinverseproblem.

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2 FIGUREI.1:Directproblemvs.inverseproblem I.2ClassicationofInverseProblemsBasedonTheUnknownFunctions Ingeneral,twoclassesofproblemsareoftendenotedasinverseproblems:inoneclass ofproblems,partoftheboundaryofthedomain,ortheinitialcondition,isunknown,andfora fulldescriptionofthesystem,theinverseproblemseekstorecovertheunknownboundaryorthe initialcondition.Thesecondclassistheidenticationofamaterialpropertyincludingthe distributionofthermalconductivity,absorptioncoecient,anddielectricconstant.Thisthesisis focusedonthesecondclassoftheinverseproblems.Forsimplicity,wewillusethefollowing exampletodescribetheclassicationofinverseproblemsbasedontheunknownfunction. Consideraboundeddomain 2 R 2 or R 3 withtheboundary)-278(= @ ,thepropagationof acousticwaveisgovernedby[4]: c )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 x u tt = 4 u )-222(r ln x : r u + h x ;t ; x 2 ; I.1 alongwiththeinitialandboundaryconditionsgivenby: u x ; 0= x ;;u t x ; 0= x ; x 2 @ ; I.2 @u @ = g x ;t ; x 2 @ ; I.3 where, c x isthespeedofsoundinthemedium, x isthedensityofthemedium, and h x ;t isthesourcefunction.Ifthefunctions c x , x ,and h x ;t areprovidedandthe

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3 boundary/initialconditionsaregiven,thenauniquesolutionfor u x ;t canbeobtained[5].In aninverseproblem,amaterialproperty,oraboundaryconditionisunknownandthegoalisto recovertheunknownbasedonadditionalmeasurementssuchas: u j )]TJ/F15 10.9091 Tf 8.817 1.688 Td [(= f x ;t x 2 @ : I.4 Inverseproblemsofmathematicalmodelscanbeclassiedintothefollowinggroupsdepending onthefunctionsthatareunknown[4]: Theinverseretrospectiveproblem: TheinverseproblemI.1-I.4issaidtoberetrospectiveifitisrequiredtodeterminethe initialconditions,i.e.,thefunctions x and x inI.2. Theinverseboundaryproblem: TheinverseproblemI.1-I.4iscalledaboundaryproblem,ifitisrequiredtodetermine thefunctionintheboundaryconditionthefunction g x;t . Theinversecontinuationproblem: TheinverseproblemI.1-I.4iscalledacontinuationproblemiftheinitialconditions I.2areunknownandtheadditionalinformationI.4andtheboundaryconditionsI.3 arespeciedonlyonacertainpart)]TJ/F34 7.9701 Tf 177.273 -1.636 Td [(1 )-333(oftheboundaryofthedomain.Itis requiredtondthesolution u x;t oftheequationI.1extendthesolutiontothe interiorofthedomain. Theinversesourceproblem: TheinverseproblemI.1-I.4iscalledasourceproblemifitisrequiredtodetermine thesource,i.e.,thefunction h x;t inEq.I.1. Theinversemediumproblem: TheinverseproblemI.1-I.4iscalledacoecientinverseproblemorinversemedium problemifitisrequiredtoreconstructthecoecients c x and x inthegovern

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4 equation. Itisimportanttomentionthatthisclassicationisstillincompletebecausetherearecaseswhere bothinitialandboundaryconditionsareunknown,andcaseswherethedomainorapartof itsboundaryisunknown[4]. I.3ExamplesforInverseProblems Wenextconsideranumberofspecicexamples. I.3.1TheGravimetryInverseProblem Thegravitationaleldinaboundeddomain 2 R 2 or R 3 isgivenbythePoisson equation[6]: u x = S x ; x 2 ; I.5 where, u x isthegravitationaleldandlim u x =0,as j x j! + 1 ; S x isthemass distributionwhichgeneratesthegravitationalforce 5 u .Thefunction S x canbeassumedtobe zerooutsideaboundeddomain.Thedirectproblemofgravimetryistondthegravitational eld u inthedomainforgiven S .Thedirectproblemisawell-posedproblem.Thesolution existsforanyintegrable S andisuniqueandstable.Thesolutionincase 2 R 2 canbeobtained bytheNewtonianpotential: u x = Z x;y S y dy; x;y = 1 j x )]TJ/F36 10.9091 Tf 10.909 0 Td [(y j : I.6 Thegravitationalforce 5 u canbemeasured.Thegoalofthe gravimetryinverseproblem isto recover S given 5 u onapartoftheboundary @ of.Ingeophysics,thisinverseproblemis fundamental,sinceitisusedtosimulaterecoveryoftheinterioroftheearthfromboundary measurementsofthegravitationaleld.Itiswell-knownthatsuchproblemishighlyill-posed andinaddition S isnotuniqueforagivengravitationalpotentialoutside[6].However,if

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5 additionalconditionson S areimposed,suchas S beingaharmonicfunction,then S canbe recovereduniquelyfrom u givenon @ [6]. I.3.2TheAdmittivityInverseProblem Consideraboundeddomain 2 R 2 ,or R 3 ,withacontinuousboundary.The forwardproblemistondtheelectricpotential u 2 suchthat: 5 x ;! 5 u =0 ; x 2 ; I.7 where, x ;! istheadmittivityproleingivenby x ;! = x ;! + i! x ;! ,where is theelectricconductivity, istheelectricpermittivity,and ! istheangularfrequencyofthe appliedcurrent.Auniquesolutionoftheforwardproblemcanbeobtainedbyprescribingthe Dirichletconditiongivenby: u x = ' x ; x 2 @ : I.8 Thefunction ' x 2 C 2 @ isthecurrentdensity.Then u 2 C 1 near @ ,sotheNeumanndata arewelldened: x ;! u x @ = x ; x 2 )]TJ/F36 10.9091 Tf 6.819 0 Td [(; I.9 where,)-334(isapartof @ 2 C 2 and istheoutwardunitvectorto @ . Thegoalofthe admittivityinverseproblem istond ,given ,foroneboundary measurement ' or,formanyboundarymeasurements ' .Insomecases ' isprescribedand is measuredon.Theadmittivityinverseproblemisusedasamathematicalbasisforelectrical impedancetomography[7,8,6]. I.3.3InverseScatteringProblem Generally,the inversescatteringproblem goalistodeterminethelocation,geometry,or materialproperty,fromtheknowledgeofthescatteredeld.The totaleld u isgivenasthesum

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6 ofthe incidenteld u i anda scatteredeld u s gureI.2, u = u i + u s .Theincidenteldis givenby:[9,10,11].Forsimplicity,consideraboundedregion 2 R 2 ,or R 3 ,withasmooth boundary @ thatisthescatteringobject. u i = exp ik ^ : x ; I.10 where k> 0denotesthewavenumberand ^ isaunitvectorthatdescribesthedirectionofthe incidentwave[12].ThetotaleldcanbedeterminedbysolvingtheHelmholtzequation[6]: 4 u x + k 2 u x =0 ; x 2 R n n ; I.11 withthehomogeneousDirichletboundarycondition[6]: u x =0 ; x 2 @ softobstacle ; I.12 orwiththeimpedanceboundarycondition[6]: @u @ + iku =0 ; x 2 @ hardobstacle : I.13 Thegoaloftheaboveinverseproblemistodeterminetheshapeofwhenthefareldpattern ismeasured. FIGUREI.2:Inversescatteringproblem[12]. I.3.4IdenticationofFlexuralRigidityInKirchho-LovePlates Flexuralrigidityisalsocalledthebendingstinessofaplate.Theforwardproblemfor aplateistonditsdeectionwhereaforceandbasicparameterssuchasthegeometryofthe

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7 structureandthemechanicalpropertiesofthematerialsusedareknown.Usually,itisimpossible tomeasurethechangeofthepropertiesofmaterialdirectly.Thesechangesofpropertiesof materialleadtochangeintheforwardsolution[13].Theproblemisill-posedinthesenseof stability.Theill-posednessleadstoanunrealisticinversesolutionofexuralrigiditywhenone treatstheproblemwithoutanyregularizationprocedure[14].Consideraboundeddomain 2 R 2 withcontinuousboundary @ .Let p x beareal-valuedfunctionin,andconsiderthe Kirchho-Loveequationwithboundaryconditionsgivenby: 4 p x 4 u x = S x ; x 2 : 4 p x 4 u x + p x 4 2 u x = S x ; x 2 ; I.14 where,theboundaryconditionsaregivenby: u x = D x ; x 2 @ ; I.15 4 u x = Q x ; x 2 @ : I.16 Theadditionalboundaryconditionfortheeldisgivenby: 5 u x = R x ; x 2 @ ; I.17 where, u x isthetransversedeectionofthebeam, p x istheexuralrigidity,whichisthe productofthemodulusofelasticity E andthemomentofinertia I ofthecross-sectionofthe beamaboutanaxisthroughitscentroidatrightanglestothecross-section.Thefunction S x representsthetransverselydistributedload[15].Theidenticationproblemistondthe unknowncoecient p x inEq.I.14fromthegivendatainEq.I.17. I.4PracticalExamplesofNon-InvasiveImagingTechniques Inthissectionweconsideranumberofpracticalexamplesforinverseproblems.

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8 I.4.1Vibrothermography ThehealthofastructurecanbeexaminedbyVibrothermographywhichisa nondestructivetestingtechnique.Vibrothermographymonitorstheheatproducedbydamage undervibrationand/orultrasonicexcitation[16].ConsiderthesampleshowningureI.3of thickness e ,innitein x and y directions.Itcontainsaninnerheatsource,modulatedat frequency f ,spreadoveranarea,inplane x =0,whichwewillcallplane,perpendicularto thesamplesurface z =0. Theheatsourcecausesariseinthetemperaturedistributioninthesample.Itis assumedthatthissamplehasnoheatlosesatthesurface.Itisalsoassumedthatthesurface temperaturedistributioni.e.,vibrothermographycanbemeasured.Thegoalistorecoverthe heatsourcedistribution[17].Thetemperaturedistributionisgivenbythefollowingmodied Helmholtzequationwithadiabaticboundaryconditions: 4 u f x )]TJ/F36 10.9091 Tf 10.91 0 Td [(q 2 f u f x = S x ; in 2 R 2 or R 3 ; I.18 andthesourcetermS x isgivenby: S x = Q f x K x ; 2 R 2 or R 3 I.19 where, u f x isthetemperatureatmodulationfrequency f ,and p 2 fD isthethermalwave vectoratfrequency f .Thefunction Q f x istheux,i.etheenergygeneratedperunittimeand unitareawithinplane.Thefunctions D and K arethethermaldiusivityandconductivityof thesample,respectively.Thefunction x istheDiracdeltafunctionforplane. I.4.2ElectricImpedanceTomography ElectricalimpedancetomographyEITcanbeconsideredasanonlinearill-posed inverseproblem[18,19].Thegoalofsolvingsuchaninverseproblemistoreconstructthe

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9 FIGUREI.3:"Geometryoftheproblem:aburiedheatsourceofareacontainedintheplane x =0,perpendiculartothesamplesurface z =0"[17]. electricalconductivitydistributionfromtheboundarypotentialsasaresultofinjectinga constantcurrentthroughthesurfaceelectrodes.Theboundarypotentialscanbecollectedbyan EITinstrumentationattachedtotheboundary[20]asitisshowningureI.4.Inelectric impedancetomographyEIT[21,22,23]oneseekstoprobetheinternalstructureofabiological system,i.e.,humanbody,byrecoveringitsconductivity/permittivitydistributionfromthe voltagemeasurementsontheboundary.FigureI.5illustratestheEITofchestimaging.The potentialequationisgivenby: 5 x 5 u =0 ; x in 2 R 2 or R 3 ; I.20 where,thegoalistorecoverthediusioncoecient x . I.4.3MicrowaveImaging Inmicrowaveimaging[26]thesystemisexcitedbytransversemodeTMof electromagneticwaveandtheresponseiscollectedattheboundaries.Theelectriceldandthe magneticeldareperpendiculartooneanother,andtheyaretransversetothedirectionof travel.FigureI.7showsthemechanismofmicrowaveimaging.Thebiologicaltissueis surroundedbyantennas.Whileamicrowavesignalistransmittedbyaselectedantenna,the signalsreceivedbyotherantennasarecollected.Thetransmittingantennasaresuccessively changedtoobtaindierentsetofdata.Thecollecteddataisusedtoreconstructtheinternal

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10 FIGUREI.4:"Schematicoftheconstantcurrentinjectionandtheboundarydatacollectionin electricalimpedancetomographyEITforacirculardomain"[24]. FIGUREI.5:"SchematicofEITimagingofahumanchest"[25]. structure[27].ThecomplexelectriceldisdescribedbythescalarHelmholtzequation: 4 u + p x u = g x ; in 2 R n ; n =2or3 ; I.21 where, g x isthesourcefunction.Theinterestistorecoverthecomplexwavenumber p x .The complexwavenumberisgivenby: p x = ! 2 0 0 h x i + x ! 0 i ; I.22

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11 where, istherelativepermittivity,and istherelativeconductivity. 0 and 0 arethe permittivityandconductivityofthebackground. FIGUREI.6:Schematicofbreastscreeningbymicrowaveimaging[28]. I.4.4Acousto-ElectromagneticTomography Acousto-electromagnetictomographyisahybridimagingmethodwherehighresolution andhighcontrastimagescanbeobtainedbycombiningtwodierenttechniques.Basically,the twodierenttechniquesmeantwodierentwavesarecombined.Thehighresolutionisgivenby oneofthemandthehighcontrastisgivenbytheothertechnique.FigureI.9showsthe principleofmagneto-acoustictomographywithmagneticinduction.Changingthe electromagneticpropertiesofthemediumcanbeachievedbyusingthehighresolutionof ultrasounds[30].Inacousto-electromagnetictomography[31,32],oneisalsofacedwithsolving aninverseproblemforascalarHelmholtzequationgivenby: 4 u + 2 + iq x u =0 ; in 2 R N N =2or3 : I.23

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12 FIGUREI.7:"ChestMicrowaveimaging"[29].

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13 TheRobinboundaryconditionisgivenby: : r u )]TJ/F36 10.9091 Tf 10.909 0 Td [(iu = g; on @ ; I.24 where, > 0istherealwavenumber, g istheboundarydata,nad istheoutwardunitvector normalto @ .Theinverseproblemofinterestistorecovertheabsorptioncoecient q x . FIGUREI.8:"PrincipleofMagneto-AcousticTomographywithMagneticInduction"[33]. I.4.5BioluminescenceTomography Inbioluminescencetomography[35,36,37],biologicaltestsubjectse.g.tumourcells aretaggedwithluciferaseenzymesandimplantedinasmallanimal.Thistechnologyprovidesa waytorevealcellularandmolecularfeaturesinbiologyanddiseaseinrealtime.Inessence,the goalofbioluminescencetomographyBLTistolocatethedistributionandquantitatively reconstructtheintensityoftheinternallightsourceusingthetransmittedandscattered bioluminescentsignalonthesurfaceofthesmallanimal.Oneapproachtomodelthetransferof

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14 FIGUREI.9:Acousto-electromagneticimagingdevice[34].

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15 lightinbiophotonicsistheRadiativeTransferEquationRTEapproximatedfromtheMaxwells Equations.Inmanyapplications,itissucienttoconsiderthediusionequationgivenby )-222(5 5 u x + a u x = S x ; x 2 ; I.25 where u x denotesthephotondensity, isthediusioncoecientand a representsabsorption coecients.Theterm S x isthesourcedistributionofgeneexpression.Theappropriate boundaryconditionisgivenby 5 u x )]TJ/F36 10.9091 Tf 10.909 0 Td [(u x =0 ; x 2 @ ; I.26 where, isacoecientrelatedtotheinternalreectionattheboundary.Inthisproblem,the parameters , a and areknown,andthebioluminescencetomographyBLTiscomposedof recoveringthesourcedistributionfromthecollecteddataattheboundary.Mathematically,BLT isthesourceinversionproblemthatrecovers S x fromopticalmeasurementonthedomain boundary @ .Inadditiontobeinghighlyill-posed,thesolutiontotheaboveBLTinverse problemisingeneralnotunique[37]. I.5Well-PosedandIll-PosedProblems Forsimplicity,wewillusethefollowingproblemtodiscusstheconceptofwell-posed andill-posedproblems: A x = y ; I.27 where A isanotnecessarilylinearoperatoractingfrom X into Y . X and Y aretwovector spacesover R . x 2 X isthesolutionfortheproblemI.27foragiven y 2 Y .Theequation I.27iscalledawell-posedprobleminthesenseofHadamardiftheoperator A hasacontinuous inversefrom Y onto X .ThethreeconditionsthatHadamardintroducedcanberewritten: Forany y 2 Y ,asolution x 2 X existsexistenceofasolution. Thesolution x isuniqueuniquenessofthesolution.

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16 Thesolutionisstablewithrespecttoperturbationsin y stabilityofasolution. ThestabilityconditioncanbeexplainedasthesolutionofequationI.27mustbestablewith respecttoperturbationontheright-handside.Ontheotherhand,theoperator A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 mustbe denedthroughoutthesubset Y andbecontinuous.Formostill-posedproblemstheinverse operator A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 isnotcontinuousinitsdomain A x Y [38,39,40,41].Generally,the approximatesolutionofanill-posedproblemdoesnotnecessarilydependcontinuouslyonthe measureddata.Also,thesolutioncanhaveaweaklinktothemeasureddata.Smallerrorsinthe measurementsleadtounacceptableperturbationsinthesolution[38,42].However,itispossible toconstructecientnumericalalgorithmsifthereisapriorinformationaboutthesolution,such assmoothness[12].TableI.1showsafewexamplesforill-posedandwell-posedproblems [6,42].Inthenextsection,someoftheseproblemsarediscussed. TABLEI.1:Afewexamplesforill-posedandwell-posedproblems[6]. Well-posedproblems Ill-posedproblems MultiplicationbyasmallnumberA Aq = f Divisionbyasmallnumber q = A )]TJ/F15 10.9091 Tf 7.085 -3.958 Td [(1 A 1 Multiplicationbyamatrix Aq = f q = A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 f A isanill-conditioneddegenerate orrectangular m n matrix Integration f x = f + R x 0 q d Dierentiation q x = f 0 x Ellipticequation 4 u =0 ;x u j )]TJ/F15 10.9091 Tf 8.817 1.689 Td [(= g or @u @n = f or u + @u @n j )]TJ/F15 10.9091 Tf 8.817 1.689 Td [(= h DirichletorNeumannproblem, Robinproblemmixed 4 u =0 ;x Cauchyproblem Initial-boundaryvalueproblem withdatagivenonapartoftheboundary )]TJ/F34 7.9701 Tf 6.819 -1.636 Td [(1 )-278(= @ Directproblems u t = 4 u )]TJ/F36 10.9091 Tf 10.91 0 Td [(q x u; u j t =0 =0 ;u j )]TJ/F15 10.9091 Tf 8.817 1.688 Td [(= f Coecientinverseproblems inversemediumproblem u t = 4 u )]TJ/F36 10.9091 Tf 10.909 0 Td [(q x u; u j t =0 =0 ;u j )]TJ/F15 10.9091 Tf 8.817 1.689 Td [(= f; @u @n = g Directproblems 5 q x 5 u =0 ;x 2 ; u j )]TJ/F15 10.9091 Tf 8.817 1.689 Td [(= g or @u @n = f Coecientinverseproblems 5 q x 5 u =0 ;x 2 ; u j )]TJ/F15 10.9091 Tf 8.817 1.689 Td [(= g , @u @n = f

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17 I.6ExamplesofIll-PosedProblems I.6.1SystemsofLinearEquations Consideralinearsystemofequationsgivenby: Ax = y ; I.28 where, A 2 R m n n;m 2 N isamatrix,and x 2 R n ,and y 2 R m .Assumethattherankof A isequaltomin m;n .If m>n ,thesystemmayhavenosolutions.If m
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18 haverelativelylargevariationsiftherearesmallvariationsin y unstablesolutions.Next,ifthe matrix A isdegeneratedet A =0and m = n ,thesystemI.28mayeitherhavenosolutionsor morethanonesolution.Ifthisisthecase,thesystemI.28isill-posedbasedonHadamard conditions[4,43]. I.6.2CauchyProblemforALaplaceEquation Considerthefollowingdirectproblem: 4 u =0 ;x> 0 ;y 2 [ )]TJ/F36 10.9091 Tf 8.485 0 Td [(l;l ] ; 0
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20 ill-conditionedmatrixequations.SingularvaluedecompositionSVDofamatrix A isgivenby: A = UV T ; I.43 where, U =[ u 1 ; u 2 ;:::; u m ]and V =[ v 1 ; v 2 ;:::; v n ]aretheleftandrightmatriceswhose columnareorthonormal.Thematrix =diag 1 ; 2 ;:::; n isadiagonalmatrixwith non-negativediagonalelementsinnon-increasingorder.Thediagonalelementsof arethe singularvaluesof A ,whicharethesquarerootofeigenvaluesof A T A .ThesolutiontoEq. I.42canbeobtainedasalinearcombinationoftherightandleftsingularvectors: x = k X i =1 u i y i v i ; I.44 where k istherankofthematrix A .Since A isanill-conditionedmatrix,therearemanysmall singularvalueswhichareclosetozero.Anapproximatesolutioncanbeobtainedaccordingto: [43,45,46]: x approx = N X i =1 u i y i v i ; I.45 where, N 2 N isthetruncationparameterdeterminedwhensmallsingularvaluesareleftout. Thesolutionforanill-posedproblembythetruncatedSVDregularizationmethoddependson thevalueof N .Ifthetruncationparameter N isequalto k ,noregularizationisapplied.Ifthe truncatedparameter N isequaltozero,thisleadstoneglectofallthesingularvalues,andthere isnoapproximationsolutionforthesystemI.42.ThetruncatedSVDregularizationwithtoo small N valueresultsamoreoscillatorysolution[47,25,48].Manyinvestigatorshavestudied severalapproachestochoosetheappropriatetruncatedSVDregularizationincludingthe generalizedcross-validationGCV[48,49],discrepancyprinciple[45,50],andL-curvecriterion [48,51,52]. I.7.2TikhonovRegularization Tikhonovregularizationiswidelyusedtostabilizevariousill-posedproblems.Inthis method,thesolutiontothelinearsystemgiveninEq.I.42isobtainedbyminimizingthecost

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21 functionalgivenby: min 8 x 2 X J = k Ax )]TJ/F25 10.9091 Tf 10.909 0 Td [(y k 2 + k x k 2 ; I.46 whereaboundonthesolutionisintroducedby > 0.Astablesolutionfor x 2 X canthenbe obtainedbyleast-squareminimization.Incaseofadierentialoperator,onecanconsiderthe minimizationproblemgivenby: min 8 x 2 X J = k Ax )]TJ/F25 10.9091 Tf 10.909 0 Td [(y k 2 + k x k 2 ; I.47 where oftenrepresentstherstorderderivative. I.8SomeExistingMethodsforInverseEllipticProblems Ifthegoverningequationforaninverseproblemisanellipticsystem,itisrefereedto anellipticinverseproblem.Itiswell-knownthatthisclassofproblemsarehighlyill-posed [6,53,54]andvariousmethodshavebeendevelopedtoovercomeit.Themethodsforsolvingthe inverseproblemscanbeiterativeornon-iterative.Thefundamentalsolutions[55]isameshless non-iterativemethod.Layer-stripping[56,57]methods,levelsetmethods[26,58,59,60],and theD-barmethod[25,61]arealsonon-iterative.Somenon-iterativemethodsarebasedon least-squareminimizationofacostfunctional[62,63,64].Sincecomputingpowerhasincreased, dierentapproachesfortheiterativemethodshavebeenintroducedincludingmethodswhichare basedonBornapproximations[65],theregularizedNewtonmethod[66],andmultipleforward solutionmethod[67].Additionaliterativemethodstreattheintegralformulationofthe scatteringproblem[68,69,70].Inrecentyears,methodsfortheidenticationofmaterial parametersincludingthedistributionofYoungsmodulusorexuralrigidityintheplateequation haveincreased.Forexamplein[71]authorsusetheso-calledMethodofVariationalImbedding MVIwhichisbasedonaproposedtechniquebyChristovin[72].Anotherexampleofexisting methodsisoptimizationmethodusingtheoutputleast-squareformulation[15].Next,ve methodsforinverseproblemsarediscussedandsomeissuesassociatedwiththesemethodsare

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22 mentioned. I.8.1OptimizationMethod Theoptimizationmethodsarewidelyusedtosolvemanyinverseproblems.Typically, oneseekstorecoveramodel x basedonobservationsofaeld u ,where u isrelatedto x bya forwardproblem.Theforwardprobleminpracticecanbewrittenas: A x u = y ; I.48 where, A 2 R n n isanonsingularmatrix,and y isaright-handsidethatconsistingofsource termsandboundaryconditionvalues.Thedataontheboundaryisgivenby: Bu = g ; I.49 where,thevector g containsthedata,andthematrix B representsthemeasurementoperator thatselectseldsatdatameasurementlocations.Asolutionto x canbeobtainedbyminimizing theerrordierences: min u ; x J = 1 2 k Bu )]TJ/F25 10.9091 Tf 10.909 0 Td [(g k 2 ; suchthat A x u = y : I.50 Sincetheinverseproblemgenerallyisill-posed,theregularizationisneededtostabilizethe solution.ThentheconstrainedoptimizationproblemwithTikhonovregularizationisgivenby: min u ; x J = 1 2 k Bu )]TJ/F25 10.9091 Tf 10.91 0 Td [(g k 2 + 1 2 k x k 2 ; I.51 SincethesystemI.48islinearin u ,thenonecanarriveat: u = A x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 y : I.52 Thisleadstotheunconstrainedoptimizationproblem: min x J = 1 2 k B A x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 y )]TJ/F25 10.9091 Tf 10.909 0 Td [(g k 2 + 1 2 k x k 2 : I.53

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23 ThisunconstrainedoptimizationproblemcanbesolvedbyGauss-Newtonmethod[73,74]. Theoptimizationmethodsworkwellformosthyperbolicsystems,butithaspoor performanceforellipticandparabolicsystems.Ingeneral,theremaybemanylocalminimums and,theconvergencecanalsobeverypoor. I.8.2Themethodoffundamentalsolutions Themethodoffundamentalsolutionsisameshlessmethodforsolvingforward problems.Itcanalsobeusedforsolvinginverseproblems.Thismethodisalsocombinedwitha regularizationmethod[45].Toexplainthismethod,Letbeanopenboundeddomainin R 2 and)-278(= @ beitsboundary.Thesteady-stateheatconductioninanisotropicmediumis describedbythePoissonequation: 4 u x = f x ; x 2 : I.54 Theheatux q x throughtheboundary @ isgivenby: q x = @u @ ; x 2 )]TJ/F36 10.9091 Tf 6.818 0 Td [(; I.55 where, istheoutwardunitvectornormalto @ .Thegoalistorecoverthesourceterm f x . Thetemperatureandheatuxcanbemeasuredonanaccessiblepartoftheboundary)]TJ/F34 7.9701 Tf 425.334 -1.636 Td [(1 . Themethodoffundamentalsolutionscanbeappliedasfollows[45]: ThefundamentalsolutionoftheLaplaceoperator 4 isgivenby: u x = )]TJ/F15 10.9091 Tf 12.986 7.381 Td [(1 2 ln r; x 2 R 2 ; I.56 where, r = jj x jj 2 istheEnclideannorm.Thefundamentalsolutiontothebiharmonic operator 4 2 isgiveby: u 4 2 = )]TJ/F15 10.9091 Tf 12.986 7.38 Td [(1 2 r 2 ln r x 2 R 2 : I.57

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24 Thesolutionofthepartialdierentialequationisapproximatedbyalinearcombination ofthefundamentalsolutions,namely: u x = n s X j =1 a j G j x + n s X j =1 b j H j x x 2 ; I.58 where, n s isthenumberofsourcepointslocatedonactitiousboundaryoutsidethe solutiondomainand a j ;b j areunknowncoecientstobedetermined. Thebasisfunctions G j x and H j x aredenedasfollows: G j x = u x )]TJ/F25 10.9091 Tf 10.909 0 Td [(y j ; I.59 H j x = u 4 2 x )]TJ/F25 10.9091 Tf 10.909 0 Td [(y j ; I.60 where, y j aresourcepointslocatedonactitiousboundaryoutsidethedomain.The approximatesolution u x automaticallysatisesthepartialdierentialequation. Theboundaryconditionssatisfyingcanbeachievedbychoosingasetofpoints f x i g on theaccessiblepartoftheboundary)]TJ/F34 7.9701 Tf 178.818 -1.636 Td [(1 . g x i = n s X j =1 a j G j x i + n s X j =1 b j H j x i ;i =1 ; 2 ;:::;n b ; I.61 h x i = n s X j =1 a j @ G j x i + n s X j =1 b j @ H j x i ;i =1 ; 2 ;:::;n b ; I.62 where, n b isthenumberofcollocationpointsontheaccessiblepartoftheboundary)]TJ/F34 7.9701 Tf 363.515 -1.637 Td [(1 and @ denotestakingthenormalderivative. Eq.I.61andEq.I.62canberewritteninmatrixformas: A = ; I.63 where =[ a 1 ;a 2 ;:::;a n s ;b 1 ;b 2 ;:::;b n s ] T istheunknowncoecientvectorand isthe datavector.

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25 ApplyingtheLaplaceoperator 4 totheapproximatesolution u x whichisthe approximationsolution f x totheheatsource: f x = n s X j =1 a j 4 G j x + n s X j =1 b j 4 H j x ; x 2 : I.64 Therearesomeissueswiththemethodoffundamentalsolutionsincludingthecomputational costwhicharisesfromtheuseofanonlinearleastsquareminimizationalgorithm.Itreliesonthe fundamentalsolutionoftheproblemunderconsiderationwhichcanbehardtoobtainforother ellipticproblems.Resultscanalsodependonthelocationofthecollocationpoints. I.8.3Layer-strippingmethod TheLayer-strippingmethodisanoniterativemethod.Theideaofthismethodisto startrstbyndingvaluesoftheelectricalconductivityontheboundary,andthenusethese computedvaluesoftheelectricalconductivityto"stripaway"theoutermostlayer.Themedium isstrippedaway,layer-by-layer,byrepeatingtheprocess[57].Toillustratethismethod,consider thereconstructionoftheelectricalconductivity,intheinteriorofabody.Weassumethatwecan applyanycurrentdensitytotheboundaryandmeasurethecorrespondingvoltageateverypoint ontheboundary.Thentheboundaryvalueproblemforthebody,whichisadiskofradius r , canbegivenby: 5 r; 5 u r; =0 ;r r 0 ; and0 2 ; I.65 r; @u r; @r = j; on @ ; I.66 where, u isthepotential, istheelectricconductivity,and j isthecurrentdensity. Onecancollectthesemeasurementsinordertoconstructamap R r 0 .Thismaptakes themeasuredcurrent j onthesurfacetotheboundaryvaluesofthepotential u .Theelectric impedancetomographyonlytheelectricalconductivityforsimplicityassociatedwiththeabove modelistorecovertheconductivityprole insidethedomain.Thelayer-strippingmethod canbesummarizedinstepsasfollows:

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26 Makemeasurementsontheboundaryofthebodyandconstructthemap R r 0 .Then foranygiven r and wehave: R r r; @u r; @r = u r; : I.67 Findtheelectricalconductivity ontheboundaryat r 0 . R canbepropagatedintotheinteriorby[56,75]: R r 0 )-222(4 r = R r 0 )-222(4 r @ R @r ; I.68 where, @ R @r = 1 + R r + R r 2 @ @ @ R @ : I.69 Replace r 0 with r 0 )-222(4 r andrepeatstep. Thelayer-strippingmethodhassomeadvantagesovertheoptimizationmethods.Themethod canbeusedtosolvethefullnonlinearproblem.Thelayer-strippingmethodcanbemodiedfor dierentgeometries.Thecomputationalcostandthestoragerequirementsarelessthanthe optimizationmethod.However,thelayer-stripmethodhassomeissues.Oneofthemisthefact thatmorerealisticmodelsoftheelectrode-bodyinterfacecannotbesimplyincorporatedinto thealgorithm.Theotherissueiswhilemarchinginsidethedomaintheerrorsaccumulate [57,56,75,76]. I.8.4Levelsetmethod ThelevelsetmethodwasintroducedbyOsherandSethian[59,77]forproblems involvingthemotionofcurvesandsurfaces.Considerthefollowinginverseproblemwherethe unknownfunctionistheboundaryofaregion D in R 2 or R 3 : A q = g ; I.70

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27 with, q D x = 8 > > > > > > > < > > > > > > > : q int interiorfor x 2 D 0for x 2 @D q ext exteriorfor x 62 D ; I.71 where,thevector g containsthedata, q isthemodelparametersthedesired unknown,andtheoperator A isamapfrommodeltothedata.Inthismethodasequenceof functionsdenotedby k x isgeneratedsuchthat[58]: D k ! D;@D k = f x : x =0 g ; I.72 where, k maybeacontinuousparameterrepresentingtime. Theunknownfunction x canbeintroducedby: q D x = q int ; for f x : x < 0 g ; q D x = q ext ; for f x : x > 0 g : I.73 ThentheoutcomeoftheinverseproblemgivenbyI.70-I.73isnowtondthesetof unknowncharacteristicfunctionsbasedonmeasureddata.Therearedierentapproachesto obtaintheseunknownfunctions x includingoptimizationapproachesandevolution approaches[58,59,60].FigureI.10showsthesetoffunctionsforanunknownshape. FIGUREI.10:Thesetoffunctionsforunknownshape.

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28 I.8.5D-barmethod Nachman'suniquenessproofforthe2-Dinverseconductivityproblemoutlinesadirect procedureforreconstructingtheconductivityprolefromknowledgeoftheDirichlet-to-Neumann map[25,61].ConsidertheconductivityinverseproblemwhichisgivenbyEq.I.7-I.9 5 x ;! 5 u =0 ; x 2 R 2 ; I.74 theDirichletboundaryconditionontheboundaryisgivenby: u x = ' x ; x 2 @ : I.75 Thefunction ' x isaknownvoltage.Thegoalistorecovertheconductivity bymeasuring thecurrentdensity ontheboundary: x ;! u x @ = x ; x 2 @ : I.76 IntheD-barmethod,therearethreeintermediatesteps.Thefunction: q z = z )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 = 2 4 z 1 = 2 ;z = x + iy; 2 C 2 ;> 0 ; I.77 isusedtotransformEq.I.74intotheSchrodingerequation: )-222(4 + q ~ u =0in ;u = )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 = 2 ~ u;> 0 : I.78 Theconductivity isassumedtobeconstantinaneighborhoodof @ .Thisleadsto q tobe zeroinaneighborhoodoftheboundary.Thenonecansmoothlyextend =1and q =0outside . Thefunction z;k , k 2 C n 0,iscalledcomplexgeometricalopticssolutionsCGOs. Thefunction z;k areexponentialgrowingsolutionsforEq.I.78,andtheyareasymptoticto

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29 exp ikz .ThesolutionsCGOshavethemostsignicantroleinthismethod.Thesesolutions areusedtocomputethescatteringtransformfromtheboundarydata[25,61,78].Theboundary conditioncanbesatisedbythefollowingequation[79,61]: z;k j @ = exp )]TJ/F36 10.9091 Tf 8.485 0 Td [(ikz )-222(S )]TJ/F15 10.9091 Tf 10.909 0 Td [( 1 z;k ; I.79 where, istheDirichlet-to-NeumannDNmaptakingthevoltagetocurrentdensity,and 1 is theDirichlet-to-NeumannmapDNtakingthevoltagetocurrentdensitywhen =1.The operator S isasingle-layeroperator[79,61].Sincetheoperator I + S )]TJ/F15 10.9091 Tf 10.909 0 Td [( 1 isinvertible [79,61],thenthefunctions z;k ontheboundarycanbeobtained.Thescatteringtransform denotedby t k isthethirdintermediatefunctioninthismethod.Thescatteringtransformisa complex-valuedfunction.ScatteringtransformcanalsobedescribedasanonlinearFourier transformoftheSchrodingerpotential q givenby[61]: t k = Z R 2 exp i kz + k z z;k q z d x ; exp )]TJ/F36 10.9091 Tf 8.485 0 Td [(ikz z;k : I.80 Thescatteringtransformcanbeobtainedfromthedataontheboundarybythefollowing integralequation[61,79,80]: t k = Z @ exp k z )]TJ/F15 10.9091 Tf 10.909 0 Td [( 1 z;k d z ; I.81 where, d z isthemeasureofarclengthon @ . ByusingthecomputedscatteringtransformonecanarrivetotheD-barequation: @ @ k z;k = t k 4 k exp )]TJ/F36 10.9091 Tf 8.485 0 Td [(i kz + k z z;k ; @ @ k = )]TJ/F36 10.9091 Tf 11.402 -1.457 Td [(@ @k 1 + i @ @k 2 I.82 TheD-barequationhasauniquesolutionwithlarge j k j asymptotics x ;k 1[79]. Thelaststep,theconductivityisrecoveredby[25,61]: p z =lim k ! 0 z;k ; I.83 becausetakingthelimitof exp )]TJ/F36 10.9091 Tf 8.485 0 Td [(ikz ~ u when k ! 0leadstothethesolutionofthe SchrodingerequationI.78whichisunique,and p satisesEq.I.78.Weremarkthatthe methodisnotanumericaltechnique,butratheramethodofsolutionforcertaininverse boundaryvalueproblems.

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30 I.8.6Aniterativemethodbasedonmultipleforwardsolutions Consideraclosedboundeddomain 2 R 2 anda2-DHelmholtzequationgivenby[67]: 4 u + k 2 g x u =0 ; x 2 ; I.84 whereDirichletdataisgivenattheboundaryaccordingto: u x = ' x ; x 2 @ : I.85 Thevariable u x denotestheelectriceld,theparameter k denotesthefrequencyofthe incidentwaveandthefunction g x isaphysicalparameter.Thegoalistorecoverthefunction g x basedonboundarymeasurementsgivenby: @u @n = x ; x 2 @ : I.86 Thismethodconsistsofthefollowingsteps: UsingtheinitialguessandthegivenDirichletconditionsgenerateabackgroundeld: 4 ^ u + k 2 ^ g x ^ u =0 ; x 2 ; ^ u x = ' x ; x 2 @ : I.87 Theerroreldgivenby e x = u x )]TJ/F15 10.9091 Tf 11.607 0 Td [(^ u x ,andcanbeobtainedbysubtractingthe equationI.84fromI.87: 4 e + k 2 g x ^ u )]TJ/F36 10.9091 Tf 10.909 0 Td [(k 2 ^ g x ^ u =0 ; x 2 ; e x =0 ; x 2 @ : I.88 Theactualfunctionisrelatedtotheassumedvalueaccordingto g x =^ g x + x . Afterlinearizing,theerroreldisgivenintermoftheunknowncorrectionterm x accordingto: 4 e + k 2 ^ g x e + k 2 x ^ u =0 ; x 2 ; e x =0 ; x 2 @ : I.89

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31 Considertwowell-posedproblemsfortheerroreldasshowninguresI.11andI.12. Obtaintwoequationsforthetwounknowns x and e x .Eliminate e x andobtain oneequationfortheunknown x . Updatetheassumedvalueandrepeatstepstoaccordingto g x =^ g x + x [67]. I.9Organizationofthedissertation Therestofthisdissertationisorganizedasfollows:ChapterIIisconcernedwithan iterativealgorithmforinverseevaluationofthesourcefunctionfortwoellipticsystems.We consider2-Daswellas3-Ddomains.InchapterIIIandchapterIV,weapplyasecondmethod totwoinverseellipticproblemsforaHelmholtzequation.Itassumesaninitialvalueforthe unknownfunctionandobtainscorrectionstotheassumedvalue.SectionIII.3andsection IV.3presenttheupdatingformulationbasedonthe propersolutionspace .Weconsiderthe evaluationofrealandcomplexgeneralnon-separablewavenumbers.Anumberofnumerical examplesareusedtostudytheapplicabilityandeectivenessoftheproceduresandtoshowthe presentedmethodcanalsobeappliedtovariousdomainsusingexistingmappingtechniques.In chapterVthemethodbasedonthepropersolutionspaceissuccessfullyappliedtoafourthorder ellipticinverseproblem.Thegoalofthisinverseproblemistoidentifytheexuralrigidity functionforaplate.TheplategoverningequationisgivenbyKirchho-Loveplatesequation.

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32 4 e + k 2 ^ ge + k 2 ^ u =0 ProblemI e =0 e =0 e =0 e =0 FIGUREI.11:Therstellipticproblem. 4 e + k 2 ^ ge + k 2 ^ u =0 ProblemII e =0 e =0 e x = q y e =0 FIGUREI.12:Thesecondellipticproblem.

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33 CHAPTERII INVERSESOURCEPROBLEMFORPOISSONANDHELMHOLTZ EQUATIONS Abstract [81] Thispaperisconcernedwithaniterativealgorithmforinverseevaluationofthesourcefunction fortwoellipticsystems.Thealgorithmstartswithaninitialguessfortheunknownsource function,obtainsabackgroundeldand,obtainstheworkingequationsfortheerroreld.The correctiontotheassumedvalueappearsasasourcetermfortheerroreld.Itformulatestwo well-posedproblemsfortheerroreldwhichmakesitpossibletoobtainthecorrectionterm.The algorithmcanalsorecoverthesourcefunctionwithpartialdataattheboundary.We consider2-Daswellas3-Ddomains.ThemethodcanbeappliedtobothPoissonandHelmholtz operators.Numericalresultsindicatethatthealgorithmcanrecovercloseestimatesofthe unknownsourcefunctionsbasedonmeasurementscollectedattheboundary. II.1Introduction Inthischapterwedevelopanumericalmethodtorecoverthesourcefunctionforan ellipticsystemintwoandthreedimensions.Considertheinverseproblemofidentifyingasource function, S x ,foranellipticsystemgivenby: 4 u x = S x in : II.1 TheproposedmethodcanalsobeappliedtoaninversesourceproblemforaHelmholtzoperator givenby: 4 u x + $ 2 u x = S x in ;

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34 where $ isthewavenumber.Theboundaryofthedomain,i.e., @ ,isaccessibleandcanbe usedtocollectmeasurements.Itiswellknownthatsuchproblemsarehighly ill-posed [82].A numberofinvestigatorshavedevelopedvariousmethodstodealwiththeinherentill-posednessof suchproblems[seesectionsI.7andI.8].Amajordicultywiththeinversesourceproblems forPoissonandHelmholtzequationsisthatthesolutionsarenot unique [83].Itispossibleto obtainuniquesolutionsifonecanintroduceadditionalrequirementsontheunknownfunction. Oneoftheearlycomputationalmethodsthatwasspecicallydevisedfortheinversesource problemforPoissonequationisthe minimumsupport functional[84].Inthismethod,the algorithmseeksasourcefunctionwithminimumvolumecompactnesstorestrictthesolutionof theinverseproblem.Inotherwords,thesourceiszeroeverywhereexceptwithinanunknown domainsupportandtheminimumsupportfunctionalseekstominimizethevolumeenclosedby thefunctionandthebase.Anaprioriestimateofthisvolumewasusedtoensurethe convergenceofthealgorithmtoanonzerofunction.Thecostfunctionalwaslateronmodiedto minimizetheareaoverwhichthesourcefunctionhasanonzerogradient,alsoknownas minimumgradientsupportfunctional [85].Anotherapproachtodealwiththenon-uniquenessof thisproblemistosimplyrestricttheformofthesourcefunction.In[86],theauthorsdeveloped adirectinversionalgorithmtoidentifyaone-point-mass-likesourceusingboundaryelement approach.In[87],itwasshownthatasourcefunctioncanbeuniquelyrecoveredifsomeapriori informationisgiven.Itwasshownthatauniquesourcefunctioncanbeobtained,1if separationofvariablesarepossibleandonefactoroftheproductisknown;2ifthedomainand thesourceisinacylindricalgeometry.InanothercomputationaleortforthePoissonequation [88]thesourcefunctionisassumedtobezeroexceptwithinanunknowndomainwithinwhich it'svalueisequaltoone.Inotherwordstheunknownsourcefunctionisacharacteristicfunction givenby: X ! x = 8 > > > < > > > : 1if x 2 ! 0otherwise ;

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35 onadomain ! .Infactuniquenessresultsforsourcefunctionswithstarshaped characteristicfunctionshavebeenpresentedin[6].Inanotherformulation[45],theauthors assumethatthesourcefunctionsatisesthehomogeneousLaplaceequation,i.e.,thesourceis harmonicin.Itisoftenthecasethattheunknownsourcefunctioniscomposedofauniform backgroundandasmallperturbationwithcompactsupport[89],Eq..Thisisthecasefor thesourcefunctionsthatareunderstudyinthischapter.Suchasourcefunctionappearsin electromagneticsourceimaging[90],andinparticularinnon-destructiveevaluationofmaterials usinglock-invibrothermography[17],Eq..Theellipticoperatorstudiedin[17]isthe modiedHelmholtzequationwithasimilargeometrythatisunderstudyinthischaptergure II.1.Inthischapterweconsidersourcefunctionscomposedofaknownuniformbackgroundand anunknownperturbationthatisseparable.Thistypeofsourcefunctionissimilartothe unknownfunctionin[89],exceptthattheunknownperturbationpartisaseparablefunction. Thevalueoftheknownuniformbackgroundcanbezero,similartothesourcefunctionsina numberofreferencesmentionedforinstance,[17,6].However,herewecanhaveanonzerovalue, simplybecauseourmethodtreatstheequationfortheerroreld.Sincethebackgroundeldis assumedtobeknownthen,theunknownsourcethecorrectionfortheerroreldhasazero backgroundvalue.Inthischapter,weassumethattheunknownpartofthesourcefunctionis separable ,whichisappropriateforsomeperturbationswithsmallcompactsupport.Thepresent algorithmisiterativeinnatureandiscomposedofthreesteps: Assumeavalueoftentheuniformvalueofthesourceforthesought-aftersource functionandusetheDirichletconditionstoobtainabackgroundeld. UsethecollecteddataandtheknowledgeoftheDirichletboundaryconditionandobtain workingequationfortheerroreld.Thisleadstoanerroreldthatis over-specied and thereforeis ill-posed .Inthischapter,weintroducenewmethodstoobtainsolutionsto theerroreld.Inessence,themethodconsiderstwowell-posedformulationsoftheerror eld.Itisthenpossibletoobtaincorrectionstotheassumedvaluesofthesource

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36 4 e =^ g x % y e =0 e =0 e =0, e x = & w y e =0 ;e x = & e y FIGUREII.1:Erroreldwithover-speciedboundaryconditions. functioninstep. Updatetheassumedvaluesandrepeatstepsand. InsectionII.2,wedevelopthemethodfora2-Ddomain.InsectionII.3,weconsidera3-D domain,anddevelopaslightlydierentapproachtoachievestepoftheabovealgorithm.In sectionII.4,weuseanumberof2-Dand3-Dexamplestostudytheapplicabilityofthe method. II.2Inversesourceproblemin2-D Consideraboundeddomainin R 2 ,withacontinuousboundary @ .Let S x =1+ g x f y representsthesourcefunctionin.Foraninversesourceproblem,thegoal istorecoverthesourcetermforanellipticsystemgivenby: u xx + u yy =1+ g x f y in ; II.2

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37 whereaunitsquareandthevariable u x canbetheelectricpotentialorthematerial temperature.Theboundaryisaccessible,anditispossibletomeasurebothuxorNeumann boundaryandDirichletconditionsontheboundary @ .Startingwithanassumedvalueforthe sourceterm1+^ g x ^ f y andusingtheDirichletcondition,itispossibletoobtainabackground eld,^ u x ,givenby: ^ u xx +^ u yy =1+^ g x ^ f y in : II.3 SubtractingEq.II.3fromEq.II.2leadstotheerroreld, e x ,givenby: e xx + e yy = g x f y )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g x ^ f y in ; II.4 where, e x = u x )]TJ/F15 10.9091 Tf 11.607 0 Td [(^ u x .Theactualsourcefunctionarerelatedtotheassumedvalues accordingto: g x =^ g x + x ;f y = ^ f y + % y ; II.5 wherenow,thecorrectionterms x and % y areunknowns.Itispossibletorecoverthese correctionsonetermatatimeaccordingtothefollowing.Thisisthestepofthealgorithm outlinedinsectionII.1. II.2.1Updateinydirection Firstassumethat g x ^ g x then,theaboveequationfortheerroreldsimpliesto: e xx + e yy =^ g x % y in : II.6 ThebackgroundeldsatisestheDirichletcondition,andtherefore e =0ontheboundary @ . ThedataisintermsofuxattheboundariesasshowngureII.1. Inordertorecoverthecorrectioninthesourceterm % y itispossibletodividethe domainintheydirectioninto n e equalininterval,i.e., y ,andusethecentraldierencinginthe ydirectionaccordingto: e xx + 1 y 2 e j )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 )]TJ/F15 10.9091 Tf 10.909 0 Td [(2 e j + e j +1 =^ g x % y j =^ g x % j : II.7

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38 Itispossibletowritetheabovesysteminavectorformgivenby: d 2 e x dx 2 + 1 y 2 2 6 6 6 6 6 6 6 6 6 6 6 6 4 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2100 ::: 1 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2100 ::::::::::::::: ::: 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(21 :::::: 01 )]TJ/F15 10.9091 Tf 8.485 0 Td [(2 3 7 7 7 7 7 7 7 7 7 7 7 7 5 e =^ g x d ; II.8 where, e x T =[ e 2 x ;e 3 x ;:::;e n e x ] T 2 R n )]TJ/F34 7.9701 Tf 6.586 0 Td [(2 and d T =[ % 2 y ;% 3 y ;:::;% ne y ] T 2 R n )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 are theunknownerroreldandtheunknownsourceterm,respectively.Notethatdividingthe domain y =[0:1]into n e equalelementsleadsto n = n e +1nodes.However,duetothezero boundaryconditionsfortheerroreld e 1 = e n =0,thereareonly n )]TJ/F15 10.9091 Tf 10.909 0 Td [(2interiornodesatwhich theunknownsourcefunctions.Also,itisreasonabletoassumethattheunknownsource functionsareknownattheboundaryandasaresultitisreasonabletoassumethatthe correctionsarealsozeroattheboundaries.Itisduetothezeroboundaryconditionfortheerror eldthatthecoecientmatrixinEqII.8. Thedataiscollectedattheboundariesandasaresulttheabovesystemisrequiredto satisfytheboundaryconditionsgivenby: e x j x =0 =0 ; d e x dx j x =0 = w y j ; e x j x =1 =0 ; d e x dx j x =1 = e y j : II.9 SincethematrixinEq.II.8issymmetric,itispossibletodecoupletheabovesystemusingthe matrixofeigenvectorsaccordingto e = v where 2 R n )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 n )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 istheorthogonalmatrixof theeigenvectorsofthematrixinEq.II.8with T = I .Theabovesystemsimpliesto: d 2 v i x dx 2 + i v i x =^ g x c i ; II.10 where, v T =[ v 2 x ;v 3 x ;:::;v n e x ] T , i aretheeigenvaluesofthematrixinEq.II.8and

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39 c = T d .Inaddition,theboundaryconditionsinthemodalspacearegivenby: v x j x =0 = v x j x =1 =0 ; d v x dx j x =0 = T d e x dx j x =0 = l; d v x dx j x =1 = T d e x dx j x =1 = ^ l; II.11 where,wearealsodeningnewvariables l and ^ l forsimplicity.Theabovesystemisnowfully decoupled.Everyequationinthesystemisasecondorderdierentialequationthatisrequired tosatisfytwoboundaryconditionsateachend.Eachequationalsoincludesanunknown constant c i .Therefore,wearefacedwithasystemofill-posedlinearboundaryvalueproblem. Aftersolvingtheaboveequationsandobtainingthevaluesofunknownconstants c i ,the correctiontotheassumedvalueofthesourcetermcanbecomputedusingthetransformation d = c . Itispossibletoapplythemethodusedin[67]toobtainsolutionstotheabove boundaryvalueproblems.Theapproachistoconsidertwo well-posed boundaryvalueproblems andnotethattheydescribethesamesolution.Inotherwords,insteadofEqs.II.10andII.11, weconsidertwowell-posedproblemsgivenby: d 2 v i x dx 2 + i v i x =^ g x c i ; v i x j x =0 =0 ; dv i x dx j x =1 = ^ l i ProblemI ; II.12 d 2 v i x dx 2 + i v i x =^ g x c i ; dv i x dx j x =0 = l i ;v i x j x =1 =0 ; ProblemII ; II.13 for i =1 ; 2 ;:::; n )]TJ/F15 10.9091 Tf 10.909 0 Td [(2.Notethatthereare n )]TJ/F15 10.9091 Tf 10.909 0 Td [(2decoupleequations.Bothoftheabove problemsarewell-posed,andonecanproceedtoobtainsolutionsusingnitedierence approximations.Ifonedividesthe x domaininto n e equalintervalsandusecentraldierencing

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40 toapproximatethesecondderivativetheyleadto: 2 6 6 6 6 6 6 6 6 4 10 :::::: 1 x 2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 x 2 + i 1 x 2 ::: :::::::::::: ::: 1 2 x )]TJ/F34 7.9701 Tf 6.586 0 Td [(4 2 x 3 2 x 3 7 7 7 7 7 7 7 7 5 | {z } A 1 2 6 6 6 6 6 6 6 6 4 v 1 i v 2 i ::: v n i 3 7 7 7 7 7 7 7 7 5 | {z } w = 2 6 6 6 6 6 6 6 6 4 0 0 ::: ^ l i 3 7 7 7 7 7 7 7 7 5 | {z } b 1 + 2 6 6 6 6 6 6 6 6 6 6 6 6 4 0 ^ g x 2 ::: ^ g x n e 0 3 7 7 7 7 7 7 7 7 7 7 7 7 5 | {z } ^ g c i : ProblemI ; II.14 where,forsimplicity,wearealsodeningnewvariablesaccordingto w T =[ v 1 i ;v 2 i ;:::;v n i ] T and b T 1 =[0 ; 0 ; 0 ;:::; ^ l i ] T ,and g T =[0 ; ^ g x 2 ;:::; ^ g x n e ; 0] T ,andsimilarly, A 1 .Notethatthelastrow equationenforcestheslopeconditionat x =1.Similarly,anitedierenceapproximationto thesecondproblemisgivenby: 2 6 6 6 6 6 6 6 6 4 )]TJ/F34 7.9701 Tf 6.586 0 Td [(3 2 x 4 2 x )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 2 x 0 1 x 2 )]TJ/F34 7.9701 Tf 6.587 0 Td [(2 x 2 + i 1 x 2 ::: :::::::::::: :::::: 01 3 7 7 7 7 7 7 7 7 5 | {z } A 2 2 6 6 6 6 6 6 6 6 4 v 1 i v 2 i ::: v n i 3 7 7 7 7 7 7 7 7 5 | {z } w = 2 6 6 6 6 6 6 6 6 4 l i 0 ::: 0 3 7 7 7 7 7 7 7 7 5 | {z } b 2 + 2 6 6 6 6 6 6 6 6 6 6 6 6 4 0 ^ g x 2 ::: ^ g x n e 0 3 7 7 7 7 7 7 7 7 7 7 7 7 5 | {z } ^ g c i : ProblemII ; II.15 where,forsimplicity,weareagaindeningnewvariablesaccordingto b T 2 =[ l i ; 0 ;:::; 0] T ,and, similarly, A 2 .Theabovetwoproblemsarewell-posedandthecoecientmatricescanbestably inverted.Eliminatingtheunknownvector, w ,fromtheabovetwoequationsleadsto A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 b 1 + A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 ^ g c i = A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 2 b 2 + A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 2 ^ g c i ; or A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 1 b 1 )]TJ/F25 10.9091 Tf 10.909 0 Td [(A )]TJ/F34 7.9701 Tf 6.586 0 Td [(1 2 b 2 | {z } r ; = A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 2 )]TJ/F25 10.9091 Tf 10.909 0 Td [(A )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 1 | {z } s ^ g c i ; II.16 where,wearealsodeningnewvectors r and s forsimplicityandspaceconsiderations.Itisnow possibletosolvefortheunknownscalaraccordingto: c i = s T r s T s ; II.17

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41 where,thepositivescalar isused0 < andissetbythedesigner.Wewillexplainthis parameterinmoredetails.Oncethescalars c i ;i =1 ; 2 ;:::; n )]TJ/F15 10.9091 Tf 10.909 0 Td [(2arecomputed,thenthe correctiontotheassumedvalueofthesourcetermcanbesolvedforusingthetransformation d = c .Wecanapplyasimilarproceduretoupdatetheassumedvalueofthesourcetermin the x direction. II.2.2Updateinxdirections Afterupdatingthesourcetermintheydirection,itispossibletoassumethat f y = ^ f y .ThenequationII.4fortheerroreldsimpliesto: e xx + e yy = x ^ f y in : II.18 ThebackgroundeldsatisestheDirichletcondition,andthereforeattheboundary e =0.The dataisintermsofuxattheboundariesasshowningureII.2. Thesameprocedurecanbeusedtorecoverthecorrectioninthe x direction. II.2.3Recoverywithpartialdata Itispossibletoapplythepresentmethodwhenthedataiscollectedatonlytwosides ofthedomain.ConsiderthesameinverseproblempresentedinEq.II.2andassumethatthe dataiscollectedatonlytwosidesofthedomain.Followingsimilarstepsandobtainingtheerror eldleadsto: e xx + e yy = g x f y )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g x ^ f y ; in ;e =0on @ ; II.19 inadditionto: e x ;y = y ;e y x; 0= x : II.20 Forupdatingin y direction,onecanassumethat g x ^ g x andarriveatthesystemgivenin gureII.3.

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42 4 e = ^ f y x e =0, e x = & n x e =0, e x = & s x e =0 e =0 FIGUREII.2:Erroreldwithover-speciedboundaryconditions. 4 e =^ g x % y e =0 e =0, e y = s x e =0 e x = w y e =0 FIGUREII.3:Erroreldwithpartialdataattheboundary.

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43 Afterdiscretizingthesecondderivativeandusingtheeigenvectorstodecouplethe workingequationsonecanarriveatsimilarequationsgivenbyEq.II.10withboundary conditionsgivenby: v x j x =0 = v x j x =1 =0 ; d v x dx j x =0 = T d e x dx j x =0 = t : II.21 SimilartoEqs.II.12andII.13,itispossibletoformulatetwowell-posedproblemsforthe erroreldwithpartialdataaccordingto: d 2 v i x dx 2 + i v i x =^ g x c i ; v i x j x =0 =0 ;v i x j x =1 =0P.I ; II.22 d 2 v i x dx 2 + i v i x =^ g x c i ; dv i x dx j x =0 = t i ;v i x j x =1 =0 ; P.II : II.23 Bothoftheaboveproblemsarewell-posedandonecanfollowthesameformulationandobtain theunknownconstant c i .Thesameprocedurecanbeusedtorecoverthecorrectioninthe x direction.Wenextproceedtoconsiderinversesourceproblemin3-D. II.3Inversesourceproblemin3-D Consideraboundeddomainin R 3 ,withacontinuousboundary @ .Let S x =1+ g x f y h z representthesourcefunctioninwiththenon-uniformportionbeing separable.Thegoalistorecoverthesourcetermforanellipticsystemgivenby: u xx + u yy + u zz =1+ g x f y h z in ; II.24 wherenowisaunitcube.Theboundaryisaccessibleanditispossibletomeasureboth NeumannandDirichletboundaryconditionsontheboundary @ .Startingwithanassumed valueforthesourceterm g x f y h z andusingtheDirichletcondition,itispossibletoobtain abackgroundeldgivenby: ^ u xx +^ u yy +^ u zz =1+^ g x ^ f y ^ h z in ; II.25

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44 SubtractingEq.II.25fromII.24leadstotheerroreldgivenby: e xx + e yy + e zz = g x f y h z )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g x ^ f y ^ h z in ; II.26 forwhichwehavebothDirichletandNeumannboundaryconditions.Theactualsource functionsarerelatedtotheassumedvaluesaccordingto: g x =^ g x + x ; f y = ^ f y + % y ; h z = ^ h z + z : II.27 Similartotheproblemin2-D,itispossibletorecoverthecorrectionstotheassumedvaluesone termatatime.Wecanproceedasfollows. II.3.1Updateinzdirection Wecanassumethat g x ^ g x and f y ^ f y .Then,theequationfortheerror simpliesto: e xx + e yy + e zz =^ g x ^ f y z in : II.28 Since e x;y;z =0attheboundaries x 2 @ ,itispossibletoassumeanexpansionoftheform givenby: e x;y;z = 1 X l =1 A l x;y sin lz ; A l x;y =2 Z 1 0 e x;y;z sin lz dz; II.29 wherethesuperscript l simplydenotesthecomponentinthefunctionexpansion.Byselecting sin : functionstheDirichletboundaryconditionsinthe z directionaresatised.Substituting theaboveexpansionintheerrorequationleadsto: 1 X l =1 [ A l xx + A l yy )]TJ/F15 10.9091 Tf 10.909 0 Td [( l 2 A l ] sin lz =^ g x ^ f y z in : II.30

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45 Usingtheorthogonalitypropertyofthesinefunctions,onecanarriveatthefollowingequation foreachcomponent: A l xx + A l yy )]TJ/F15 10.9091 Tf 10.909 0 Td [( l 2 A l =2^ g x ^ f y Z 1 0 z sin lz dz in ; II.31 for l =1 ; 2 ;:::; 1 .Retainingtherst L componentsleadsto: A l xx + A l yy )]TJ/F15 10.9091 Tf 10.909 0 Td [( l 2 A l =^ g x ^ f y l ; l =2 Z 1 0 z sin lz dz; in ; II.32 for l =1 ; 2 ;::;L .ThecollecteddatacanprovideNeumannboundaryconditionsfortheabove equationswhichareoftencalledmodiedHelmholtzequation.Theaboveequationsarestill ill-posedinthesensethatevery A l x;y isrequiredtosatisfybothDirichletzeroandNeumann conditions.Inaddition,everyequationincludesanunknownscalar l .Oncethesescalarsare computed,thenthecorrectiontotheassumedvalueofthesourcefunctioninthe z directioncan becomputedaccordingto: z = 1 X l =1 l sin lz L X l =1 l sin lz : II.33 InordertosolvetheabovemodiedHelmholtzequations,onecanfollowtheapproachthatwas developedforthe2-DprobleminsectionII.2.1andconsidertwowell-posedproblemsgivenby where)]TJ/F37 7.9701 Tf 38.364 -1.778 Td [(l = @ @x 2 + @ @y 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( l 2 isthemodiedHelmholtzoperator.Thecollecteddataappearsas boundaryconditionsaccordingto: A l x ;y = y =2 Z 1 0 e x ;y;z sin lz dz; II.34 A l x ;y = y =2 Z 1 0 e x ;y;z sin lz dz; II.35 A l y x; 0= x =2 Z 1 0 e y x; 0 ;z sin lz dz; II.36 A l y x; 1= x =2 Z 1 0 e y x; 1 ;z sin lz dz: II.37 Theaboveproblemsarebothwell-posed,andonecanobtainnitedimensionalapproximations usingcentraldierencing.SimilartoEqs.II.14andII.15,nitedierenceapproximationsof

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46 theaboveproblemsleadto: A 3 a = b 3 + ^ d l ; II.38 A 4 a = b 4 + ^ d l ; II.39 where, A 3 and A 4 arethenitedierenceapproximationoftheHelmholtzoperatorsinproblem IandproblemIIshowninguresII.4andII.5,respectively.Thevector a isthenodalvaluesof theunknownvariable A l x;y , b 3 and b 4 aretheknownright-handsidesofthenitedierence approximationstoproblemsIandIIshowninguresII.4andII.5,andthevector ^ d isthe nodalvaluesoftheknownproduct^ g x ^ f y .Notethatthecollecteddata,whichappearas Neumannboundaryconditions,showupinthevectors b 3 and b 4 .Also,theassumedvaluesfor thesourcefunction^ g x ^ f y appearinthevector ^ d .Bothproblemsarewell-posed,asaresult A )]TJ/F51 7.9701 Tf 7.594 0 Td [(1 3 and A )]TJ/F51 7.9701 Tf 7.594 0 Td [(1 4 exist.Itispossibletoeliminate a fromtheaboveequationsandarriveat: A )]TJ/F51 7.9701 Tf 7.594 0 Td [(1 3 b 3 )]TJ/F53 10.9091 Tf 10.909 0 Td [(A )]TJ/F51 7.9701 Tf 7.594 0 Td [(1 4 b 4 | {z } w = A )]TJ/F51 7.9701 Tf 7.593 0 Td [(1 4 )]TJ/F53 10.9091 Tf 10.909 0 Td [(A )]TJ/F51 7.9701 Tf 7.594 0 Td [(1 3 ^ d | {z } v l ; II.40 whereforsimplicityandspaceconsiderations,weareredeningthevectors w and v .Solvingfor thescalar l leadsto: l = w T v v T v ; II.41 where,thescalar > 0ischosenbythedesigner.Aftercomputingthescalars l ,for l =1 ; 2 ;:::;L ,thecorrectiontotheassumedvaluecanbeobtainedusingEq.II.33.The updatinginotherdirectionscanalsobeaccomplishedbyusingasimilarapproach. II.3.2Updatein x and y directions Forupdatinginthe y direction,wecanassumethat g x ^ g x and h z ^ h z . Then,theequationfortheerrorsimpliesto: e xx + e yy + e zz =^ g x % y ^ h z in : II.42

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47 )]TJ/F37 7.9701 Tf 6.818 -1.777 Td [(l A l =^ g x ^ f y l ProblemI A l =0 A l y = A l x = A l =0 FIGUREII.4:Therstwell-posedproblemforthecomponentsoftheerroreld. )]TJ/F37 7.9701 Tf 6.818 -1.777 Td [(l A l =^ g x ^ f y l ProblemII A l y = A l =0 A l =0 A l x = FIGUREII.5:Thesecondwell-posedproblemforthecomponentsoftheerroreld.

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48 TheaboveequationissimilartoEq.II.28,andthesameapproachcanbeapplied.Likewise,for updatinginthe x direction,wecanassumethat f y ^ f y and h z ^ h z .Then,the equationfortheerrorsimpliesto: e xx + e yy + e zz = ^ f y x ^ h z in : II.43 Again,theaboveequationissimilartoEq.II.28,andthesameapproachcanbeapplied. II.4Helmholtzinversesourceproblem ThesameproceduredescribedinsectionII.3canbeappliedtotheinversesource problemfortheHelmholtzequation.Startingwithaninitialguessandfollowingthealgorithm leadstotheequationfortheerroreldgivenby: e xx + e yy + e zz + $ 2 e = g x f y h z )]TJ/F15 10.9091 Tf 11.282 0 Td [(^ g x ^ f y ^ h z ; in : II.44 RelatingtheactualvaluestotheassumedvaluesaccordingtoEq.II.27andproceedingto recoverthecorrectiontermsforonedirectionatatimeleadto: e xx + e yy + e zz + $ 2 e =^ g x ^ f y z ; in : II.45 Followingthesameprocedure,theover-speciedmodiedHelmholtzequationgiveninEq. II.32simpliesto: A l xx + A l yy + $ 2 )]TJ/F15 10.9091 Tf 10.909 0 Td [( l 2 A l =^ g x ^ f y l ; l =2 Z 1 0 z sin lz dz; II.46 for l =1 ; 2 ;:::;L .TheaboveequationissimilartoEq.II.32exceptforthenonzerowave number $ 2 .Forlowtomoderatewavenumberstheaboveequationscanbesolvedvery accurately.Thesameprocedurecanbeappliedtosolvetheaboveill-posedproblem.Wenext applytheabovealgorithmstoanumberofnumericalexamples.

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49 II.5Numericalexperiments Inthissectionweuseanumberofnumericalexamplestostudytheapplicabilityofthe method. II.5.1Example1 Firstconsideraninversesourceproblemintwodimensionaldomain.Assumethatthe actualsourceterminEq.II.2isgivenby: g x f y = h 1+ exp x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 3 4 0 : 000001 ih 1+2 exp y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 65 4 0 : 00001 i : II.47 Wecandividethedomainineachdirectioninto100equalintervals.Thedataiscollectedinthe formofNeumannconditionattheboundaries.Thedataisalsocontaminatedwithnoise.The noiseisazeromeanGaussianwhitenoisewithsignal-noiseratioof6%.Beforeusingthedataa threepointaveragingisperformedtosmoothoutthedata.Theactualsourcefunctionhasa nominalwhichisequalto1.Sinceitsvaluecanbemeasuredattheboundary,itisreasonableto starttheiterationsfrom^ g x = ^ f y =1.Thealgorithmseekstorecoverthecorrectiontothe assumedvaluesrstinthe y direction,andthenin x direction.ItisinstructivetoconsiderEq. II.6whichisusedtoupdatethesourceterminthe y direction.Itispresentedhereagain: e xx + e yy =^ g x | {z } previousiteration % y in : Asimilarequation,i.e.,Eq.II.18,isusedtoupdatethesourceterminthe x direction. Consideringtheaboveequationitisclearthat,inordertobeabletorecovertheunknown function % y ,oneneedstohavethefunction^ g x awayfromzero.Forthisexample,the nominalvalueoftheeldisequaltoone,andtheinversioncanbeaccomplished.Asaresultfor thisexample,theparameter inEq.II.17issetequaltoone,i.e., =1.Ifthenominalvalue oftheeldisclosetozero,whichisthecaseinmostapplications,thentheaboveequationtends to over-estimate theunknownfunction.Asaresultasmallervalueof shouldbeused.Thiswill becomeclearinthenextexample.

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50 FIGUREII.6:Reductionintheerrorfortheexample1. FigureII.6presentsthereductionintheerrorforthisexample.Theerroristhe dierencebetweenthemeasureddataandthecomputedvalueattheboundariesgivenby: Error= Z @ r u )-222(r ^ u 2 d x : FigureII.7andgureII.8showtherecoveredfunctionsafterafewiterationsandcompare themtotheactualsourcefunctions.Forthisexample,thealgorithmcanrecoverthesource functionverycloselyafteronlyafewiterations.Wenextassumethatonlypartialdatais availableforinverseevaluationofthesourcefunction. Weconsiderthesameproblemandassumethatdataisavailableononlytwosides givenbyEq.II.20.FigureII.9showsthereductioninthetotalerror.Onlytheerrorgivenin isusedintherecovery.Withthislimiteddata,itisstillpossibletoformulatetwowell-posed problemsgiveninEqs.II.22andII.23.FiguresII.10andII.11showconvergenceofthe sourcefunctionwhenonlypartialdataisavailable. II.5.2Example2 Wenextconsidertherecoveringofasourcefunctionwithanominalvalueoftheeld closetozero.Inthisexample,theactualsourcetermisgivenby: g x f y = h 0 : 05+ exp x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 3 4 0 : 000001 ih 0 : 05+2 exp y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 65 4 0 : 00001 i : II.48

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51 FIGUREII.7:Convergenceoftherecoveredsourcefunctioninthe x directionforexample1. FIGUREII.8:Convergenceoftherecoveredsourcefunctioninthe y directionforexample1. FIGUREII.9:Reductionintheerrorfortheexample1withpartialdata.

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52 FIGUREII.10:Convergenceoftherecoveredsourcefunctioninthe x directionforexample1 withpartialdata. FIGUREII.11:Convergenceoftherecoveredsourcefunctioninthe y directionforexample1 withpartialdata.

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53 Forthisexample,sincethenominalvalueoftheeldisveryclosetozero : 05,and0 : 0025in 2-Dthealgorithmover-estimatesthesoughtafterfunction.Itisthennecessarytouseasmaller valuefortheparameter .Forthiscaseweuseavalueof =0 : 02.Usingasmallervaluefor willsimplyrequiremoreiterations.FigureII.12showsthereductioninerrorforthiscase.The sameamountofnoiseisusedtomodelarealisticdata.FiguresII.13andII.14showthe actualsourcefunctionandtherecoveredsourcefunctionafter140iterations.Thepresent algorithmcanrecoveraverycloseestimateofthesourcefunction.Wenextproceedtoincrease thenoiseinthedata.FigureII.15showsthereductioninerrorwhenthelevelofnoiseis increasebyafactorofve.FigureII.16showstherecoveredsourcefunctionafter120 iterations.Thepresentalgorithmcanstillrecoverareasonablycloseestimateofthesource function. FIGUREII.12:Reductionintheerrorfortheexample2.

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54 FIGUREII.13:Actualsourcefunctionfortheexample2.Thebackgroundeldis 0 : 0025. FIGUREII.14:Recoveredsourcefunctionfortheexample2after140iterations. FIGUREII.15:Reductionintheerrorfortheexample2withhighernoiselevel.

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55 FIGUREII.16:Recoveredsourcefunctionfortheexample2after120iterations. II.5.3Example3 Wenextconsiderrecoveringofasourcefunctionin3-D.Considerrecoveringasource functionthatisgivenby: g x f y h z = h 0 : 1+ exp x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 3 4 0 : 00002 i h 0 : 1+ exp y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 75 4 0 : 00002 i h 0 : 1+ exp z )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 5 4 0 : 00002 i : II.49 Forthiscase,thebackgroundeldis1.0andthenominalorapparentvalueofthefunctionis 0.001,andthereisonetargetthatislocatedat x;y;z : 3 ; 0 : 75 ; 0 : 5.Wecandividethe domaininto50equalintervalsineachdirection.Thissizemeshisrequiredtocalculatetheeld variablesin3-Dwithhighaccuracy.Inordertoprojecttheerrordowntothespaceofsine functionsinEq.II.29,weuse40termsintheseries.However,whenrecoveringthefunction backusingEq.II.33weusetherst L =18terms.Thisisduetothefactthatthenoiseinthe datapollutesthehigherfrequencycomponentsandincludingthemleadstoinstability.The parameter =30.FigureII.17showsthelocationofthetargetwithinthe3-Ddomain.Figure II.18presentsthelocationoftherecoveredfunction.FigureII.19throughII.21presentthe individualcomponentsofthesourcefunctionandcomparethemtotheactualfunction.Forthis case,60iterationsareusedtorecoverthetargetpresentedinFigureII.17.

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56 FIGUREII.17:Actualtargetwithina3-Ddomainfortheexample3. FIGUREII.18:Recoveredtargetwithina3-Ddomainfortheexample3after60iterations.

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57 FIGUREII.19:Comparisonoftheactualfunctionandtherecoveredsourcefunctionin x forthe example3. FIGUREII.20:Comparisonoftheactualfunctionandtherecoveredsourcefunctionin y forthe example3.

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58 FIGUREII.21:Comparisonoftheactualfunctionandtherecoveredsourcefunctionin z forthe example3.

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59 II.5.4Example4 Wenextconsiderrecoveringofmultipletargetsina3-Ddomain.Considerthe recoveringofasourcefunctiongivenby S x =1+ g x f y h z where: g x = h 0 : 1+ exp x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 3 4 0 : 00002 + exp x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 65 4 0 : 000015 i ; II.50 f y = h 0 : 1+ exp y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 75 4 0 : 00001 + exp y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 65 4 0 : 000015 i ; II.51 h z = h 0 : 1+ exp z )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 4 4 0 : 00001 + exp z )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 7 4 0 : 000015 i : II.52 TheabovesourcefunctionrepresentsfourtargetswithinthedomainasisshowningureII.22. FigureII.23showstherecoveredtargetsafter60iterations.Alloftheparametersaresettobe thesameasinExample3.Also,thesameamountofnoiseisused.Thealgorithmcanrecover multipletargetswithinthesourcefunction. FIGUREII.22:Actualsourcefunctionwith4targetsfortheexample4. II.5.5Example5 Wenextconsiderrecoveringofmultipletargetsina3-DdomainfortheHelmholtz operatorwiththewavenumber $ 2 =4 : 5.Theunknownsourcefunctionisthesameasin

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60 FIGUREII.23:Recoveredsourcefunctionafter60iterationsfortheexample4.

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61 example4.FigureII.24presentstherecoveredfunctionafter80iterations.Alltheparameters andthelevelofnoisearesetequaltotheirvaluesinexample4. FIGUREII.24:Recoveredsourcefunctionafter80iterationsfortheHelmholtzoperatorinexample5. II.6Summary Inthischapterwepresentedacomputationalalgorithmfortheinverseevaluationof thesourcefunctionforPoissonandHelmholtzequations.Thesourcetermisassumedtobe composedofauniformbackgroundvalueandaseparablefunction.Thealgorithmisiterativein nature.Itupdatesthesourcefunctionforeachcoordinateseparately.Thealgorithmshows reasonablerobustnesstonoise.Itcanalsorecoverthesourcefunctionwithpartialdata.A numberof2-Dand3-Dexampleswereusedtostudytheapplicabilityoftheproposedalgorithm. WeconsideredbothPoissonandHelmholtzoperators.

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62 CHAPTERIII ONIDENTIFICATIONOFAWAVENUMBERFORHELMHOLTZ EQUATION Abstract [91] ThisnoteisconcernedwithinverseevaluationofanunknownwavenumberinaHelmholtz equation.Thealgorithmassumesaninitialguessfortheunknownfunctionandobtains correctionstotheguessedvalue.Theupdatingstageisaccomplishedbygeneratingasetof functionsthatsatisfysomeoftherequiredboundaryconditions.Werefertothisspaceasproper solutionspace.Thecorrectiontotheassumedvaluecanthenbeobtainbyimposingthe remainingboundaryconditions.Weconsidertheevaluationofrealandcomplexwavenumbers. Anumberofnumericalexamplesareusedtostudytheapplicabilityandeectivenessofthe procedures. III.1Introduction Inthischapter,weconsideraninversereconstructionmethodfortheHelmholtz equation.Itiswell-knownthatthisclassofproblemsarehighlyill-posed[6,53,54]andvarious methodshavebeendevelopedtoovercomeit[seesectionsI.7andI.8]. Inthischapterweapplyanewiterativemethodtorecoverageneralwavenumber. Thenoveltyofthepresentmethodisthattheupdatingisachievedbylookingforthecorrection termwithinasetofpreviouslygeneratedfunctionsthatsatisfyessentialboundaryconditionsof theproblem,andthereby,limitingthesearchspace.Forupdatingtheinitialguess,thepresent algorithmseeksthesolutiontotheerroreldinthespaceofproperfunctions.Theupdating procedurepresentedhereismostlybasedonsimpleargumentsinlinearalgebraandinvolves matrixvectormultiplicationsthat,foractualimplementations,caneasilybeperformedon

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63 parallelmachines.Ourmethodisfullycomputationalandcanbeappliedtovariouselliptic equation 1 .Forthiswork,weassumethattheunknownwavenumberisknownattheboundaries equaltooneforsimplicity.Wealsoassumethatthesystemcanbeexcitedbyincomingwaves attheboundaries.Inaddition,datacanbecollectedintheformoftheeldgradient,orthereal andimaginarypartsoftheeldattheboundaries.Forsimplicity,wepresentthemethodforthe evaluationofarealwavenumberforarealHelmholtzequation.Wethenpresentnumerical resultsfortheevaluationofbothrealandcomplexwavenumbers. InsectionIII.2,wepresentthebasiciterativealgorithm.Itassumesaninitialvalue fortheunknownfunctionandobtainscorrectionstotheassumedvaluewhichwehaverecently usedinchapterII.Thenewfeatureofthepresentworkistheupdatingstage.SectionIII.3 presentstheupdatingformulationbasedonthe propersolutionspace .InsectionsIII.4weapply themethodforacomplexwavenumber.InsectionIII.5,wepresentthepracticalsteps necessaryfortheimplementationofthemethod.InsectionIII.6,weuseanumberofnumerical examplestostudytheapplicabilityoftheproposedalgorithm. Notation: Weuselowercasestandardlettersforvariouseldvariables.Weusebold upper-caseletterstodenoteamatrixi.e. S ,boldlower-caseletterstodenotevectorswhose entriesarethenodalvaluesofeldvariablesi.e., s ,for s x , x 2 .Weusesecond-order nite-dierencemethodtoapproximatevariousdierentialoperator,andRe s todenotethereal partofacomplexfunctions x . III.2Problemstatementandtheidenticationalgorithm Let 2 R 2 beaclosedboundedset.Considera2-DHelmholtzequationgivenby: 4 u x + 2 p x u x =0 ; x 2 ; III.1 1 Forexampleforanonhomogenousellipticsystem, O : x O u + k 2 p x u + u =0 ; x 2 ; @u @ )]TJ/F73 8.9664 Tf 9.516 0 Td [(ik x u = 0 ; x 2 @ ,knownfunction x whereGreen'sfunctionforthebackgroundeldmaynotbereadyavailable.

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64 whereDirichletandNeumannboundaryconditionsaregivenattheboundaryof,denotedby @ ,accordingto: u x = ' x ; x 2 @ : III.2 Measurementsintheformofnormalderivativeattheboundariescanbecollectedandprovided forthepurposeofinversion: O u x = & x ; x 2 @ : III.3 Wewilluseotherformsofmeasurementsinthenumericalexamples.Thevariable u x denotes theelectriceld,theparameter k = 2 forsimplicitydenotesthefrequencyoftheincident waveandthefunction p x isaphysicalparameter.Thegoalistorecoverthefunction p x basedonboundarymeasurements.Theproposedreconstructionmethodisaniterativeinnature andiscomposedofthreesteps. Assumeaninitialvaluefortheunknownfunction,i.e.,^ p x ,andusingtheDirichlet boundaryconditions,obtainabackgroundeldsatisfyingthesystem: 4 ^ u x + 2 ^ p x ^ u x =0 ; x 2 ; ^ u x = ' x ; x 2 @ : III.4 SubtractthebackgroundeldfromEq.III.1,andobtaintheerroreld, e x = u x )]TJ/F15 10.9091 Tf 11.607 0 Td [(^ u x givenby: 4 e x + k p x u x )]TJ/F15 10.9091 Tf 11.835 0 Td [(^ p x ^ u x =0 ; x 2 : III.5 SincethebackgroundeldsatisestheDirichletboundarycondition,theboundary conditionsfortheerroreldaregivenby e =0,and O e = O u )]TJ/F76 10.9091 Tf 10.909 0 Td [(O ^ u 8 x 2 @ where O denotesthenormalderivative. Theassumedvalueisrelatedtotheactualvalueofthewavenumberaccordingto p x =^ p x + x where x istheunknowncorrectionterm.Obtaincorrectiontothe assumedvalue,i.e., x fortheunknownandupdatetheassumedvalueaccordingto

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65 p x =^ p x + x ,andgotostep[1].Stoptheiteration,whennoreductionintheerror isrecorded. Wehaverecentlyusedtheabovealgorithmfortheinverseidenticationofasourcetermfor HelmholtzequationinchapterII.Thethirdstepintheabovealgorithmisthenewfeaturein thepresentwork.Inthenextsection,wedevelopthemethodof propersolutionspace toevaluate thecorrectiontotheassumedvalueinthethirdstepoftheabovealgorithm. III.3Propersolutionspacefortheidenticationofawavenumber Inthissectionwepresentourmethodwhichisbasedontheideaofexpandingtheerror eldina propersolutionspace .Thismethodwasintroducedandappliedtoan ill-posed heat conductionproblemin[92],andaparabolicproblemin[93]. Theunknownfunctionisrelatedtotheguessedvalueaccordingto p x =^ p x + x where x istheunknowncorrectionterm.Torecoverthecorrectionterm,weassumethat e x issmallandlinearizetheerrorEq.III.5aroundthebackgroundeldandarriveat: 4 e + k ^ p x e + k x ^ u =0 ; x 2 ; III.6 withtheappropriateboundaryconditionsgivenby: e =0 ; O e = O u )]TJ/F76 10.9091 Tf 10.909 0 Td [(O ^ u; 8 x 2 @ : III.7 Wecanproceedasfollow.Consideralinearlyindependentsetoffunctions l x ;l =1 ; 2 ;:::;N , over x 2 ,andassumethattheunknownfunction x canbeexpressedasalinear combinationofthesefunctions,i.e., x = 1 1 x + ::: + N N x forsomeconstants 1 ; 2 ;:::; N .Next,generateasetoffunctionsthatsatisfytheerroreldequationwiththe knownzeroDirichletboundarycondition,i.e., 4 l x + k ^ p x l x + k l x ^ u =0 ; x 2 ; l x =0 ; x 2 @ : III.8

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66 Therefore,everyfunction l x satisestheDirichletboundaryconditionfor x 2 .Itisthen possibletoexpandtheactualandunknownerroreld e x inthespanofthespacegenerated by l x ;l =1 ; 2 ;:::;N accordingto: e x = N X j = l l l x ; III.9 wherethefunctions l x areknown,buttheconstants l areunknown.Wenextarguethatthe erroreld e x mustsatisfythegradientconditionthatisfurnishedbythemeasurements,i.e., 5 e x = 5 u x )-222(5 ^ u x ; 8 x 2 @ : III.10 Thegradientconditioncanbeexpressedbytheoperator B andtheerroreldisrequiredto satisfytheconditiongivenby: B e x = B N X l =1 l l x = N X l =1 l B l x = 5 u x )-222(5 ^ u x ; x 2 @ : III.11 Theaboveequationcanbeusedtoobtaintheunknowncoecients.Thisstepwillbeexplained indetailsinthenextsection.Oncetheunknowncoecients 1 ; 2 ;:::; N areobtained,the unknowncorrectiontermcanbeobtainedbysubstitutingfortheerroreldinEq.III.6with expansiongiveninEq.III.9.Itleadsto: 4 N X l =1 l l x + k ^ p x N X l =1 l l x + k x ^ u = N X l =1 l 4 l x + k ^ p x l x + k x ^ u =0 ; III.12 for x 2 .UsingEq.III.8andexpandingtheunknowncorrectionterm x inthespaceof l x accordingto x = P N l =1 l l x leadsto: )]TJ/F36 10.9091 Tf 8.485 0 Td [(k N X l =1 l l x ^ u x + k N X l =1 l l x ^ u x =0 ; or k N X l =1 l )]TJ/F36 10.9091 Tf 10.909 0 Td [( l l x ^ u x =0 : III.13 Wecansatisfytheaboveequationbysetting l = l ,whichleadsto: x = N X l =1 l l x : III.14 Thiscompletesthethirdstepofthealgorithm.

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67 III.4Complexwavenumber Considerasimilarboundeddomain 2 R 2 withcontinuousboundary @ Let q x be areal-valuedfunctioninandconsiderthecomplexHelmholtzequationwithimpedance boundaryconditiongivenby[94]: 4 u x + 2 p x u x =0 ; x 2 ; III.15 : r u x + iu x = g x ; x 2 @ ; III.16 where p x =1+ iq x , istheunitnormaltotheboundary,and g x istheboundarydata. Measurementsintheformoftherealpartoftheeld u x attheboundariescanbecollected andprovidedforthepurposeofinversion.Thevariable u x denotestheelectriceld,the parameter k = 2 forsimplicitydenotesthefrequencyoftheincidentwaveandthefunction p x isaphysicalparameter.Wecanstartwithaguessfortheunknownfunction ^ p x =1+ i ^ q x andlookforthecorrectionterm x = q x )]TJ/F15 10.9091 Tf 11.722 0 Td [(^ q x .Applyingthepresent algorithmandlinearizingaroundthebackgroundeldleadstotheerroreldgivenby: 4 e + 2 ^ p x e + 2 x ^ u =0 ; x 2 ; where ;e x = u x )]TJ/F15 10.9091 Tf 11.607 0 Td [(^ u x ; III.17 withtheboundaryconditionsgivenby : O e + ie =0 ; 8 x 2 @ : III.18 TheadditionalboundaryconditionfortheerroreldisgivenbyRe e =Re u )]TJ/F15 10.9091 Tf 10.909 0 Td [(Re^ u 8 x 2 @ , whereRedenotestherealpartofthecomplexfunction.Themethodofpropersolutionspace cannowbeappliedtorecoverthecorrectionterm x . III.5Numericalimplementation Inthissectionwepresentthepracticalstepsnecessaryfortheimplementationofthe abovemethod.Considertheinverseevaluationofawavefunctioninaunitsquare

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68 =[0 ; 1] [0 ; 1]givenby: p x;y =1+ g x f y ; g s = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 6 4 0 : 00002 i ; f s = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 6 4 0 : 00002 i : III.19 Theboundarycanbeexposedtoanincomingwavegivenby: u x; 0= cos xcos ; u x; 1= cos xcos + sin ; u ;y = cos ysin ; u ;y = cos cos + ysin : III.20 Onecangeneratethedataandaddarealisticlevelofnoise.Theaddednoiseisazeromean randomlygeneratedwhitenoisewithnoise-to-signalratioof4%.Inthissection,westudythe reductionintheerrorwhichisthedierencebetweenthegivendataandthecalculatedvalueat agiveniteration,normalizedwithrespecttoitsvalueattherstiteration,i.e., Error-2= R p [ O u x )]TJ/F76 10.9091 Tf 10.909 0 Td [(O ^ u x ] 2 d x R p [ O u x )]TJ/F76 10.9091 Tf 10.909 0 Td [(O ^ u x ] 2 d x j rstiteration ; 8 x 2 @ : III.21 Propersolutionspacemethodcallsforthecomputationofthepropersolutionspacewhichisthe spanoffunctionsthatarethesolutionofEq.III.8.Forthecorrectionterm x ,an appropriatesetoflinearlyindependentfunctionsforthegivendomaincanbeconstructed accordingto 2 l x = sin ix sin jy ;i =1 ; 2 ;:::;M;j =1 ; 2 ;:::;M;l = i )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 M + j; III.22 where, l x =0 8 x 2 @ for l =1 ; 2 ;:::;N ,with N = M 2 .Next,usingtheindividual l x and thezeroboundaryconditioninEq.III.8,onecangeneratethesolutionsetrequiredinEq. III.9.FinitedierenceapproximationstotheellipticsystemsinEq.III.8simplifytolinear systemsgivenby: F l + k ^ U l =0 ; or l = )]TJ/F36 10.9091 Tf 8.485 0 Td [(k F )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 ^ U l ;l =1 ; 2 ;:::;N; III.23 2 Sincetheunknownwavenumberisconsideredtobeknownattheboundaries, x =0 8 x 2 @

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69 where F isthenite-dimensionalapproximationofthe2-DHelmholtzoperatorwithaunique inversesincetheproblemiswell-posed, l isthevectorcontainingthenodalvaluesofthe function l x and l isthevectorthatcontainsthenodalvaluesofthefunction sin ix sin jy for l = i )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 M + j .Thematrix ^ U isadiagonalmatrixthatcontainsthe nodalvaluesof^ u x .Ingeneral,^ u x 6 =0,andsince l x arelinearlyindependent,itfollows that l arelinearlyindependent.Oncethepropersolutionspaceiscomputedonecanarriveat Eq.III.11whichisgivenby: N X j = l l B l x = 5 u x )-222(5 ^ u x ; x 2 @ ; or B l = b ; III.24 where 2 R N istheunknownvectorcontainingthecoecients l 8 l =1 ; 2 ;:::;N: .Theoperator B representstherst-orderoperatorattheboundaryandthematrix B representsits nite-dimensionalapproximation.Theright-handsideoftheaboveequationisthedierence betweenthegivenmeasurementandthecomputedgradientsattheboundariesandareknown. Aftertheunknowncoecientsaresolved,thecorrectiontotheassumedvaluecanbe computedusingEq.III.14.Thenite-dimensionalapproximationoftheaboveequationisa nonsquareover-determinedsystem.Asexpected,thecoecientmatrixisrankdecientandthe linearsystemcanbesolvedaftertheintroductionofsomeregularization.AspointedoutinEq. III.14,thecorrectiontermcanbeexpandedaccordingto x = P N l =1 l sin ix sin jy for i =1 ; 2 ;:::;M and l = i )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 M + j .Itispossibletostabilizetheinversionbyintroducingsome formofregularizationaccordingto: 2 6 6 6 6 6 4 B l 1 2 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 b 0 0 3 7 7 7 7 7 5 ; III.25 where,thematrices 'srepresenttherst-derivativeoperatorsgivenby: 1 = @ x @x j x = x k ;y = y m = N X l =1 i l cos ix k sin iy m ; 2 = @ x @y j x = x k ;y = y m = N X l =1 j l sin ix k cos iy m ; III.26

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70 for l = i )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 M + j ,and x k and y m arediscretelocationsinsidethedomain.Forsimplicity,we areusingthesamenite-dierenceinteriornodesfor x k and y m .Theconstant > 0issetby thedesigner,andtheabovelinearsystemcanbesolvedfortheunknowncoecients l .Dividing theunitsquareinto n e =60equalintervalsleadsto L = n e +1=61nodesineachdirection,and thesizeoftheonedimensionalvectorsinthesolutionset, l ,inEq.III.9is L 2 .Using M =20 notethat M
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71 FIGUREIII.1:TheactualwavenumbergiveninIII.19toberecoveredforexample1. FIGUREIII.2:Thereductioninerrorasafunctionofthenumberofiterationsforexample1. FIGUREIII.3:Therecoveredfunctionafter400iterationsforexample1.

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72 FIGUREIII.4:Comparisonoftherecoveredfunctionwiththeactualfunctionattwocrosssectionsforexample1.

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73 III.6.2Example2 Considertherecoveringofawavenumbergivenby: p x;y =1+ f x g y + f y g x ; f s = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 23 4 0 : 00002 i ; g s = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 76 4 0 : 00002 i ; III.27 whichisshowningureIII.5.Theparameter issetat =0 : 4.FigureIII.6showsthe reductioninerrorasfunctionsofthenumberofiterations.FigureIII.7showstherecovered functionafter350iterations,andgureIII.8comparestheactualfunctionwiththerecovered functionatadiagonalcross-section. FIGUREIII.5:TheactualwavenumbergiveninIII.27toberecoveredforexample2.

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74 FIGUREIII.6:Thereductioninerrorasafunctionofthenumberofiterationsforexample2. FIGUREIII.7:Therecoveredfunctionafter350iterationsforexample2. FIGUREIII.8:Comparisonoftherecoveredfunctionwiththeactualfunctionatadiagonal cross-sectionsforexample2.

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75 III.6.3Example3 Considertherecoveringofawavenumbergivenby: p x;y =1+ f 1 x f 1 y + f 2 x f 3 y ; f 1 s = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 2 4 0 : 000018 i ; f 2 s = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 6 4 0 : 000018 i ; f 3 s = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 78 4 0 : 000015 i ; III.28 whichisshowningureIII.9.FigureIII.10showstherecoveredfunctionafter200iterations whentheparameter issetat =1 : 2.FiguresIII.11andIII.12showthesamerecovered functionafter200iterationswhentheparameter isreducedto =0 : 7andthen =0 : 2.As thisparameterisreducedthealgorithmpenalizes less thehighgradientregionsintherecovered function,andtheresultsgetbetter.However,theeectofnoisebecomesabitmorevisible. FigureIII.13showsthereductionintheerrorasafunctionofthenumberofiterationsforthe threedierent 's.WenextconsideraHelmholtzequationwithcomplexwavenumber. FIGUREIII.9:TheactualwavenumbergiveninIII.28toberecoveredforexample3.

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76 FIGUREIII.10:Therecoveredfunctionwith =1 : 2after200iterationsforexample3. FIGUREIII.11:Therecoveredfunctionwith =0 : 7after200iterationsforexample3. FIGUREIII.12:Therecoveredfunctionwith =0 : 2after200iterationsforexample3.

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77 FIGUREIII.13:Thereductionintheerrorasafunctionofthenumberofiterationsforthethree dierent 'sforexample3. III.6.4Example4 Considertheinverseevaluationofacomplexwavefunctioninaunitsquare =[0 ; 1] [0 ; 1]givenby: p x;y =1+ i 1+ exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 25 2 0 : 01 ] exp [ )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 75 2 0 : 01 i : III.29 Theboundarycanbeexposedtoanincomingwavegivenby: : O u + iu =cos k x cos + y cos : III.30 Weusesixteensetsofdatawith =2 ; 3 ; 4 ; 5.Foreach ,weusefouranglesofincidents,i.e., = = 3 : 2, = = 4 : 2, = = 5 : 7, = = 7 : 2.Theregularizationparameterissetat =0 : 05.We arealsousingbothpartsoftheeldvariableattheboundaries.FigureIII.14presentsthe actualfunctionandgureIII.15showstherecoveredfunctionafter600iterations.Figure III.16comparestheactualfunctionwiththerecoveredfunctionatadiagonalcross-section. FigureIII.17comparestheactualfunctionwiththerecoveredfunctionatadiagonal cross-sectionfordierentiterations.

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78 FIGUREIII.14:TheactualfunctiongiveninIII.29toberecoveredforexample4. FIGUREIII.15:Therecoveredfunctionwith =0 : 0005after600iterationsforexample4.

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79 FIGUREIII.16:Comparisonoftherecoveredfunctionwiththeactualfunctionatadiagonal cross-sectionsforexample4. FIGUREIII.17:Convergenceoftherecoveredwavefunctionalongthediagonalinexample4.

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80 III.6.5Example5 Considertheinverseevaluationofacomplexwavefunctioninaunitsquare =[0 ; 1] [0 ; 1]givenby: q x;y =1+ exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.91 0 Td [(0 : 75 2 0 : 01 i exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 25 2 0 : 01 i + exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 25 2 0 : 01 i exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 75 2 0 : 01 ] ; III.31 whichisshowningureIII.18.Theboundarycanbeexposedtoanincomingwavegivenby: : O u + iu =cos k x cos + y cos : III.32 Weusethesameparametersinexample4.Wearealsousingbothpartsoftheeldvariableat theboundaries.FigureIII.18presentstheactualfunctionandgureIII.19showsthe recoveredfunctionafter600iterations. FIGUREIII.18:TheactualfunctiongiveninIII.31toberecoveredforexample5.

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81 FIGUREIII.19:Therecoveredfunctionwith =0 : 0005after600iterationsforexample5.

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82 III.6.6Example6 Thepresentedmethodcanalsobeappliedtovariousdomainsusingexistingmapping techniquesforexample[95],chapter10.Considertherecoveringofawavenumbergivenby: p r; =1+ i 1+ exp h )]TJ/F15 10.9091 Tf 10.909 0 Td [(20 rcos )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 5 2 + rsin 2 i ; for 0 r< 1 ; for 0 < 2 ; III.33 whichisshowningureIII.20.Theboundaryconditionscanbeexposedtoanincomingwave givenby: u ; =10+ cos cos 0 + :for 0 < 2 : III.34 Weusesixteensetsofdatawith =[2 ; 3 ; 4 ; 5]tosamplethedomain.Thefouranglesof incidenceare 0 =[ = 3 : 2 ;= 4 : 2 ;= 5 : 7 ;= 7 : 2].Theparameter issetat =0 : 05.FigureIII.21 showstherecoveredfunctionwith4%signal-to-noiseratio.FigureIII.22comparestheactual functionwiththerecoveredfunctionatadiagonalcross-section.FigureIII.23comparesthe actualfunctionwiththerecoveredfunctionatadiagonalcross-sectionfordierentiterations.We willusethesameparametersinthenextexamples. III.6.7Example7 Considertherecoveringofawavenumbergivenby: q r; =1+ exp h )]TJ/F15 10.9091 Tf 10.909 0 Td [(20 rcos )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 5 2 + rsin 2 i + exp h )]TJ/F15 10.9091 Tf 10.909 0 Td [(20 rcos +0 : 5 2 + rsin 2 i ; for0 r< 1 ; for0 < 2 : III.35 FigureIII.24showstheactual q r; whichisgivenbyEq.III.35.FigureIII.25presentsthe recoveredfunctionafter2500iterationswith4%signal-to-noiseratio.FigureIII.26compares theactualfunctionwiththerecoveredfunctionatadiagonalcross-section.FigureIII.27

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83 FIGUREIII.20:Therecoveringofawavenumberforexample6 FIGUREIII.21:Therecoveredfunctionafter2500iterationsforexample6

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84 FIGUREIII.22:Comparisonoftherecoveredfunctionwiththeactualfunctionatadiagonal cross-sectionsforexample6. FIGUREIII.23:Convergenceoftherecoveredwavefunctionalongthediagonalinexample6.

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85 comparestheactualfunctionwiththerecoveredfunctionatadiagonalcross-sectionfordierent iterations. FIGUREIII.24:TheactualfunctiongiveninIII.35toberecoveredforexample7. III.6.8Example8 FigureIII.28presentstheactualfunctionwhereonetargethasarelativelysmaller width,andisgivenby: q r; =1+ exp h )]TJ/F15 10.9091 Tf 10.909 0 Td [(20 rcos )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 5 2 + rsin 2 i + exp h )]TJ/F15 10.9091 Tf 10.909 0 Td [(60 rcos +0 : 5 2 + rsin 2 i ; for0 r< 1 ; for0 < 2 : III.36 FigureIII.29showstherecoveredfunctionafter5000iterationswith4%signal-to-noiseratio. FigureIII.30comparestheactualfunctionwiththerecoveredfunctionatadiagonal cross-section.FigureIII.31comparestheactualfunctionwiththerecoveredfunctionata diagonalcross-sectionfordierentiterations.

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86 FIGUREIII.25:Therecoveredfunctionwith =0 : 05after2500iterationsforexample7. FIGUREIII.26:Comparisonoftherecoveredfunctionwiththeactualfunctionatadiagonal cross-sectionsforexample7.

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87 FIGUREIII.27:Convergenceoftherecoveredwavefunctionalongthediagonalinexample7. FIGUREIII.28:TheactualfunctiongiveninIII.36toberecoveredforexample8.

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88 FIGUREIII.29:Therecoveredfunctionwith =0 : 05after5000iterationsforexample8. FIGUREIII.30:Comparisonoftherecoveredfunctionwiththeactualfunctionatadiagonal cross-sectionsforexample8.

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89 FIGUREIII.31:Convergenceoftherecoveredwavefunctionalongthediagonalinexample8.

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90 III.7Summary Inthischapterwepresentedacomputationalalgorithmforinverseevaluationofan unknownwavenumberforaHelmholtzequation.Themethodisacompletedeparturefromthe existingmethods.Anumberof2-Dexamplesindierentdomainswereusedtostudythe applicabilityoftheproposedalgorithms.Themethodcanbeusedtorecoverbothrealand complexwavenumbers.Thealgorithmshowsgoodrobustnesstonoise.Themethodcanbe combinedwithothercomputationalmethodstoimprovethequalityoftherecoveredfunction.

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91 CHAPTERIV INVERSESCATTERINGPROBLEMSBASEDONPROPERSOLUTION SPACE Abstract [96] Thispaperisconcernedwithaninversescatteringprobleminfrequencydomain,whenthe scatteredeldisgovernedbytheHelmholtzequation.Thealgorithmisiterativeinnature.It introducesanewapproachwhichwerefertoaspropersolutionspace.Itassumesaninitialguess fortheunknownfunctionandobtainscorrectionstotheguessedvalue.Theupdatingstageis accomplishedbygeneratingasetoffunctionsthatsatisfysomeoftherequiredboundary conditions.Werefertothisspaceaspropersolutionspace.Thecorrectiontotheassumedvalue canthenbeobtainedbyimposingtheremainingboundaryconditions.Anumberofnumerical examplesareusedtostudytheapplicabilityandeectivenessofthenewapproach. IV.1Introduction. Inthischapterthegoalistoreconstructawavenumberwhichisafunctionofa material'srelativepermeabilityandpermittivityinthescatteringequation.Itiswellknownthat thisclassofproblemsishighlyill-posed[6]andvariousmethodshavebeendevelopedto overcomeit[seesectionsI.7andI.8].Themethodproposedinthischapteristhesame methodthatappliedinchapterIIItoidentifyawavenumberforHelmholtzequation.Material propertyisknownattheboundariesequaltooneforsimplicity.Wealsoassumethatdatacan becollectedattheboundaries.WeconsiderbothDirichletandNeumanndata. InsectionIV.2,wepresentthebasiciterativealgorithmagainbutitisfora scatteringinverseproblem.SectionIV.3,presentstheupdatingformulationbasedonthe propersolutionspaceforthescatteringinverseproblem.InsectionIV.4,wepresentthe

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92 practicalstepsnecessaryfortheimplementationofthemethod.InsectionIV.5,weusea numberofnumericalexamplestostudytheapplicabilityoftheproposedalgorithm. IV.2Problemstatementandtheidenticationalgorithm Let 2 R n n =2 ; 3beaclosedboundedset.ConsideraHelmholtzequationgiven by[97]: )-222(5 c x 5 u )]TJ/F36 10.9091 Tf 10.909 0 Td [(! 2 u = S x ; x 2 ; IV.1 @u @ )]TJ/F36 10.9091 Tf 10.909 0 Td [(ik x u =0 ; x 2 @ ; IV.2 where ! denotestheconstantfrequency, c x > 0denotesthesquareofthewavespeed, k x = != p c x ,and c x isthewavenumberwhichinthischapterisnon-separablewave number.EquationIV.2istheimpedanceboundarycondition[98,99]and istheoutward normaldirectionattheboundary.Forthepurposeofevaluation,thedomainisilluminatedby thesourceterm S x .Measurementsintheformoftherealpartofthenormalderivativeofthe eld u x attheboundariesarecollectedandprovided.Wealsostudycaseswheremeasurements intheformoftherealpartoftheeld u x attheboundariesarecollectedandprovided. Thegoalistorecoverthefunction c x basedonboundarymeasurements.Here,we assumethat c x attheboundaryisknownhere=1,theinterestistheevaluationofthis functionatsubsurfacelocation.Theproposedalgorithmcanbeappliedtothescatteringinverse probleminfourstepswhicharesimilartothestepsweappliedinsectionIII.2: Assumeaninitialvaluefortheunknownfunction,i.e.^ c x andobtainabackgroundeld satisfyingthesystem: )-222(5 ^ c x 5 ^ u )]TJ/F36 10.9091 Tf 10.909 0 Td [(! 2 ^ u = S x ; x 2 ; @ ^ u @ )]TJ/F36 10.9091 Tf 10.909 0 Td [(ik x ^ u =0 ; x 2 @ : IV.3 SubtractthebackgroundeldfromEq.IV.1,andobtaintheerroreld,

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93 e x = u x )]TJ/F15 10.9091 Tf 11.607 0 Td [(^ u x ,givenby: )-222(5 c x 5 u + 5 ^ c x 5 ^ u )]TJ/F36 10.9091 Tf 10.909 0 Td [(! 2 e =0 ; x 2 ; @e @ )]TJ/F36 10.9091 Tf 10.909 0 Td [(ik x e =0 ; x 2 @ : IV.4 Theadditionalboundaryconditionfortheerroreldisgivenby 5 Re e = 5 Re u )-222(5 Re^ u ,where 5 denotesthenormalderivativeandRe denotestherealpartofthecomplexfunction. Linearizetheaboveerroreldaroundthebackgroundeld.Assumethattheunknown function c x isrelatedtotheassumedvalueaccordingto c x =^ c x + x . Substitutingfor c x andignoringthequadratictermsintheunknowns,i.e. x and e x ,leadsto: ^ c 4 e + 5 ^ c 5 e + 4 ^ u + 5 5 ^ u + ! 2 e; x 2 ; @e @ )]TJ/F36 10.9091 Tf 10.909 0 Td [(ik x e =0 ; x 2 @ : IV.5 Obtainthecorrectiontotheassumedvalue x ,andupdatetheguessedvalueinstep ,andrepeat. Inthenextsection,wedevelopthemethodofpropersolutionspacetoevaluatethe correctiontermtotheassumedvalue. IV.3Propersolutionspaceforthescatteringinverseproblem Inthissection,weapplythemethodofpropersolutionspaceintroducedinsection III.2tothescatteringinverseproblem.Consideralinearlyindependentsetoffunctions l x ;l =1 ; 2 ;:::;N overx 2 andassumethattheunknownfunction x canbeexpressedas alinearcombinationofthesefunctions,i.e. x = 1 1 x + + N N x forsomeconstants 1 ; 2 ;:::; N .Next,generateasetoffunctionsthatsatisfytheerroreldequationwiththe

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94 knownimpedanceboundarycondition,i.e., ^ c 4 l + 5 ^ c 5 l + l 4 ^ u + 5 l 5 ^ u + ! 2 l =0 ; x 2 ; @ l @ )]TJ/F36 10.9091 Tf 10.909 0 Td [(ik l =0 ; x 2 @ : IV.6 Therefore,everyfunction l x satisestheimpedanceboundaryconditionfor x 2 @ .Itisthen possibletoexpandtheactualandunknownerroreld e x inthespanofthespacegenerated by l x ;l =1 ; 2 ;:::;N accordingto e x = N X l =1 l l x ; IV.7 wherethefunctions l x areknown,buttheconstants l areunknown.Wenextarguethatthe erroreld e x mustsatisfythegradientconditionthatisfurnishedbythemeasurements,i.e, 5 Re e x = 5 Re u x )-222(5 Re^ u x ; x 2 @ : IV.8 Thegradientconditioncanbeexpressedbytheoperator B andtheerroreldisrequiredto satisfytheconditiongivenby: B Re e x = B N X l =1 l Re l x = N X l =1 l B Re l x = 5 Re u x )-222(5 Re^ u x ; x 2 @ : IV.9 Theaboveequationcanbeusedtoobtaintheunknowncoecients.Oncetheunknown coecients 1 ; 2 ;:::; N areobtained,theunknowncorrectiontermcanbeobtainedby substitutingfortheerroreldinEq.IV.5withexpansiongiveninEq.IV.7.Itleadsto: N X l =1 l ^ c 4 l x + 5 ^ c 5 l x + ! 2 l x + 4 ^ u + 5 5 ^ u =0 ; x 2 : IV.10 UsingEq.IV.6andexpandingtheunknowncorrectionterm x whichisthespaceof l x accordingto x = P N l =1 l l x leadsto: )]TJ/F37 7.9701 Tf 14.396 13.636 Td [(N X l =1 l l 4 ^ u + 5 l 5 ^ u + N X l =1 l l 4 ^ u + N X l =1 l 5 l 5 ^ u =0 : IV.11 Groupingvarioustermsleadsto: N X l =1 l )]TJ/F36 10.9091 Tf 10.909 0 Td [( l l 4 ^ u + N X l =1 l )]TJ/F36 10.9091 Tf 10.909 0 Td [( l 5 l 5 ^ u =0 : IV.12

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95 Wenextarguethatingeneral,^ u 6 =0, l x arelinearlyindependent, 5 l x arelinearly independentfor l =1 ; 2 ;:::;N andtherefore, l = l and x = N X l =1 l l x : IV.13 Thiscompletesthefourthstepofthealgorithm. IV.4Numericalimplementationfortheinversescatteringproblemmethod Inthissectionwepresentthepracticalstepsnecessaryfortheimplementationofthe aboveprocedurefortheinversescatteringproblem.Considertheinverseevaluationofawave functioninaunitsquare=[0 ; 1] [0 ; 1]givenby: c x;y =1+ p x p y ; p s = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 3 2 0 : 01 i : IV.14 Thedomaincanbeilluminatedbythefunctions S j x .Wecanconstructaseriesofexcitations by: g s;s i = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F36 10.9091 Tf 10.909 0 Td [(s i 2 0 : 02 i ; s j x;y = g x;x i g y;y j ; IV.15 where s j x;y representsaGaussiansignalcenteredat x i ;y j .WeareusingeightGaussian inputslocatedat : 3 ; 0 : 3 ; : 3 ; 0 : 4 ; : 4 ; 0 : 4 ; : 3 ; 0 : 5 ; : 5 ; 0 : 5 ; : 4 ; 0 : 6 ; : 6 ; 0 : 6and : 7 ; 0 : 7.Itispossibletonumericallygeneratethedataandaddarealisticlevelofnoise.The addednoiseisazeromeanrandomlygeneratedwhitenoisewithsignal-to-noiseratioof4%.The erroristhedierencebetweenthegivendata,whichisthenormalderivativeoftheeldvariable attheboundary,andthecalculatedvalueatagiveniteration,normalizedwithrespecttoits valueattherstiteration,i.e., Error= R r Re u x )-222(r Re^ u x 2 d x R r Re u x )-222(r Re^ u x 2 d x j rstiteration ; 8 x 2 @ : IV.16

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96 WearealsostudyingthealgorithmwhenthedataaretherealpartofthefunctionDirichlet data.Forthiscase,theerrorisgivenby: Error= R Re u x )]TJ/F15 10.9091 Tf 10.91 0 Td [(Re^ u x 2 d x R Re u x )]TJ/F15 10.9091 Tf 10.909 0 Td [(Re^ u x 2 d x j rstiteration ; 8 x 2 @ : IV.17 Propersolutionspacemethodcallsforthecomputationofthepropersolutionspacewhichisthe spanoffunctionsthatarethesolutionofEq.IV.6.Forthecorrectionterm x ,an appropriatesetoflinearlyindependentfunctionsforthegivendomaincanbeconstructed accordingto 1 : l x = sin ix sin jy ;i =1 ; 2 ;:::;M;j =1 ; 2 ;:::;M;l = i )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 M + j; IV.18 where, l x =0 8 x 2 @ for, l =1 ; 2 ;:::;N ,with N = M 2 .Forapplicationswithirregular boundaries,thedomaincanbemappedintoaunitsquareorcubeSec.5.6ofRef.[95].These functionscanthenbetransformedbacktothephysicaldomainaftertheconstruction. Next,usingtheindividual l x andtheboundaryconditioninEq.IV.6,onecan generatethesolutionsetrequiredinEq.IV.7.Finite-dierenceapproximationstotheelliptic systemsinEq.IV.6simplifytolinearsystemsgivenby: F l + E l =0 ; or l = )]TJ/F53 10.9091 Tf 8.485 0 Td [(F )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 E l ;l =1 ; 2 ;:::;N; IV.19 where l isthevectorcontainingthenodalvaluesofthefunction l x and l isthevectorthat containsthenodalvaluesofthefunction sin ix sin jy for l = i )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 M + j .Theterm F l representsthenite-dimensionalapproximationofthedierentialterm: ^ c 4 + 5 ^ c + ! 2 l with @ @ )]TJ/F36 10.9091 Tf 10.909 0 Td [(ik l =0 ; x 2 @ ; IV.20 and E representsthedierentialoperator 4 ^ u + 5 ^ u .TheellipticsysteminEq.IV.6iswell posedandthematrix F hasauniqueinversesincetheproblemiswellposed.Thefunctions l x arelinearlyindependentandfollowthat l arelinearlyindependent.Oncetheproper 1 Sincetheunknownwavenumberisconsideredtobeknownattheboundaries, x =0 8 x 2 @ .

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97 solutionspaceiscomputed,onecanarriveatEq.IV.9whichisgivenby: N X l =1 l B Re l x = 5 Re u x )-222(5 Re^ u x ; x 2 @ ; or B l = b ; IV.21 where 2 R n istheunknownvectorcontainingthecoecients l ; 8 l =1 ; 2 ;:::;N .Theoperator B representstherst-orderoperatorattheboundaryandthematrix B representsits nite-dimensionalapproximation.Theright-handsideoftheaboveequationisthedierence betweenthegivenmeasurementandthecomputedgradientsattheboundariesandareknown. Aftertheunknowncoecientsaresolved,thecorrectiontotheassumedvaluecanbecomputed usingEq.IV.13.Thenite-dimensionalapproximationoftheaboveequationisanonsquare over-determinedsystem.Asexpected,thecoecientmatrixisrankdecientandthelinear systemcanbesolvedaftertheintroductionofsomeregularization.AspointedoutinEq. IV.18,thecorrectiontermcanbeexpandedaccordingto x = P N l =1 l sin ix sin jy for i =1 ; 2 ;:::;M and l = i )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 M + j .Itispossibletostabilizetheinversionbyintroducingsome formofregularizationaccordingto 2 6 6 6 6 6 4 B l 1 2 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 b 0 0 3 7 7 7 7 7 5 ; IV.22 where,thematrices 'srepresenttherst-derivativeoperatorsgivenby 1 = @ x @x j x = x k ;y = y m = N X l =1 i l cos ix k sin iy m ; 2 = @ x @y j x = x k ;y = y m = N X l =1 j l sin ix k cos iy m ; IV.23 for l = i )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 M + j ,and x k and y m arediscretelocationsinsidethedomain.Forsimplicity,we areusingthesamenite-dierenceinteriornodesfor x k and y m .Theconstant > 0issetby thedesigner,andtheabovelinearsystemcanbesolvedfortheunknowncoecients l for l =1 ; 2 ;:::;N . Dividingtheunitsquareinto n e =60equalintervalsleadsto L = n e +1=61nodesin eachdirection,andthesizeoftheonedimensionalvectorsinthesolutionset, l ,inEq.IV.7is

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98 L 2 .Using M =20notethat M
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99 FIGUREIV.1:TheactualfunctiongiveninEq.IV.14toberecoveredinexample1. FIGUREIV.2:Therecoveredfunctionforexample1after20iterations =0 : 2 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10. FIGUREIV.3:Therecoveredfunctionforexample1after20iterations =0 : 1 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(11.

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100 FIGUREIV.4:Comparisonoftheactualfunctionwiththerecoveredfunctionsat y =0 : 3for threedierentvaluesof . FIGUREIV.5:Thereductioninerrorforexample1asfunctionofthenumberofiterations = 0 : 1 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(11. FIGUREIV.6:Therecoveredfunctionforexample1after50iterationswithDiricheltdataand =0 : 1 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10.

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101 FIGUREIV.7:Thereductioninerrorforexample1asfunctionofthenumberofiterationswith Dirichletdata. IV.5.2Example2 Considertherecoveringofawavenumbergivenby: c x;y =1+ p x p y + q x q y ; p s = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 25 2 0 : 02 i ; q s = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 75 2 0 : 02 i ; IV.24 whichisshowningureIV.8.WeconsiderbothNeumanndataandDirichletdata.Figure IV.9showstherecoveredfunctionwhenNeumanndataareused,andgureIV.10showsthe recoveredfunctionwhenDirichletdataareused.FigureIV.11comparesthetworecovered functiontotheactualfunctionalongthediagonal x = y andgureIV.12showsthereduction intheerrorforthetwocasesasfunctionsofthenumberofiterations.Wenextconsiderthesame problemwithNeumanndataandincreasethelevelofnoise.FigureIV.13showsthereduction inerrorforthreedierentlevelsofnoise.FigureIV.14showstherecoveredfunctionfor% noiseafter75iterations.

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102 FIGUREIV.8:TheactualfunctiongiveninEq.IV.24toberecoveredinexample2. FIGUREIV.9:Therecoveredfunctionforexample2after75iterationswithNeumanndata = 0 : 2 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10.

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103 FIGUREIV.10:Therecoveredfunctionforexample2after75iterationswithDirichletdata = 0 : 2 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10. FIGUREIV.11:Comparisonoftheactualfunctionwiththerecoveredfunctionsalongthediagonal y = x forNeumannandDirichletdata. FIGUREIV.12:Thereductioninerrorforexample2asfunctionofthenumberofiterations.

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104 FIGUREIV.13:Thereductioninerrorforexample2asfunctionofthenumberofiterationsfor threedierentlevelsofnoise. FIGUREIV.14:Therecoveredfunctionforexample2after75iterationswithNeumanndata =0 : 2 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10with8%noise.

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105 IV.5.3Example3 Considertherecoveringofawavenumbergivenby: c x;y =1+ p x q y + q x p y ; p s = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 72 4 0 : 000025 i ; q s = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 28 4 0 : 000025 i ; IV.25 whichisshowningureIV.15.Theunknownfunctionhasregionswithsharpgradients.For thisproblem,wegenerate12GaussianinputsanduseNeumanndatafortheinversion algorithm.Thelocationsaregivenby : 2 ; 0 : 2 ; : 4 ; 0 : 4 ; : 6 ; 10 : 6 ; : 8 ; 0 : 8 ; : 8 ; 0 : 2 ; : 2 ; 0 : 8 ; : 6 ; 0 : 4 ; .4,0.6,.5,0.2,.2,0.5,.8,0.5 and .5, 0.8 :Forthiscase;wedividethedomaininto 70 equalintervalsineachdirection:Figure IV: 16 showstherecoveredfunctionafter 30 iterationswith 4% noiseandwith = 0 : 8 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10.OuralgorithmreliesonTikhonovregularizationwhichputsaboundontherst derivativeoftheunknownfunction.Forwavenumberswithregionswithlargegradient,the presentalgorithmcanstilllocatethetarget.Theconvergerateisimprovedbyusingmoredata. Therecoveredfunctionafter30iterationsiftheunknownwavenumberisgivenby: c x;y =1+ p x q y + q x p y ; p s = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 72 2 0 : 01 i ; q s = exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( s )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 28 2 0 : 01 i ; IV.26 showningureIV.17.ThisfunctionisverysimilartotheunknownwavenumbergiveninEq. IV.25.Theonlydierenceistheregionsoflargegradients.Ouralgorithmcanrecoverabetter estimate.FigureIV.19showsthereductionintheerrorforbothcases. IV.6Summary Inthischapter,weappliedthepropersolutionspacemethodtoaninversescattering problem.Anumberof2Dexampleswereusedtostudytheapplicabilityoftheproposed

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106 FIGUREIV.15:TheactualfunctiongiveninEq.IV.25regionsoflargegradientstoberecoveredinexample3. FIGUREIV.16:TherecoveredfunctionforExample3after30iterationswithNeumanndata =0 : 8 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10with4%noise.

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107 FIGUREIV.17:TheactualfunctiongiveninEq.IV.26toberecoveredinexample3. FIGUREIV.18:Therecoveredfunctionforexample3Eq.IV.26after30iterationswithNeumanndata =0 : 8 E )]TJ/F15 10.9091 Tf 10.909 0 Td [(10with4%noise. FIGUREIV.19:Thereductioninerrorforexample3asfunctionofthenumberofiterationsfor thetwodierentfunctions.

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108 algorithms.ThealgorithmcanaccommodatebothNeumannandDirichletdata.Weconsidered thesameproblemwithNeumanndataandincreasethelevelofnoise.Itshowsgoodrobustness todierentlevelofthenoise.Next,wewillapplythepropersolutionspacemethodtoafourth orderellipticsystem.

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109 CHAPTERV IDENTIFICATIONOFFLEXURALRIGIDITYINKIRCHHOFF-LOVE PLATES V.1Introduction InthischapterthegoalistoidentifyaexuralrigidityinKirchho-Loveplates.The problemisill-posedinthesenseofstability.Applicationofthisinverseproblemcanbe measuringdevicesofmicromechanicsandawidenticationbynon-destructivemethods[100]. Theill-posednessleadstoanunrealisticinversesolutionofexuralrigiditywhenonetreatsthe problemwithoutanyregularizationprocedure[14].Forsolvingtheinverseproblemfor identicationofthebendingstiness,weusedthesecondmethodwhichisbasedonproper solutionspace.Thesecondmethodissuccessfullyappliedtotheinverseproblemfor identicationofacoecientina4thellipticpartialdierentialequation. InSectionV.2,wepresentthebasiciterativealgorithmagainbutitisforidentifying oftheexuralrigidity.SectionV.3presentstheupdatingformulationbasedontheproper solutionspaceforidenticationoftheexuralrigidity.InsectionV.4andV.5wepresentthe practicalstepsnecessaryfortheimplementationofthemethodandthenweuseanumberof numericalexamplestostudytheapplicabilityoftheproposedalgorithmwhenthegoverning equationisgivenbya4thorderellipticsystem. V.2Problemstatementandtheidenticationmethod ConsidertheidenticationoftheexuralrigidityinKirchho-Loveplatesinthe equationsI.14-I.17whichispresentedhereagain: 4 p x 4 u x = S x ; x 2 4 p x 4 u x + p x 4 2 u x = S x ; x 2 ; V.1

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110 wheretheboundaryconditionsaregivenby: u x = D x ; x 2 @ ; V.2 4 u x = Q x ; x 2 @ ; V.3 5 u x = R x ; x 2 @ ; V.4 where u x isthetransversedeectionofthebeam, p x istheexuralrigidity,whichisthe productofthemodulusofelasticity E andthemomentofinertia I ofthecross-sectionofthe beamaboutanaxisthroughitscentroidatrightanglestothecross-section.Thefunction S x representsthetransverselydistributedload[15]. P x isthetransversedeectionofthebeamat theboundaries; istheunitnormaltotheboundary. Q x isthecurvatureofthedeected longitudinalaxis.Therotationofacrosssection R x canbecollectedandprovidedforthe purposeofinversion.Theproblemisoverdeterminedbecausethedataboundaryconditionneeds tobesatised.Thegoalistorecoverthefunction p x basedonmeasurements.Theproposed reconstructionmethodcanbeappliedtothefourthorderellipticinverseproblemsinfoursteps: 1.Assumeaninitialvaluefortheunknownfunction,i.e.,^ p x andusingtheboundary conditionsinEq.V.2andEq.V.3obtainabackgroundeldsatisfyingthesystem: 4 ^ p x 4 ^ u x +^ p x 4 2 ^ u x = S x ; x 2 ; V.5 ^ u x = D x ; x 2 @ ; V.6 4 ^ u x = Q x ; x 2 @ : V.7 2.Theactualunknownfunction p x isrelatedtotheassumedvalueaccordingto p x =^ p x + x where x istheunknowncorrectionterm.ThenEq.V.1canberewritten

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111 as: 4 ^ p x + x 4 u x +^ p x + x 4 2 u x = S x ; in x 2 : V.8 3.SubtractthebackgroundeldfromEq.V.8andobtaintheerroreld, e x = u x )]TJ/F15 10.9091 Tf 11.607 0 Td [(^ u x givenby: 4 ^ p x 4 e x +^ p x 4 2 e x + 4 x 4 u x + x 4 2 u x =0 ; x 2 : V.9 4.Torecoverthecorrectionterm,weassumethat p x issmallandlinearizetheerroreld,i.e., EqV.9aroundthebackgroundeldandarriveat: 4 ^ p x 4 e x +^ p x 4 2 e x + 4 x 4 ^ u x + x 4 2 ^ u x =0 ; x 2 ; V.10 withtheappropriateboundaryconditionsgivenby: e x =0 ; x 2 @ ; V.11 4 e x =0 ; x 2 @ : V.12 Theadditionalboundaryconditionfortheerroreldisgivenby: 5 e x = 5 u x )-222(5 ^ u x ; x 2 @ : V.13 5.Obtainthecorrectiontotheassumedvalue^ p x ,andupdatetheguessedvalueinstep, andrepeat. V.3Propersolutionspaceforthereconstructionoftheexuralrigidity Inthissection,werepresentourmethodwhichisintroducedinchapterIII,IV. Consideralinearlyindependentsetoffunctions l x ;l =1 ; 2 ;:::;N ,over x 2 ,andassumethat

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112 theunknownfunction x canbeexpressedasalinearcombinationofthesefunctions,i.e., x = 1 1 x + ::: + N N x forsomeconstants 1 ; 2 ;:::; N .Next,generateasetoffunctions thatsatisfytheerroreldequationandthegivenboundarycondition,i.e., 4 ^ p x 4 l x +^ p x 4 2 l x + 4 l x 4 ^ u x + l x 4 2 ^ u x =0 ; x 2 ; V.14 l x =0 ; x 2 @ ; V.15 4 l x =0 ; x 2 @ : V.16 Itisthenpossibletoexpandtheactualandunknownerroreld e x inthespanofthespace generatedby l x ;l =1 ; 2 ;:::;N ,accordingto: e x = N X l =1 l l x ; V.17 wherethefunctions l x areknown,buttheconstants l areunknown.Wenextarguethatthe erroreld e x mustsatisfytheNeumannconditionthatisfurnishedbythemeasurements,i.e., 5 e x = 5 u x )-222(5 ^ u x ; x 2 @ : V.18 TheNeumannconditioncanbeexpressedbytheoperator B andtheerroreldisrequiredto satisfytheconditiongivenby: B e x = B N X l =1 l l x = 5 n u x )-222(5 n ^ u x ; x 2 @ : V.19 Theaboveequationcanbeusedtoobtaintheunknowncoecients.Oncetheunknown coecients l ;l =1 ; 2 ;:::;N areobtained,theunknowncorrectiontermcanbeobtainedby substitutingfortheerroreldinEq.V.10withtheexpansiongiveninEq.V.17.Itleadsto: 4 ^ p x 4 N X l =1 l l x +^ p x 4 2 N X l =1 l l x + 4 x 4 ^ u x + x 4 2 ^ u x =0 ; x 2 ; V.20

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113 4 ^ p x N X l =1 l 4 l x +^ p x N X l =1 l 4 2 l x + 4 x 4 ^ u x + x 4 2 ^ u x =0 ; x 2 ; V.21 for x 2 .UsingEq.V.14andexpandingtheunknowncorrectionterm x inthespaceof l x accordingto x = N P l =1 % l l x leadsto: 4 ^ p x N X l =1 l 4 l x +^ p x N X l =1 l 4 2 l x + N X l =1 % l 4 l x 4 ^ u x + N X l =1 % l l x 4 2 ^ u x =0 ; x 2 ; V.22 N X l =1 l 4 ^ p x 4 l x +^ p x 4 2 l x + N X l =1 % l 4 l x 4 ^ u x + N X l =1 % l l x 4 2 ^ u x =0 ; x 2 ; V.23 )]TJ/F37 7.9701 Tf 14.397 13.636 Td [(N X l =1 l 4 l x 4 ^ u x + l x 4 2 ^ u x + N X l =1 % l 4 l x 4 ^ u x + N X l =1 % l l x 4 2 ^ u x =0 ; x 2 ; V.24 N X l =1 l )]TJ/F36 10.9091 Tf 10.909 0 Td [(% l 4 l x 4 ^ u x + l x 4 2 ^ u x =0 ; in x 2 : V.25 Wenextarguethatingeneral,^ u 6 =0and l x for l =1 ; 2 ;:::;N ,arelinearlyindependentand therefore, l = l and x = N X l =1 l l x : V.26 Thiscompletesthefthstepofthealgorithm. V.4Numericalimplementationfortheidenticationofexuralrigiditycoecient Inthissectionwepresentthepracticalstepsnecessaryfortheimplementationofthe aboveprocedure.Consideranelasticplatesuchthattheendsoftheplatearexed.Thegoalis

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114 toreconstructtheexuralrigidityfunction p x inEq.V.1foraunitsquare=[0 ; 1] [0 ; 1] givenby: p x;y =1+ exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 75 2 0 : 008 i exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 25 2 0 : 008 i : V.27 Thesourceterm S x inEq.V.1givenby: S x;y =1+ exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F36 10.9091 Tf 10.909 0 Td [(x 0 2 0 : 008 i exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F36 10.9091 Tf 10.909 0 Td [(y 0 2 0 : 008 i ; V.28 wherethetransverselydistributedload S x;y isgivenbyaGaussiansignalcenteredat x 0 ;y 0 . Sincetheendsofthebeamassumedtobexed,theboundaryconditionsaregivenby: u x =0 ; x 2 @ ; V.29 4 n u x =0 ; x 2 @ : V.30 Weusevesetsofdatatosamplethedomain.TheNeummandatacanbecollectedatthe boundaryandisprovidedforidenticationpurposes.Thevesetsofdatacanbegeneratedby locatingtheloadat : 25 ; 0 : 67 ; : 5 ; 0 : 5 ; : 67 ; 0 : 25 ; : 35 ; 0 : 65 ; and : 65 ; 0 : 35.Itispossible tonumericallygeneratethedataandaddarealisticlevelofnoise.Theaddednoiseisazero meanrandomlygeneratedwhitenoisewithsignal-to-noiseratioof4%. Forthecorrectionterm x ,anappropriatesetoflinearlyindependentfunctionsfor thegivendomaincanbeconsideredaccordingto: l x = sin ix sin jy ; V.31 i =1 ; 2 ;:::;M;j =1 ; 2 ;:::;M;l = i )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 M + j; V.32 where l x =0 ; 8 x 2 @ ,for l =1 ; 2 ;:::;N ,with N = M 2 .Next,usingtheindividual l x and theboundaryconditionintheequationsV.15andV.16,onecangeneratethesolutionset requiredinEq.V.17.Finitedierenceapproximationstothefourth-orderellipticsysteminEq. V.14simplifytolinearsystemsgivenby: F l + 4 ^ U 4 l + 4 2 ^ U l =0 ; V.33

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115 l = )]TJ/F25 10.9091 Tf 8.485 0 Td [(F )]TJ/F34 7.9701 Tf 6.587 0 Td [(1 4 ^ U 4 l + 4 2 ^ U l ;l =1 ; 2 ;:::;N; V.34 where F isthenite-dimensionalapproximationofthe2-D 4 ^ p 4 +^ p 4 2 operatorwithaunique inversesincetheproblemiswell-posed, l isthevectorcontainingthenodalvaluesofthe function l x and l isthevectorthatcontainsthenodalvaluesofthefunction sin ix sin jy for l = i )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 M + j .Thematrix ^ U isadiagonalmatrixthatcontainsthe nodalvaluesof^ u x 6 =0,andsince l x arelinearlyindependentitfollowsthat l arelinearly independent.Oncethepropersolutionspaceiscomputed,onecanarriveatEq.V.19whichis givenby: B e x = B N X l =1 l l x = 5 n u x )-222(5 n ^ u x ; x 2 @ : V.35 B l = b ; 8 x 2 @ ; V.36 where 2 R N istheunknownvectorcontainingthecoecients l ; 8 l =1 ; 2 ;:::;N .Theoperator B representstheboundarycondition,andthematrix B representsitsnitedimensional approximation.Therighthandsideoftheaboveequationisthedierencebetweenthegiven measurementandthecomputedvaluesattheboundariesandareknown.Aftertheunknown coecientsaresolvedfor,thecorrectiontotheassumedvaluecanbecomputedusingEq. V.26.Thenitedimensionalapproximationoftheaboveequationisanon-square over-determinedsystem.Asexpected,thecoecientmatrixisrankdecientandthelinear systemcanbesolvedaftertheintroductionofsomeregularization.AspointedoutinEq.V.26, thecorrectiontermcanbeexpandedaccordingto x = N P l =1 l sin ix sin iy for l = i )]TJ/F15 10.9091 Tf 10.216 0 Td [(1 M + j . Itispossibletostabilizetheinversionbyintroducingsomeformofregularizationaccordingto: 2 6 6 6 6 6 4 B l 1 2 3 7 7 7 7 7 5 = 2 6 6 6 6 6 4 b 0 0 3 7 7 7 7 7 5 ; V.37 where,the 'srepresenttherst-derivativeoperatorsgivenby: 1 = @ x @x j x = x k ;y = y m = N X l =1 i l cos ix k sin iy m ; V.38

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116 2 = @ x @y j x = x k ;y = y m = N X l =1 j l sin ix k cos iy m ; V.39 for l = i )]TJ/F15 10.9091 Tf 10.909 0 Td [(1 M + j ,and x k and y m arediscretelocationsinsidethedomain.Forsimplicity,we areusingthesamenite-dierenceinteriornodesfor x k and y m .Theconstant > 0issetby thedesigner,andtheabovelinearsystemcanbesolvedfortheunknowncoecients l for l =1 ; 2 ;:::;N . Dividingtheunitsquareinto n e =59equalintervalsleadsto L = n e +1=60nodesin eachdirection,andthesizeoftheonedimensionalvectorsinthesolutionset, l ,inEq.V.17is L 2 .Using M =14notethat M
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117 FIGUREV.1:Theexactfunctionofexuralrigiditycoecientforexample1. FIGUREV.2:Thenumericallyrecoveredfunctionofexuralrigiditycoecientforexample1.

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118 FIGUREV.3:Thecomparisonoftheactualfunctionwiththerecoveredfunctionalongthediagonalforexample1. FIGUREV.4:Convergenceoftherecoveredexuralrigiditycoecientalongthediagonalinexample1.

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119 V.5.2Example2 Considertherecoveringofaexuralrigiditycoecientgivenby: p x;y =1+ exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 25 2 0 : 008 i exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 75 2 0 : 008 i ; V.41 whichisshowningureV.5.Theparameter issetat =5 10 )]TJ/F34 7.9701 Tf 6.586 0 Td [(5 .FigureV.6showsthe recoveredfunctionafter1000iterations.FigureV.7showsthecomparisonoftheactual functionwiththerecoveredfunctionalongthediagonal.FigureV.8showsthecomparisonof theactualfunctionwiththerecoveredfunctionalongthediagonalfordierentiterations. FIGUREV.5:Theexactfunctionofexuralrigiditycoecientforexample2. V.5.3Example3 Considertherecoveringofaexuralrigiditycoecientgivenby: p x;y =1+ exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 25 2 0 : 008 i exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.91 0 Td [(0 : 25 2 0 : 008 i + exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( x )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 75 2 0 : 008 i exp h )]TJ/F15 10.9091 Tf 8.485 0 Td [( y )]TJ/F15 10.9091 Tf 10.909 0 Td [(0 : 75 2 0 : 008 i ; V.42 whichisshowningureV.9.Theparameter issetat =5 10 )]TJ/F34 7.9701 Tf 6.587 0 Td [(5 .FigureV.10showsthe recoveredfunctionafter1000iterations.FigureV.11showsthecomparisonoftheactual

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120 FIGUREV.6:Thenumericallyrecoveredfunctionofexuralrigiditycoecientforexample2. FIGUREV.7:Thecomparisonoftheactualfunctionwiththerecoveredfunctionalongthediagonalforexample2.

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121 FIGUREV.8:Convergenceoftherecoveredexuralrigiditycoecientalongthediagonalinexample2.

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122 functionwiththerecoveredfunctionalongthediagonal.FigureV.12showsthecomparisonof theactualfunctionwiththerecoveredfunctionalongthediagonalfordierentiterations. FIGUREV.9:Theexactfunctionofexuralrigiditycoecientforexample3. V.6Summary Inthischapterweappliedthesecondmethodtoreconstructtheexuralrigidity coecientinKirchho-Loveplates.Severalof2-Dnumericalexperimentswereruntostudythe applicabilityofthemethod.Thenumericalresultsofthemethodwhichisbasedontheproper solutionspaceareacceptableforfourthorderellipticsystems.

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123 FIGUREV.10:Thenumericallyrecoveredfunctionofexuralrigiditycoecientforexample3. FIGUREV.11:Thecomparisonoftheactualfunctionwiththerecoveredfunctionalongthediagonalforexample3.

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124 FIGUREV.12:Convergenceoftherecoveredexuralrigiditycoecientalongthediagonalin example3.

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125 CHAPTERVI CONCLUSION Inthisdissertationweappliedtwonewcomputationalmethodsforinverseelliptic problems.TheseinverseellipticproblemsincludeinversesourceproblemsforPoissonand Helmholtzequations,aninverseproblemforidenticationofawavenumberforHelmholtz equation,aninversescatteringproblem,andidenticationofexuralrigidityforKirchho-Love plates.Thenewcomputationalmethodsareiterativealgorithms.Thealgorithmsstartwithan initialguessfortheunknownfunction,obtainabackgroundeld,andobtaintheworking equationsfortheerroreld. Therstmethodisparticularlysuitableforidentifyingasingleanomalywithcompact support.Forthecasesthattheactualperturbationcannotbemodelledbyaseparablefunction, theproposedmethodwillconvergetotheclosestapproximationofthatanomalythatcanbe describedbyaseparablefunction.Thisisnotamajorweakness,simplybecausetheuniqueness issuesthatareinherentforbothoftheaboveproblemsinversesourceproblemslimitsthe actualsourcefunctionsthatcanberecovered. Therstmethodconsidersaunitsquareorcube.Forothergeometriesthemethodcan bemodiedinanumberofways.Onepossibleapproachistousethemethodofimmersed boundarytoconvertanirregulargeometryintoacube.Anotherapproachthatcanbeusedis theexactmappingofageneraldomainontoasquare/cube.Forthismethod,theworking equationsinthetransformedspacewillincludeanumberoftermswhichwillnotbeamendable todiagonalization.Weenvisionthatforthesecases,itwillbepossibletosetupaninternal iterationtohandlesuchterms.Forgeometriesthatareclosetocube/square,variablemeshnite dierencemethodcanbeused.Again,an internaliteration canbeusedtohandlethetermsthat arenotamendabletodiagonalization. Wepresentedthesecondmethodwhichwecalled thepropersolutionspace .Theproper

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126 solutionspacemethodisacompletedeparturefromtheexistingmethods.Weappliedthesecond methodtoinverseevaluationofanon-separableunknownwavenumberforaHelmholtzequation. Successfully,weappliedthepropersolutionspacemethodalsotoaninversescatteringproblem andtoreconstructthebendingstinesscoecientinKirchho-Loveplates. Inthepropersolutionspacemethod,weneedtoassumethattheunknownfunctionis knownattheboundary.Inotherwords,thecorrectiontermneedstohaveazerovalueatthe boundary,otherwise,theexpansioninEq.III.14willnotbevalid.Ifweneedtoexpandthe correctionterminaspacedierentthanthefunctionspacethatweusetogeneratethesolution set,thenadditionalcareneedstobetaken.Thisissueneedscarefulanalysis,andwillbepartof theextensionsthatwillbefullystudied. Overallthealgorithmisverysimpletoimplement,andtheresultsareacceptable.The methodcanbeappliedtoproblemsinvariousgeometries.Itcanalsobeappliedtorealaswellas complexformofHelmholtzequation.Theresultscanbeimprovedifitispossibletoprovide bothrealandimaginarypartsofthedataattheboundaries..Anumberof2-Dexamplesin dierentdomainswereusedtostudytheapplicabilityoftheproposedalgorithm.Themethod canbeusedtorecoverbothrealandcomplexwavenumbersandaccommodatebothNeumann andDirichletdata.Thealgorithmshowsgoodrobustnesstonoise.Thepropersolutionspace methodcanbecombinedwithothercomputationalmethodstoimprovethequalityofthe recoveredfunction. Lastly,thealgorithmshowsgoodrobustnesstonoise.Forthiswork,wehavenoactual estimateoftheconvergencerate.However,resultsandconvergenceratecanbeimprovedby increasingthenumberofcollectedmeasurements.Forourcalculationswith12datasets,each iterationtakesabout3minutesonasinglenode2.6GHzchip.Thiscanbesignicantly improvedifthealgorithmisimplementedonaparallelmachine.Ingeneral,theresultscanbe improvedbyusinghigherfrequencies.However,forhigherfrequencies,oneneedstouseamuch nnermeshfortheHelmholtzequation.Thatwouldincreasethecomputationalcost.

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