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A Convergence Study of Spectrally Matched Grids in the Presence of Non-Smooth Data and Anisotropy

Permanent Link: http://ufdc.ufl.edu/UFE0022060/00001

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Title: A Convergence Study of Spectrally Matched Grids in the Presence of Non-Smooth Data and Anisotropy
Physical Description: 1 online resource (106 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: anisotropy, fd, geophysics, grids, optimal, pde, remes, spectral
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this work, we present techniques that apply to receiver-targeted problems such as in geophysical exploration. In such applications, one wishes to construct an accurate image of the earth's profile. One usually sets up a system of signal sources and receivers and the underlying pde's are solved to obtain analytic solutions at the receiver locations. These are then compared to the received data and the guess for the earth's profile is adjusted accordingly. One needs to solve these problems repeatedly and in an efficient manner. This calls for the use of non-uniform grids with some kind of spectral matching. In our work, we have analyzed the error convergence rate when such non-uniform spectrally matched grids are used for these receiver-targeted problems. We have also developed a new set of grids which we call Remes grids that prove to be extremely useful in problems over semi-infinite spectral intervals. The construction of these grids is outlined and so also their application to delta function signal source problems has been studied and analyzed to obtain the error convergence rate. Towards the end of our work, we have applied these grids to anisotropic problems with the goal of studying their convergence rates.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Gopalakrishnan, Jayadeep.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-05-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022060:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022060/00001

Material Information

Title: A Convergence Study of Spectrally Matched Grids in the Presence of Non-Smooth Data and Anisotropy
Physical Description: 1 online resource (106 p.)
Language: english
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: anisotropy, fd, geophysics, grids, optimal, pde, remes, spectral
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this work, we present techniques that apply to receiver-targeted problems such as in geophysical exploration. In such applications, one wishes to construct an accurate image of the earth's profile. One usually sets up a system of signal sources and receivers and the underlying pde's are solved to obtain analytic solutions at the receiver locations. These are then compared to the received data and the guess for the earth's profile is adjusted accordingly. One needs to solve these problems repeatedly and in an efficient manner. This calls for the use of non-uniform grids with some kind of spectral matching. In our work, we have analyzed the error convergence rate when such non-uniform spectrally matched grids are used for these receiver-targeted problems. We have also developed a new set of grids which we call Remes grids that prove to be extremely useful in problems over semi-infinite spectral intervals. The construction of these grids is outlined and so also their application to delta function signal source problems has been studied and analyzed to obtain the error convergence rate. Towards the end of our work, we have applied these grids to anisotropic problems with the goal of studying their convergence rates.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Gopalakrishnan, Jayadeep.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2009-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022060:00001


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ThereareamyriadofpeoplethatIknowandtowhomIwouldliketoextendmythanks.Firstandforemost,Iamimmenselythankfulandindebtedtomyadvisor,Dr.ShariL.Moskow,forhercontinuedsupportandconstantmentoring.Iwouldalsoliketothankherforherbeliefinme.Thisworkwouldnothavebeenpossiblewithoutherguidance.IwouldalsoliketothankVladimirDruskinforhisinvaluablesuggestionsandideasfromtimetotime.ThisworkwassupportedbytheNationalScienceFoundationundergrantsSCREMS-0619080,DMS-0605021,DMS-0713833.Next,Iwouldliketothankmyparentsforbelievinginmeandbeingpatientwithmethroughouttheseyears.Myknowledgeisincomparabletotheirvastexperienceanditiswiththisexperiencethattheyhaveguidedmeallmylifehelpingmetacklebothacademicandpersonalproblemsthatlifehasthrownatmeovertheseyears.Iamextremelygratefultomyco-advisor,Dr.JayadeepGopalakrishnan,whoextendedhisselesssupporttomethroughoutmyjourney.IamthankfultoDr.WilliamHager,Dr.SergeiPilyugin,andDr.BabetteBrumbackforservingonmycommittee.Iamalsothankfultomybeautifulwife,Aleya,whoshowedgreatpatiencewithmeinmynalstagesofcompletingmydoctoralstudies.Finally,wordsarenotenoughtodescribemygratitudeforallthefriendsthatIhavemadethroughoutmystayatGainesvilleandIwanttoacknowledgeeachandeveryoneofthemforbeingmyfriendandmakingmefeelathomefarawayfromhome! 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 11 2SPECTRALLYMATCHEDGRIDS ........................ 13 2.1Introduction ................................... 13 2.2ComputingtheGrids .............................. 13 2.3AnAttempttoFindanEquivalentFiniteElementMethod ......... 18 3SEMI-INFINITESPECTRALINTERVALS .................... 23 3.1Introduction ................................... 23 3.2Motivation:AnEllipticProblemwithNon-SmoothData .......... 23 3.2.1TheProblem ............................... 23 3.2.2TheSemidiscretization ......................... 25 3.2.3ConvergenceAnalysis .......................... 27 3.3RemesGrids ................................... 29 3.3.1TheRemesAlgorithm .......................... 29 3.3.2TheRemesGrids ............................ 31 3.3.3ConvergenceofRemesGrids ...................... 31 3.4SourceProblemonaSquare .......................... 33 3.4.1SomeNumericalResults ........................ 33 3.4.2ComparisontoPade-ChebyshevGrids ................. 36 4ANISOTROPY .................................... 39 4.1Introduction ................................... 39 4.2The1-DAnisotropicProblem ......................... 39 4.2.1Motivation ................................ 39 4.2.2TheTwo-SidedAnisotropicProblemonaFiniteInterval ...... 40 4.3The2-DAnisotropicProblem ......................... 52 5CONCLUSIONSANDFUTUREWORK ...................... 62 APPENDIX ......................................... 65 REFERENCES ....................................... 105 5

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Table page 4-1Comparisonofsolutionerrormagnitudesforthetwo-sided1-danisotropicproblemusingRemesgridsovernitespectralintervalfor=1 .............. 48 4-2Comparisonofsolutionerrormagnitudesforthetwo-sided1-danisotropicproblemusingRemesgridscomputedoversemi-innitespectralintervalsfor=105 51 7

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Figure page 2-1Aplotofthestaggeredgridovertheinterval[0;0:5]fora[4=5]Pade-Chebyshevrationalapproximation,k=5. ............................ 18 2-2Aplotofthestaggeredgridovertheinterval[0;0:5]fora[14=15]Pade-Cheby-shevrationalapproximation,k=15. ........................ 19 3-1Aplotoftheerrorbetweenthetrueimpedancefunctionandnumericallycomp-utedrationalapproximationfork=7. ....................... 32 3-2Aplotoftheerrorbetweenthetrueimpedancefunctionandnumericallycomp-utedrationalapproximationfork=8. ....................... 33 3-3Aplotoflog(abs(logerror))vs.logkfork=3;:::;17. .............. 34 3-4AplotoflogarithmoftheL2errorvs.kfork=3;:::;16usingthesolutionfork=17asabenchmark. ............................... 35 3-5AcomparisonplotoflogarithmoftheL2errorvs.kfork=3;:::;16usingthesolutionusingRemesgridsfork=17asabenchmarkandM=100;000uniformstepsalongthey-direction. ......................... 37 3-6AplotoflogarithmoftheL2errorvs.p ................................. 38 4-1Acomparisonplotoftherealpartsofthetrueandnumericallycomputedsolu-tionsforthetwo-sided1-Danisotropicproblemfork=6;=1. ......... 43 4-2Acomparisonplotoftheimaginarypartsofthetrueandnumericallycomputedsolutionsforthetwo-sided1-Danisotropicproblemfork=6;=1. ....... 44 4-3Acomparisonplotofthemagnitudesofthetrueandnumericallycomputedsol-utionsforthetwo-sided1-Danisotropicproblemfork=6;=1. ........ 45 4-4Acomparisonplotoftheerrorbetweenthetrueandnumericallycomputedsol-utionsforthetwo-sided1-Danisotropicproblemfork=6;=1. ........ 46 4-5Acomparisonplotoftherealpartsofthetrueandnumericallycomputedsolu-tionsforthetwo-sided1-Danisotropicproblemfork=13;=1. ........ 47 4-6Acomparisonplotoftheimaginarypartsofthetrueandnumericallycomputedsolutionsforthetwo-sided1-Danisotropicproblemfork=13;=1. ...... 48 4-7Acomparisonplotofthemagnitudesofthetrueandnumericallycomputedsol-utionsforthetwo-sided1-Danisotropicproblemfork=13;=1. ........ 49 8

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........ 50 4-9Spectralbehavioroftherelativeerrorincomputingthenumericalsolutionforthetwo-sided1-Danisotropicproblem. ....................... 51 4-10Acomparisonplotoftherealpartsofthetrueandnumericallycomputedsolu-tionsforthetwo-sided1-DanisotropicproblemusingRemesgridsfork=6;=1. ............................................ 52 4-11Acomparisonplotoftheimaginarypartsofthetrueandnumericallycomputedsolutionsforthetwo-sided1-DanisotropicproblemusingRemesgridsfork=6;=1. ........................................ 53 4-12Acomparisonplotofthemagnitudesofthetrueandnumericallycomputedsol-utionsforthetwo-sided1-DanisotropicproblemusingRemesgridsfork=6;=1. ........................................ 54 4-13Acomparisonplotoftheerrorbetweenthetrueandnumericallycomputedsol-utionsforthetwo-sided1-DanisotropicproblemusingRemesgridsfork=6;=1. ........................................ 55 4-14Spectralbehavioroftherelativeerrorincomputingthenumericalsolutionforthetwo-sided1-DanisotropicproblemusingRemesgrids. ............ 56 4-15Spectralbehavioroftherelativeerrorincomputingthenumericalsolutionforthetwo-sided1-DanisotropicproblemusingRemesgridsandPade-Chebyshevgridsatx=0onalog-logscale. ........................... 57 4-16Spectralbehavioroftherelativeerrorincomputingthenumericalsolutionforthetwo-sided1-DanisotropicproblemusingRemesgridsandPade-Chebyshevgridsatx=1onalog-logscale. ........................... 58 4-17AplotoflogarithmoftheL2errorvs.kfork=3;:::;15usingthesolutionfork=16asabenchmark. ............................... 59 4-18AplotoflogarithmoftheL2errorvs.kfork=3;:::;16usingthesolutionfork=17asabenchmark. ............................... 60 4-19Aplotofthecomputedsolutionforthe2-danisotropicproblemwithk=6Remesgridstepsinthex-directionandM=100gridstepsinthey-direction. 61 9

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Inthiswork,wepresenttechniquesthatapplytoreceiver-targetedproblemssuchasingeophysicalexploration.Insuchapplications,onewishestoconstructanaccurateimageoftheearth'sprole.Oneusuallysetsupasystemofsignalsourcesandreceiversandtheunderlyingpde'saresolvedtoobtainanalyticsolutionsatthereceiverlocations.Thesearethencomparedtothereceiveddataandtheguessfortheearth'sproleisadjustedaccordingly.Oneneedstosolvetheseproblemsrepeatedlyandinanecientmanner.Thiscallsfortheuseofnon-uniformgridswithsomekindofspectralmatching.Inourwork,wehaveanalyzedtheerrorconvergenceratewhensuchnon-uniformspectrallymatchedgridsareusedforthesereceiver-targetedproblems.WehavealsodevelopedanewsetofgridswhichwecallRemesgridsthatprovetobeextremelyusefulinproblemsoversemi-innitespectralintervals.Theconstructionofthesegridsisoutlinedandsoalsotheirapplicationtodeltafunctionsignalsourceproblemshasbeenstudiedandanalyzedtoobtaintheerrorconvergencerate.Towardstheendourwork,wehaveappliedthesegridstoanisotropicproblemswiththegoalofstudyingtheirconvergencerates. 10

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Remotesensingisanextremelyusefultoolforscientistsandengineers.Ithelpsinseveralareasofexplorationincludinggeophysicalexplorationwherescientiststrytoconstructimagesoftheearth'sprole.Typically,ingeophysicalexploration,onesetsupasystemofsignalsourcesandreceiversoveranareaoftheearth'ssurfacewhoseimageisdesired.Signalsaresentintotheearth'scrustandthereectedsignalsarereadbythereceivers.Basedonthereceiveddataonecanconstructanimageoftheearth'sprole.Thisinvolvessolvingcertainsetofpartialdierentialequations(PDE's)whosecoecientsdependontheearth'sprole.Usuallyonestartswithaguessfortheearth'sprole,solvestheseequations,andthencomparestheanalyticalsolutionatthereceiverlocationswiththereceiveddatatoadjusttheguessoftheearth'sproleappropriately.Assuch,oneneedstosolvetheseequationsrepeatedly,quicklyandaccuratelyatthereceiverlocations.ConventionalnitedierencetechniquesofsolvingPDE'sareslowandyieldasolutionovertheentiredomain.However,wewishtocomputerelativelyfastersolutionsthatareveryaccurateonlyatthereceiverlocations.Thisreceiver-targetedapplicationhasbeeninvestigatedbeforewhereanon-uniformdiscretizationofthedomainisapplied[ 2 8 9 ].Theideaisnotmerelytouseaverynerenementtowardsthelocationsofthesignalsourcesandreceiversbuttochoosegridswhichmatchthesolutioninthespectraldomain.Thefoundationofthistechniquehasbeenlaidin[ 8 9 ]wheretheexactconstructionofthesegridshasbeendetailed.ItisbasedonasuitablerationalapproximationoftheNeumanntoDirichletmap.Lateronthesegridshavebeenanalyzedfurtherin[ 2 12 ]whereasimpleideaoftensorproductgridsisusedtosolvemulti-dimensionalproblemsandtheerrorconvergenceratehasbeenstudiedfortheinnitespatialintervalcase.Anisotropicmediapresentchallengesintheapplicationofthesegridsandthesehavebeenstudiedinfurtherdetailin[ 4 ].Ourcurrentworkisaimedprimarilyatanalyzingtheconvergencerateoftheerrorinvolvedinapproximatingthesolutionwhenweuse 11

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Weshallbeginwithaverysimple1-DHelmholtzequationtoexplainthetheorybehindthecomputationofspectrallymatchedgrids.Considerthe1-DHelmholtzequationonthespatialinterval[0;L];L>0,withtheprescribedboundaryconditions: (2{1) Wedenetheimpedancefunctionofproblem( 2{1 )tobethesolutionattheleftend-pointx=0.Itiseasytoseethatthesolutiontoproblem( 2{1 )isgivenby, 13

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So,theimpedancefunctionisgivenby, Wewishtoapproximate( 2{1 )byatwo-pointnitedierenceschemeusingnon-uniformspectrallymatchedgrids.Inparticular,wewilldenethesolutionuat\potential"nodesxi;i=1;;k+1,withx1=0andthe\derivatives"uxatthederivativenodes^xi;i=0;;kwith^x0=0.Correspondingtothelocationofthepotentialnodes,wegetarstsetofgridstepswhichwewillcalltheprimarygridsteps.Inasimilarmanner,thelocationofthederivativenodesgiverisetoasecondsetofgridstepswhichwewillcallthedualgridsteps.Thus,denetheprimarygridsizestobehi=xi+1xi;i=1;;kandthedualgridsizestobe^hi=^xi^xi1;i=1;;k.Ourgoalistodeterminethevaluesforhi;^hiwhichleadtocertaindesiredspectralapproximationproperties.Rewriting( 2{1 )usingthisscheme,wegetthefollowingnitedierenceproblem: ^hiui+1ui ^h1u2u1 ^h1;uk+1=0: NotethatherewehaveimplementedtheNeumannboundaryconditionattheleftend-pointx=0asaghostpointcondition. Equation( 2{4 )revealsthattheFDsolutionatx=0,u1,isadiscreterationalfunctionof,fk()seeforexample[ 10 ].Thisrationalfunctiondependsontheparametershi;^hithatareyettodetermined.Alsorecallthatthesolutiontothecontinuousproblem( 2{1 )atx=0wasgivenbyacontinuousfunctionofasin( 2{3 ). 14

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2 ]. TheFDapproximation( 2{4 )canberewrittenmorecompactlyinmatrixformas whereu=(u1;;uk)TandSisasystemmatrixwhoseentriesdependonhi;^hi.ItiseasytoseethatSisnotsymmetricandsowemakeasuitabletransformationtomakeitsymmetric.Ifweintroduceanewvariablewi=^h1=2iui;i=1;;k,thenwecanwrite( 2{5 )as whereHisnowasymmetrictridiagonalsystemmatrixoftheform where 15

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SupposetheeigenvectorsofHaresiandthecorrespondingeigenvaluesarei,thenwecanwriteH=LDLTusingeigenvaluedecomposition,whereD=diagfigandL=[s1;;sk]Tistheorthogonalmatrixofeigenvectors.Wecannowsolveforwandhence,u1,usingtheabovedecomposition.Multiplying( 2{6 )byLTontheleftandusingH=LDLTcombinedwithLTL=I,weget, whereIisthecorrespondingidentitymatrix. Ifwerearrange( 2{9 ),thenweseethat isapartialfractionof[k1=k]formofourrationalfunctionwithyi=s2i=^h1;i=1;;kandsiaretherstcomponentsoftheeigenvectorssi. Theabovediscussionthensuggeststhefollowingpseudocodeforthecomputationofthespectrallymatchedgrids. Pseudocodeforcomputingthegrids: Intherststep,wecomputeaPade-Chebyshevrationalapproximation[ 6 ]ofa[k1=k]formtoourimpedancefunctionf()=u(0)=tanh(Lp 16

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2{10 )fromwhichwecanextractthevaluesforyi;i. Onceweknowthevaluesofyi;i,thenwereconstructthematrixHbysolvingtheinversespectralproblem.Weuseak-steprecursiveLanczosalgorithm[ 14 ]withreorthogonalizationtoavoidlossoforthogonalityoftheLanczosvectorsinniteprecisionarithmetic.NotethatweassumeanormalizationkXi=1s2i=1andcompute^h1=1 WethenuseEquations( 2{8 )recursivelytodeterminethevaluesofhiand^hifori=1;;k.Itiseasytoseethat AlloftheabovecalculationsaredoneinMATLAB. Wecomputethesegridsforseveralvaluesofkoverthespatialintervalx=0tox=0:5andspectralinterval=1to=100.Figures( 2-1 )and( 2-2 )showthecorrespondingPade-Chebyshevgridsfork=5andk=15gridsteps.Notethatthegridsareactuallystaggeredeventhoughthiswasnotoneoftherequirementsimposedwhenwewereconstructingthesegrids. ItcanbeshownthattheconvergenceofthePade-Chebyshevrationalapproximationissuperexponential[ 9 ].Finally,thespectralintervalofinterestmaycontainsomeresonancesioff().Ifn;0nkarethenumberofsuchresonances,thenweprescribetherstntermsoffk()tocontaintheresonances,thatis,welookforarationalapproximationoftheform 17

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Aplotofthestaggeredgridovertheinterval[0;0:5]fora[4=5]Pade-Chebyshevrationalapproximation,k=5. insteadof( 2{10 ),seeforexample[ 2 ]. 18

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Aplotofthestaggeredgridovertheinterval[0;0:5]fora[14=15]Pade-Cheby-shevrationalapproximation,k=15. such,itwouldbedesirabletoaskthequestion,\Isthereaniteelementmethodwhichisequivalenttothenitedierencescheme?"Inthissection,weattempttomathematicallyformulatetheproblemandthentryvariousapproachesinordertoanswerthisquestion. Considerthecontinuous1-DHelmholtzproblemon[0;1]withDirichletboundaryconditions: (2{13) 19

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2{13 )ledtoexponentialconvergenceattheleftend-point.Theresultingschemewasstaggeredandconsistedofasystemofprimaryanddualgridlinesgivenrespectivelybyx=xjandx=^xj.Thisnitedierenceapproximationledtoakksystemmatrixfor( 2{13 ).Oneofthekeypropertiesofthismatrixisthatitisatridiagonalmatrixthatissymmetricwithrespecttothe^h-weightednorm. Inthissection,weformulate( 2{13 )invariationalform.Thiswillenableustodevelopaniteelementtechniquewhichwillbeequivalent(inthesenseofsystemmatrices)tothenitedierenceschemethathasbeenpreviouslydeveloped. Letusbeginwithasecond-ordercontinuousvariationalformulationwherewemultiply( 2{13 )byatestfunctionv2Vandintegratebypartsover[0;1].Here Thisyieldsthefollowingequation whichcanberewrittenas wherehv;wi=Z10vwdx.Thus,acontinuoussecond-ordervariationalformulationcanbestatedas 20

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(2{17) Wenowpresentadiscretesecond-ordervariationalformulationcorrespondingtotheaboveapproach.First,letusmakeafewnotationaldenitions.LetIj=[xj;xj+1]and^Ij=[^xj1;^xj],8j=1;;kbeapartitionof[0;1]usingthesystemofprimaryanddualgridsasderivedforournitedierencescheme.Letfjgkj=1bethesetofstandardbasishatfunctionswherej(xi)=ijandjislinearoneachIi.HereijdenotesthestandardKroneckerdelta.Similarly,letf^jgkj=1bethesetofstandardbasishatfunctionswhere^j(^xi)=ijand^jislinearoneach^Ii.Now,deneVhVtobethefollowingsubspace: and,UhVtobe: Notethatj2Uhand^j2Vhandtheyformthebasisfortherespectivespaces.Thus,asecondorderdiscretevariationalformulationfor( 2{17 )is Finduh2Uhsuchthat8vh2Vhhu0h;v0hi+huh;vhi=vh(0) (2{20) Sincef^jgkj=1andfjgkj=1formabasisforVhandUhrespectively,wecanwrite 21

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(2{21) Combining( 2{20 )and( 2{21 )andusingvh=^i;8i=1;;k,wegetthefollowingsetofequations: Theabovesetofequationscanbewritteninmatrixformleadingtoakksystemmatrix.However,ifwecomputeh0j;^0iiandhj;^ii(8i=1;;k),wendthattheresultingsystemmatrixisnottridiagonal(unlikethesecondordersystemmatrixforthenitedierencescheme).Assuch,wehavenotyetbeenabletondasecondorderniteelementformulation( 2{20 )whichyieldsexactlythenitedierenceformulationfor( 2{13 ).Mostlikelywewillneedtousearstorderformulation.Thisisasubjectofongoingresearch.Noteherethatitisnotnecessarytohavethematricesexactlyalike{whatweneedisforthesolutionstobethesame. 22

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3.2.1TheProblem (3{1) 23

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4 ]: wherefistheimpedancefunctiondenedby Below,Ioutlineaquickproofofthisresult. 3{1 )andwriteitbyseparationofvariablesas Then,itfollowsfromEquation( 3{1 )that So,wecanwrite (3{7) 24

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3{3 ). 3{1 )onasystemofprimaryanddualgridlinesgivenrespectivelybyx=xjandx=^xjasdescribedinChapter2.Thisyieldsthefollowingsemidiscretizedversionof( 3{1 ): (3{9) 3{9 )toobtain[ 4 ] wherefkisthediscreteimpedancefunctiongivenby Onceagain,Ioutlineaquickproofofthisbelow: 3{9 )andwriteitbyseparationofvariablesas: Then,itfollowsfrom( 3{9 )that ^ddGj 25

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SinceWjis~L-periodiciny,itfollowsthat!j=2j=~L,forj=0;1;.Hence,wehave Usingtheotherboundarycondition,namely,(dW)j0(y)=(y)combinedwithEquation(2.8)in[ 4 ],wecansolveforGjCjandGjDjwhenceweget( 3{10 ). Thetwoproofsoutlinedaboveeasilyextendtothecaseofunboundedspectrumforthedata(y)(y)=1Xj=ajei!jy!,thatis,wehavethefollowingtworesults: 3{1 )forLaplace'sequationontherectangle(0;L)(0;~L).Supposethatfajg1j=isasequenceinl2.Assumethatthedataisgivenby: 3{4 )andtheconvergenceisinthesenseofL2(0;~L). 3{1 )givenby( 3{9 ).Supposethatfajg1j=isasequenceinl2.Assumethatthedataisgivenby:

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3{11 )andtheconvergenceisinthesenseofL2(0;~L). 4 ]: Furthermore,forellipticproblemsonanitespectralintervalofinterest[1;2]whichistotherightoftheorigin(sothatthepolesareoutsidethespectralinterval),thePade-Chebyshevnear-bestrationalapproximationhasexponentialconvergenceinkgivenby[ 6 ]: 27

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3{1 ),weseethat=2=1=!2m=!21=m2since!j=2j=~L.Thus,combining( 3{20 )and( 3{21 ),wegetthefollowingexponentialerrorboundfortheDirichletdatawhenthedataspectrumisbounded: Theobviousquestionthatarisesthen,iswhatdowedointhecaseofasemi-innitespectralintervalofinterest[1;1)?Itiscrucialtoanswerthisquestionforthecaseofunboundeddataspectrum,sinceinthiscaseourestimatefortheerrorboundin( 3{21 )isnolongervalid.InordertoanalyzetheconvergencerateofourDirichletdataerrorfortheunboundeddataspectrumcase,weneedanerrorestimateonthesemi-innitespectralinterval.Sofar,weareunawareofanerrorestimateandhence,wepresentthefollowingpropositionwhichdescribestheconvergenceanalysisoftheDirichleterrorinmoregenerality. 3{1 )wherethedatahasunboundedspectrumandisgivenby( 3{16 ),theDirichletdataerrorisboundedbyE(k;!21),thatis, 3{20 ).Inthiscase,usingLemmas 3.2.2.1 and 3.2.2.2 ,wehavethefollowingestimate: 28

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ThesecondequalityintheaboveprooffollowsfromParseval'sequalityintheunboundedspectrumcase.Further,usingthefactthattheerrorfunction,f()fk(),isL1on[!21;1),wecanestimatetheerrorboundasinthethirdinequalityshownabove.Finally,usingthemaximumerrorestimatein( 3{23 ),wearriveattheconclusion. 1 13 ]. 5 ]guaranteesthatthisapproximationisuniquelyoptimal.Althoughwewilluseaversion 29

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Suppose isanNthdegreepolynomialthatleadstoanerrorfunctionwithN+2levelextremawithvaluesatN+2giventestpointsx1;x2;;xN+2(whereusuallyx1;xN+2aretheendpointsoftheintervalofinterest).Then,weneedtosolvethefollowingsetofN+2linearequations: RemesalgorithmistypicallystartedbychoosingthemaximaoftheNthdegreeChebyshevpolynomialastheinitialsetoftestpoints.Theresultantpolynomialapproximationiscalledthe\Chebyshevapproximation"orthe\minimaxapproximation". 30

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ThisisachievedinMathematicausingthe\GeneralMinimaxApproximation"commandunderthe\NumericalMath\Approximations""package.Intherststep,arationalapproximationisconstructedusingthe\RationalInterpolation"command.ThisrstapproximationisthenusedtogenerateabetterapproximationusingaschemebasedonRemes'algorithm.Whenweusedtheabovecommandtogeneratethisapproximation,weobservedthatMathematicaforcestwoofthetestpoints,x1;xN+2,tobetheend-pointsoftheintervalofapproximation.Further,itdidnotallowforasemi-inniteintervalofapproximation.Thismadeitcleartousthatinordertondanapproximationthatisoptimalontheentiresemi-inniteinterval,wewouldneedtoincreasethelengthoftheintervalofapproximationbyshiftingtherightend-pointfarenoughsothattheapproximationerrorcurvebeginstoturnbacktowardszero.Also,ifwechoosetherightend-pointtobetoofartotherightthentheerrorcurveovershootsandtherightend-pointisnolongeranextremum.So,weneededtoadjustthelengthoftheapproximationintervalappropriately.Wewillrefertothisoptimallengthintervalasan\intervalofjustrightlength".Examplesoftheseapproximationsareillustratedinthenextsubsection. 31

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3-1 )and( 3-2 )showtheerrorplotfork=7andk=8respectively. Figure3-1. Aplotoftheerrorbetweenthetrueimpedancefunctionandnumericallycomputedrationalapproximationfork=7. Wethenusedthemaximumerrorestimatefromeachsuchplotforvariousvaluesofktocreateaplotofthelogarithmoftheabsolutevalueofthelogarithmofthemaximumerroragainstthelogarithmofthekvalues.Thisplotwasalmostastraightlinewithslopecloseto0.5indicatingthatthemaximumabsoluteerrordecaysasexponentialofthesquare-rootofthemeshsize.Figure( 3-3 )showsthisplotfork=3;:::;17. 32

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Aplotoftheerrorbetweenthetrueimpedancefunctionandnumericallycomputedrationalapproximationfork=8. 3.4.1SomeNumericalResults Considerthefollowingproblemontheunitsquare[0;1][0;1]: (3{28)

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Aplotoflog(abs(logerror))vs.logkfork=3;:::;17. Westudiedthisproblemnumericallyinordertoestimatetheerrorconvergencerateandthencompareitwithspectraltechniques.Here=1,andweusedRemesgridsfordierentvaluesofgridsizes,k,alongthex-directionandaveryneuniformgridinthey-direction. Inadditiontostudyingtheconvergencerateoftheminimaxrationalapproximationitself,wealsostudiedtheerrorconvergenceratefortheerrorincomputingthenumericalsolutiontotheproblem( 3{28 ).WeusedtheminimaxrationalapproximationtoourimpedancefunctionfromMathematicatocomputeRemesgridsforallvaluesofgridstepsizesk=3;;17.Wethenusedthesegridsalongthex-directionandveryneuniformgridalongthey-directiontocomputeanumericalnitedierencesolutiontoourproblem.Figure( 3-4 )showsaplotofthelogarithmoftheL2errorincomputingthenumericalsolutionforvariousvaluesofbothRemesanduniformgridsizes.Sincewedonothaveatrueanalyticsolutionathand,weusedthenumericalsolutionobtainedforRemesgrid 34

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Figure3-4. AplotoflogarithmoftheL2errorvs.kfork=3;:::;16usingthesolutionfork=17asabenchmark. Figure( 3-4 )indicatesaconvergencerateofexponentialinthesquare-rootoftheRemesmeshsizefortherstfewvaluesofkfromk=3;;12.Thereafter,theerrorcurvebeginstomoveconcavedownwardindicatingthatthenumberofuniformgridsteps,M,arenotenoughtocapturethespectrumofthedeltafunctionboundarydataforlargervaluesoftheRemesgridsize.Weareseeingthesameexponentialconvergenceoneexpectswithanitespectralinterval[ 12 ].Thesearesomeofthecomputationalresourceslimitationsthatwefacedincomputingthenumericalsolutiontotheproblem( 3{28 ). 35

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3-5 )and( 3-6 ). Figure( 3-6 )clearlyshowsthattheoverallL2errorincomputingthenumericalsolutionusingRemesgridsismuchlowerthanthecorrespondingerrorusingthePade-Chebyshevgrids.Also,notethattheplotisastraightlinefortheRemesgridsuptoaboutk=12whiletheoneforthePade-ChebyshevgridsisslightlyconcaveupindicatingaslowerconvergencerateforthePade-ChebyshevgridsthantheRemesgrids.Also,itindicatesaconvergencerateofexponentialinthesquare-rootofthemeshsizefortheRemesgrids.Onceagain,theplotcurvesconcavedownafterk=12becausethenumberofuniformgridstepsarenotenoughtocapturethespectrumofthedeltafunctionboundarydatawhichhasaninnitespectrum. 36

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AcomparisonplotoflogarithmoftheL2errorvs.kfork=3;:::;16usingthesolutionusingRemesgridsfork=17asabenchmarkandM=100;000uniformstepsalongthey-direction. 37

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AplotoflogarithmoftheL2errorvs.p 38

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39

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4 ]: Notethatifa=0(or==2),wegetbacktoourmodelproblem( 2{1 ).Itisthepresenceoftherstorderderivativeterm,ux,thatmakesthisproblemanisotropic.Fortheisotropiccase(a=0),itisobviousthattheaboveequationhasthelinearlyindependentsolutions Fortheanisotropiccase(a6=0),thesolutionsareoftheform, wherebsatises So,b=iap Inthenextsubsection,weconsideratwo-sidedanisotropicproblemoftheform( 4{2 )withsomeboundaryconditionsonaniteinterval.Towardstheendofthesubsection,wepresentnumericalresultsonthecorrespondingproblem. 40

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(4{6) Letusdecomposeoursolution,u,intoitsoddandevenpartsaboutx=1=2.Thenwecanwritethesolutionas whereuo;uearetheoddandevenpartsrespectively.Fora1-Disotropicproblemithasbeenshownin[ 9 ]thatifweinterchangetheprimaryanddualgridsandthenusethemtocomputethesolutiontotheNeumannproblem,thenexponentialconvergenceattheleftendpoint(x=0)isstillmaintained.Thiscan,therefore,beusedtocomputethesolutiontothetwo-sidedisotropicproblembysplittingthesolutionintoitsoddandevenparts.SincetheoddpartofthesolutionsatisestheDirichletproblemandtheevenpartofthesolutionsatisestheNeumannproblem,wecaneasilycomputebothofthesepartsusingonlyonesetofgrids.Thisfactmakesiteasytocomputethesolutiontotheanisotropictwo-sidedproblemwherewesplitthesolutionintoitsoddandevenpartsasabove.Inthiscase,pluggingbackinto( 4{6 ),weseethattheoriginalequationbecomesasetofcoupledequationsasfollows[ 4 ]: 41

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(4{8) Now,observethatifwelettheoddpartofthesolution,uo,liveontheprimarygridfxigandtheevenpartofthesolution,ue,liveonthedualgridf^xig,thentheirrstorderderivatives,uoxanduex,liveonthedualandprimarygridsrespectively.So,allofthetermsineachoftheabovesetofcoupledequationslieonthesamegridandhence,summingupthesetermsdoesmakesense.Hence,weusetheprimaryfxiganddualf^xiggridsrespectivelyfortheoddandevenpartsUoandUeofthesolutionon(0;1=2)andwritethefollowingnumericalFDapproximationto( 4{8 ). (4{9) Ithasbeenshownthatthisnitedierencesolutionwillconvergeexponentiallytothetruesolutionattheboundarypointsx=0andx=1. Here,wepresentsomeresultsfromnumericalexperimentsthatwereconductedforthecurrentproblem.WecomputedthegridstepsusingaPade-ChebyshevrationalapproximationforthespatialintervaloflengthL=1=2fork=2;:::;25andaspectral 42

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Acomparisonplotoftherealpartsofthetrueandnumericallycomputedsolutionsforthetwo-sided1-Danisotropicproblemfork=6;=1. intervaloflength1=1to2=100.Wethencomputedthesolutiontothetwo-sided1-Danisotropicproblem( 4{6 )usingthenumericalnitedierenceapproximation( 4{9 )bysplittingthetruesolutionintoitsoddandevenpartsforseveralvaluesofk.Figure( 4-1 )showsanoverlayplotoftherealpartsofthetrueandnumericallycomputedsolutions.Figure( 4-2 )showsasimilarplotfortheimaginarypartsofthetrueandnumericallycomputedsolutions,whileFigure( 4-3 )showsthesameforthemagnitudesofthetrueandnumericallycomputedsolutions.Finally,Figure( 4-4 )showsaplotoftheerrorincomputingthenumericalsolution.Alloftheseplotsaredrawnfork=6gridstepson[0;1=2]and=1.Figures( 4-5 ),( 4-6 ),( 4-7 )and( 4-8 )showsimilarplotsfork=13. 43

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Acomparisonplotoftheimaginarypartsofthetrueandnumericallycomputedsolutionsforthetwo-sided1-Danisotropicproblemfork=6;=1. Inadditiontocomparingthenumericallycomputedsolutiontothetrueanalyticsolution,wealsostudiedthespectralbehavioroftherelativeerrorincomputingthenumericalsolution.Inparticular,wecomputedthenumericalsolutionbyodd-evensplittingforthetwo-sided1-Danisotropicproblemusingk=6spectralgridstepsforseveralvaluesofovertheapproximatingspectralinterval=1to=100.TheresultsfromthisstudyareindicatedinFigure( 4-9 ).Fromtheseplots,weseethattherelativeerrorattheendpoints,x=0andx=1,getsexponentiallyworseasthespectralparametervaluerangesover=1to=100.NoteherethatthegridswerecomputedbyusingaPade-Chebyshevrationalapproximationtoourimpedancefunctionoverthisspectralintervalofapproximation. Inchapter3,weintroducedanewsetofgridswhichwecalledRemesgrids.Wehaveseenthattheyproveveryusefulforproblemsoversemi-innitespectralintervals.Earlier,weusedthesegridsonasample2-disotropicproblem.Wenowwishtoapplythese 44

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Acomparisonplotofthemagnitudesofthetrueandnumericallycomputedsolutionsforthetwo-sided1-Danisotropicproblemfork=6;=1. gridstothetwo-sided1-danisotropicproblem( 4{6 )andanalyzetheerrorconvergencepropertiesaswedidwhenPade-Chebyshevgridswereusedonthesameproblem.SinceRemesgridsareconstructedbyanoptimalrationalapproximationoftheimpedancefunction,weexpecttoseethemperformingbetterthanthetraditionalPade-Chebyshevgridswhenappliedtothe1-danisotropicproblem. Inordertoperformouranalysis,werstconstructedaRemesapproximationtotheimpedancefunction( 2{3 )overanitespectralintervalof1=1to2=100.WeusedL=1=2incomputingtheapproximation.WethenusedtheapproximationtoconstructRemesgridsforseveralvaluesofkfromk=3tok=17.Thecomputedgridswerethenusedtonumericallysolvethetwo-sided1-danisotropicproblem( 4{6 )usingthenitedierenceapproximation( 4{9 ).Weusedourusualodd-evensplittingofthesolutiontoformacoupledsystemofnitedierenceequations.ThegoalwastoperformasimilarnumericalanalysiswhentheseRemesgridsareusedtosolveourproblem.We 45

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Acomparisonplotoftheerrorbetweenthetrueandnumericallycomputedsolutionsforthetwo-sided1-Danisotropicproblemfork=6;=1. used=1;=2,anda=0:5inourcomputations.Figure( 4-10 )showsanoverlayplotoftherealpartofthetrueandnumericallycomputedsolutions.Figures( 4-11 )and( 4-12 )showsimilarplotsfortheimaginarypartandthemagnitudeofthetrueandnumericallycomputedsolutionsusingtheRemesgrids.Finally,gure( 4-13 )showsthemagnitudeofthenumericalerrorincomputingthesolutionusingRemesgrids. 46

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Acomparisonplotoftherealpartsofthetrueandnumericallycomputedsolutionsforthetwo-sided1-Danisotropicproblemfork=13;=1. Inadditiontostudyingtheconvergencepropertiesoftherealpart,imaginarypart,solutionmagnitudeanderrormagnitudeofthenumericallycomputedsolution,wealsowishedtostudythespectralbehaviorofthissolutionoverthespectralintervalofinterest.Assuch,wecomputedthesolutiontothetwo-sided1-danisotropicproblemusingRemesgridswithk=6,a=0:5,=1,=2,andL=1=2forseveralvaluesofinthespectralintervalofinterest1=1to2=100.Figure( 4-14 )showsourresultantspectralbehaviorplot. WewantedtoseehowourRemesgridsperformedincomparisontothetraditionalPade-Chebyshevgridsforthetwo-sided1-danisotropicproblem.Sincethesegridsweredesignedtogivealmostaccuratesolutionsatthetwoend-points(receiverlocations)x=0andx=1,itmadesensetocomparetheoverallerrormagnitudesattheend-pointlocations.Tothiseect,wecomputedthemagnitudesoftheerrorsatx=0andx=1for 47

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Acomparisonplotoftheimaginarypartsofthetrueandnumericallycomputedsolutionsforthetwo-sided1-Danisotropicproblemfork=13;=1. thenumericallycomputedsolutionsusingboththePade-ChebyshevandtheRemesgrids.Thefollowingtablesummarizesourconclusion. Table4-1. Comparisonofsolutionerrormagnitudesforthetwo-sided1-danisotropicproblemusingRemesgridsovernitespectralintervalfor=1 ErroratUsingPade-ChebyshevgridsUsingRemesgrids 48

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Acomparisonplotofthemagnitudesofthetrueandnumericallycomputedsolutionsforthetwo-sided1-Danisotropicproblemfork=13;=1. approximationinterval1=1to2=100,samevaluesof=1,a=0:5,=1,=2,andsamespatialintervalL=1=2. RecallthattheRemesgridsprovedtobeveryusefulinproblemsoversemi-innitespectralintervals.Assuch,itwouldbeinterestingtolookathowtheseRemesgridswhichhavebeencomputedoversemi-innitespectralintervalsperformincomparisontothetraditionalPade-Chebyshevgridsoverawiderspectralintervalofinterest.Tobetterunderstandthis,weappliedRemesgridswithk=6gridstepscomputedoverthejustrightinterval1=1to2=2:95108tooursampletwo-sided1-Danisotropicproblem( 4{6 ).Wecomputedtheoverallrelativeerrorincomputingoursolutionatthetwoend-pointsx=0andx=1andcomparedthesetotherelativeerrorwhenPade-Chebyshevgridswereused.Thiscomparisonwasmadeoverawiderspectralinterval[5;105].Wepickedequi-spacedspectralparametervaluesuntil=100andthereafterweused50equallyspacedpointsinthelogspacefrom=100to=105.Figures( 4-15 ) 49

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Acomparisonplotoftheerrorbetweenthetrueandnumericallycomputedsolutionsforthetwo-sided1-Danisotropicproblemfork=13;=1. and( 4-16 )showacomparisonbetweentherelativeerrorplotsforthesolutionerroratx=0andx=1respectively.OneeasilyseesthateventhoughthePade-Chebyshevgridsperformbetterinitially,theRemesgridsperformmuchbetterovertheentirewiderspectralinterval.Inordertoquantifyourresults,wecomputedtherelativeerrorsincomputingthesolutionattheend-pointsfor=105.Weobservedthattherelativeerrorincomputingthissolutionatx=0usingRemesgridswasonly5:9%whereasthecorrespondingrelativeerrorwhenPade-Chebyshevgridswereusedwas137:61%.Inasimilarmanner,theoverallrelativeerrorinoursolutionatx=1usingRemesgridswasonly1:48%whilethatusingthePade-Chebyshevgridswas20:97%.Wealsodidsimilarcalculationsfork=13gridstepsandthecorrespondingrelativeerrorsalongwiththepreviousonesaredescribedinTable( 4-2 ).So,forproblemsinvolving-functionsignalsourceswithinnitespectrum,usingRemesgridswillyieldmuchbetterresultsthanthePade-Chebyshevgrids. 50

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Spectralbehavioroftherelativeerrorincomputingthenumericalsolutionforthetwo-sided1-Danisotropicproblem. Table4-2. Comparisonofsolutionerrormagnitudesforthetwo-sided1-danisotropicproblemusingRemesgridscomputedoversemi-innitespectralintervalsfor=105 6x=0137:61%5:9%x=120:97%1:48% 13x=06:99%0:16%x=11:78%0:039914% 51

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Acomparisonplotoftherealpartsofthetrueandnumericallycomputedsolutionsforthetwo-sided1-DanisotropicproblemusingRemesgridsfork=6;=1. (4{10) 52

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Acomparisonplotoftheimaginarypartsofthetrueandnumericallycomputedsolutionsforthetwo-sided1-DanisotropicproblemusingRemesgridsfork=6;=1. Here,thereisa-functionsignalsourceatthepoint(0;1=2).Thesolutionisperiodicinyandthisisreectedbythelasttwoconditionsin( 4{10 ).Weuseodd-evensplittingtocomputetheoverallsolution.ThisisagainmotivatedbythefactthattheoddpartofthesolutionsatisestheDirichletproblemwhiletheevenpartofthesolutionsatisestheNeumannproblemandcomputingthegridsfortheDirichletproblemandthenusingthemfortheNeumannproblembysimplyinterchangingtheprimaryanddualgridstepsmaintainstheexponentialconvergenceatthereceiverend-points.Weemployasemi-discretizationofEquation( 4{10 )usingRemesgridsinthex-directionandaveryneuniformgridinthey-direction.TheRemesgridsarecomputedfromaRemesrationalfunctionapproximationoftheimpedancefunctiononasemi-innitespectralinterval.WeusedL=1=2,a=0:5andcomputedtheL2errorinapproximatingthesolutionattheleftedge,x=0,forseveralvaluesofkfromk=3tok=16.Figures( 4-17 )and( 4-18 ) 53

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Acomparisonplotofthemagnitudesofthetrueandnumericallycomputedsolutionsforthetwo-sided1-DanisotropicproblemusingRemesgridsfork=6;=1. showplotsofthelogarithmoftheL2erroragainstkandp Figure( 4-19 )showsasurfaceplotofthecomputedsolutiontoour2-danisotropicproblem( 4{10 )withk=6Remesgridstepsinthex-directionandM=100uniformgridstepsinthey-direction. 54

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Acomparisonplotoftheerrorbetweenthetrueandnumericallycomputedsolutionsforthetwo-sided1-DanisotropicproblemusingRemesgridsfork=6;=1. 55

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Spectralbehavioroftherelativeerrorincomputingthenumericalsolutionforthetwo-sided1-DanisotropicproblemusingRemesgrids. 56

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Spectralbehavioroftherelativeerrorincomputingthenumericalsolutionforthetwo-sided1-DanisotropicproblemusingRemesgridsandPade-Chebyshevgridsatx=0onalog-logscale. 57

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Spectralbehavioroftherelativeerrorincomputingthenumericalsolutionforthetwo-sided1-DanisotropicproblemusingRemesgridsandPade-Chebyshevgridsatx=1onalog-logscale. 58

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AplotoflogarithmoftheL2errorvs.kfork=3;:::;15usingthesolutionfork=16asabenchmark. 59

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AplotoflogarithmoftheL2errorvs.kfork=3;:::;16usingthesolutionfork=17asabenchmark. 60

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Aplotofthecomputedsolutionforthe2-danisotropicproblemwithk=6Remesgridstepsinthex-directionandM=100gridstepsinthey-direction. 61

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Inchapter2,theideaofspectrallymatchedgridswasintroduced.Wedescribedhowthesegridswerecomputedalongwithapseudocodewhichdetailsthecomputationsinvolved.Wealsosawsomeexamplesofthesegridswherethestaggerednessofthegridswasillustratedeventhoughitwasnotimposedapriori.TowardstheendofthechapterweattemptedtondanequivalentFEMwhichgivesexactlythesamesystemmatrixforourFDapproximationwhenthestandardbasishatfunctionswerecomputedoveroursystemofprimaryanddualgrids.WeconcludedthatsuchaFEMdoesn'texistsincethecorrespondingsystemmatrixwasnottridiagonal.However,wenotedthatperhapsarst-orderformulationmightbeneeded. Inchapter3,weintroducedanewsetofgridswhichwecalledRemesgrids.TheseweresubsequentlyusedincomputingnumericalnitedierenceapproximationtothesolutionofHelmholtzequationontheunitsquare.Westudiedthisproblemnumericallyingreatdetailandconjecturedthattheerrorincomputingoursolutionwasconvergingexponentiallyinthesquare-rootoftheRemesmeshsize.Thiswassimilartotheexponentialconvergenceoneseeswithanitespectralinterval.AcomparisontothePade-ChebyshevgridswasmadeandweillustratednumericallythattheRemesgridsoutperformedthePade-Chebyshevgrids.Wealsoremarkedthatsofarweareunawareofanerrorestimateintherationalapproximationoftheimpedancefunctionoversemi-innitespectralintervals.Assuch,wepresentedamoregeneralresultwhichindicatesthattheoverallrelativeDirichletdataerrorincomputingthesolutiontoEquation( 3{1 )wasboundedbythismaximumerrorestimateintherationalfunctionapproximation. Inthelastchapter,weappliedourPade-Chebyshevgridstoasimpletwo-sided1-Danisotropicproblemwhereweuseodd-evensplittingtocomputetheoverallsolution.Ournumericalstudiesexhibitedconvergenceatthetwoendpoints.Wealsostudiedthespectral 62

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Thereareseveralquestionsthatstillneedtobeanswered.Wewouldliketoworkonndingtheanswerstothesequestionsinthefuture.Forinstance,wewouldliketondoutifanequivalentFEMexists.WewouldalsoliketobeabletocomputetheRemes 63

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Alltherelevantprogramcodesareattachedbelow. 1. MAPLEcodethatcomputestheyiandivaluesintherationalfunctionapproximationoftheimpedancefunction.

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ExampleofMathematicacodethatcomputestheoptimalRemesrationalfunctionapproximationalongwiththemaximumerrorinapproximationovertheintervalofjustrightlength.ThecodelistedheredoesthisforL=1=2,k=13,andtheapproximationintervalofjustrightlengthis[1;1:551013].

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MATLABlerunscript.mwhichcomputestheprimaryanddualgridstepsgiventhevaluesofyi,iandk.

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MATLABleLanTri.mthatisneededfortherunscript.mcode.ItcomputestheLanczostridiagonalizationusingreorthogonalization.

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MATLABlebuildgrid.mthatisneededfortherunscript.mle.Thispieceofcodebuildstheprimaryanddualgridsgiventhevectorsthatformthetridiagonalmatrix.

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MATLABleanisotropic1 modied.m.Thislecomputesthesolutiontothetwo-sided1-Danisotropicproblemgiventhegridstepshi;^hi;;;a,and.

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MATLABleminimaxonlyoptsolvec.m.Thislecomputesthesolutiontothe2-Danisotropicproblemwitha-functionsignalsourceattheoriginusingRemesand/orPade-Chebyshevgrids.

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MATLABleanisotropic2d minimaxonlyoptsolvec.m.Thislecomputesthesolutiontothe2-Danisotropicproblemwitha-functionsignalsourceat(0;1=2).

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MATLABleschlum pres.m.Thislecreatesthespectralbehaviorplotfortherelativeerrorsatx=0andx=1forthenitespectralintervalforboththeRemesandPade-Chebyshevgrids.

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MATLABleschlum pres Remes PC.m.Thislecreatesthespectralbehaviorplotfortherelativeerrorsatx=0andx=1forbothtypesofgridsandthenplotsthemonaloglogscale.

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[1] N.I.Akhiezer,TheoryofApproximation,F.UngarPub.Co.,NewYork,1956. [2] S.Asvadurov,V.Druskin,andL.Knizhnerman,ApplicationofthedierenceGaussianrulestosolutionofhyperbolicproblems,JournalofComputationalPhysics,vol.158,no.1,pp.116{135,Feb.2000. [3] S.Asvadurov,V.Druskin,andL.Knizhnerman,ApplicationofthedierenceGaussianrulestosolutionofhyperbolicproblems.II.Globalexpansion,JournalofComputationalPhysics,vol.175,pp.24{49,2002. [4] S.Asvadurov,V.Druskin,andS.Moskow,Optimalgridsforanisotropicproblems,ElectronicTransactionsonNumericalAnalysis,vol.26,pp.55{81,2007. [5] K.Atkinson,AnIntroductiontoNumericalAnalysis,JohnWiley&Sons,NewYork,1989. [6] G.A.Baker,andP.Graves-Morris,PadeApproximants,Addison-WesleyPublishingCp.,London,1996. [7] L.Borcea,andV.Druskin,OptimalnitedierencegridsfordirectandinverseSturm-Liouvilleproblems,InverseProblems,vol.18,pp.979{1001,Apr.2002. [8] V.Druskin,andL.Knizhnerman,Gaussianspectralrulesforthethree-pointseconddierences:I.Atwo-pointpositivedeniteprobleminasemi-innitedomain,SIAMJ.Numer.Anal.,vol.37,no.2,pp.403{422,Dec.1999. [9] V.Druskin,andL.Knizhnerman,Gaussianspectralrulesforsecondordernite-dierenceschemes,NumericalAlgorithms,vol.25,pp.139{159,Aug.2000. [10] V.Druskin,andS.Moskow,Three-pointnite-dierenceschemes,PadeandthespectralGalerkinmethod.I.One-sidedimpedanceapproximation,MathematicsofComputation,vol.71,no.239,pp.995{1019,Nov.2001. [11] K.O.Geddes,BlockstructureintheChebyshev-Padetable,SIAMJ.Numer.Anal.,vol.18,no.5,pp.844{861,Oct.1981. [12] D.Ingerman,V.Druskin,andL.Knizhnerman,Optimalnitedierencegridsandrationalapproximationsofthesquareroot.I.Ellipticproblems,Comm.onPureandAppl.Mathematics,vol.LIII,pp.1039{1066,2000. [13] C.B.Muratov,andV.V.Osipov,Optimalgrid-basedmethodsforthinlmmicromagneticssimulations,JournalofComputationalPhysics,vol.216,pp.637{653,2006. [14] B.N.Parlett,TheSymmetricEigenvalueProblem,PrenticeHall/SIAM,Philadelphia,1998. 105

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Theauthor,AdnanH.Sabuwala,wasborninMumbai,India,on18thNovember,1978.HeistheonlychildofHatimA.SabuwalaandFatemaH.Sabuwala.Helivedtherefor22yearswherehecompletedhisB.Tech.inelectricalengineeringfromIndianInstituteofTechnology,Bombay,in2000.HethenjoinedtheElectricalandComputerEngineeringDepartmentattheUniversityofFloridainAugust,2000asamaster'sstudent.HethengraduatedwithanM.S.inelectricalandcomputerengineeringfromUniversityofFloridainDecember,2002.HeworkedwithDr.JohnG.HarrisonaprojectsponsoredbyMotorola,Inc.,forhisdegree.Thereafter,hejoinedtheMathematicsDepartmentattheUniversityofFlorida,wherehewasrstenrolledasamaster'sstudent.HegraduatedwithanM.S.inmathematicsinAugust,2004.Hehastaughtseveralclassesaspartofhisteachingassistantship.SomenotablementionsincludeCalculusII,CalculusIII,ElementaryDierentialEquations.Hehaswonthedepartmentalteachingcerticateofexcellenceintheacademicyear2004-2005andsubsequentlywontheuniversity-widegraduatestudentteachingawardintheacademicyear2005-2006.HewasadmittedtothedoctoralprograminAugust,2004andgraduatedwithaPh.D.inmathematicsfromUniversityofFloridainMay,2008.HeisnowanassistantprofessorofmathematicsatCaliforniaStateUniversity,Fresno. 106