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

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Title:
A Convergence Study of Spectrally Matched Grids in the Presence of Non-Smooth Data and Anisotropy
Creator:
Sabuwala, Adnan H.
Place of Publication:
[Gainesville, Fla.]
Publisher:
University of Florida
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Language:
english
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1 online resource (106 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Gopalakrishnan, Jayadeep
Committee Members:
Moskow, Shari
Pilyugin, Sergei
Hager, William W.
Brumback, Babette
Graduation Date:
5/1/2008

Subjects

Subjects / Keywords:
Approximate values ( jstor )
Approximation ( jstor )
Error rates ( jstor )
Logarithms ( jstor )
Mathematics ( jstor )
Matrices ( jstor )
Minimax ( jstor )
Polynomials ( jstor )
Rational functions ( jstor )
Signals ( jstor )
Mathematics -- Dissertations, Academic -- UF
anisotropy, fd, geophysics, grids, optimal, pde, remes, spectral
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Electronic Thesis or Dissertation
bibliography ( marcgt )
theses ( marcgt )
Mathematics thesis, Ph.D.

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. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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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
Statement of Responsibility:
by Adnan H. Sabuwala

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University of Florida
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University of Florida
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Copyright Adnan H. Sabuwala. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Embargo Date:
5/31/2009
Classification:
LD1780 2008 ( lcc )

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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