Stability and Convergence of a Wavelet-Galerkin Me

StabilityandConvergenceofaWavelet-GalerkinMethodfortheSidewaysHeatEquationTeresaReginskaLarsEldenyJanuary12,1999AbstractWeconsideraninverseheatconductionproblem,theSide-waysHeatEquation.ThisisaCauchyproblemfortheheatequationinaquarterplane,withdatagivenalongthelinex=1,wherethesolutioniswantedfor0x1.Theproblemisill{posed,inthesensethatthesolution(ifitexists)doesnotdependcontinuouslyonthedata.Wediscussthethestabil-ityandconvergencepropertiesofawavelet-Galerkinmethodforsolvingthesidewaysheatequation.ThewaveletsareofMeyertypethathavecompactsupportinfrequencyspace.Previousstabilityresultsforthismethodweresuboptimal.Weshowthatwithadditionalassumptionsconcerningthesmoothnessofthesolution,andconcerningthedenitionofthewavelets,wecanobtainalmostoptimalerrorestimates.1IntroductionWeconsiderthesidewaysheatequationinthequarterplane,8:uxx=ut;x0;t0,u(x;0)=0;x0,u(1;t)=g(t);t0ku(x;)kL2isboundedasx!1.(1.1)InstituteofMathematics,PolishAcademyofSciences,00-950Warsaw,Poland.PartoftheworkofthisauthorwasdonewhilevisitingtheDepartmentofMathe-matics,LinkopingUniversitywithintheframeworkofanexchangeprogrambetweenthePolishandSwedishAcademiesofScience.yDepartmentofMathematics,LinkopingUniversity,S-58183Linkoping,Sweden12AsolutionisaL2(0;1)valuedfunctionx!u(x;)2DL2(0;1)forx0,whereD=ff2L2(0;1):ddtf2L2(0;1);f(0)=0g.Physically,(1.1)correspondstoasituationinwhichtheend-pointx=0isinaccessible,butforwhichonecanmake(internal)measurementsatx=1.Ofcourse,sincegismeasured,therewillbemeasurementerrors,andwewouldactuallyhaveasdatasomefunctiongm2L2,forwhichkgmgkL2;(1.2)wheretheconstant0representsaboundonthemeasurementerror.Notethat,althoughweseektorecoveruonlyfor0x1,theproblemspecicationincludestheheatequationforx1togetherwiththeboundednessatinnity.Sincewecanobtainuforx1,alsoux(1;)isdetermined.Thuswecanconsider(1.1)asaCauchyproblemwithappropriateCauchydata[u;ux]givenonthelinex=1.Theproblem(1.1)isill-posed:thesolution,ifexists,doesnotdependcontinuouslyonthedatagm.However,weassumethatfortheexactdatagthesolutioninL2exists.Moreover,werequirefortheunknownsolutionapriorismoothnessconditionku(0;)kpM;(1.3)wherekkpdenotethenormintheSobolevspaceHp,p2R+.Inthefrequencyspacetheproblemcanbeformulatedasfollows:Letthefunctionsgm(t);g(t);andu(x;t)beextendedtothewholerealtaxisbydeninggm;g,andu(x;)tobezerofort0.ApplyingtheFouriertransformwithrespecttottotheproblem(1.1)withgreplacedbygmwegetbuxx(x;)=ibu(x;);0x;1;11;bu(1;)=bgm();andbujx!1bounded(1.4)wherebu(x;)=1p2Z11u(x;t)eitdt;2R:Ifasolutionof(1.1)exists,itisgivenbytheformulabu(x;)=e(1x)pibgm();(1.5)whereqi=(1+sign()i)qjj=2=:()3Sincetherealpartofisnonnegativeandtendstoinnityasjjtendstoinnity,theexistenceofasolutiondependsonarapiddecayofbgm()athighfrequencies.Thequantitativeaprioriinformation(1.3)concerningu,forbutakestheform(Z11(1+2)pjbu(0;)j2d)12M:(1.6)Forastablenumericalapproximationofthesolutionsomeregularizationprocedurehastobeapplied(cf.[1],[2][3][4],[5]).Optimalerrorboundswereinvestigatedin[6].Forarecentsurveyonnumericalproceduressee[7].Wavelettechniquesforapproximatingsolutionsof(1.1)andsimilarproblemshavebeenusedin[8],[9],[10],[11].Thelasttwopapersde-scribeawavelet{Galerkinmethod,whereprojectionsonMeyerwaveletsubspacesareapplied.Itwasdemonstratedtherethatusingawavelet{Galerkinapproachwecansolve(1.1)ecientlyandinanumericallystableway,withoutintroducinganyhigh-frequencycomponents.Wealsogavearuleforchoosinganappropriatewaveletsubspacedepend-ingonthenoiselevelofthedata.However,thetheoreticalresultcon-cerningerrorestimatesandconvergencewereuptonowunsatisfactory.However,inpracticalcomputations,see[11]theresultswerebetterthanpredictedbytheory.Thepurposeofthispaperistoimproveconvergenceestimates,togetalmostoptimalorderofconvergence.Tothisendtwoadditionalcon-ditionsareintroduced:anapriorismoothnesscondition(1.3)concern-ingtheunknownsolutionandanadditionalassumptionontheMeyerwaveletgeneratingsubspacesappliedinGalerkinmethod.Theoutlineofthepaperisasfollows.First,inSection2wereviewsomewavelettheoryintroducingMeyerwaveletsandlistingtheproper-tiesthatmakethemusefulforsolvingourproblem.Then,inSection3wedescribethewavelet{Galerkinmethod.Themaintheoremsconcern-ingapproximationoftheexactsolutioninthewaveletsubspacesandtheerrorboundforthedistancebetweentheexactandtheGalerkinsolutionsareprovedinSection3.3.Inordertosimplifyproofs,someauxiliarylemmasaregiveninSection3.2.Afewnumericaltestsin-vestigatingtheinuenceofthesmoothnessofthefunctionu(0;)onnumericalresultsarepresentedinSection4.42WaveletsAmulti-resolutionanalysis(MRA)ofL2(R)isdenedtobeasetofincreasing,closedsubspacesV1V0V1suchthat(a)T11Vj=0;andS11Vj=L2(R);(b)f()2V0ifandonlyiff(2j)2Vjforallj2Zandf()2V0,(c)f()2V0ifandonlyiff(k)2V0forallk2Z,(d)thereexistsascalingfunction2V0suchthatf(k)gk2ZformsanorthogonalbasisinV0.ItfollowsthatVj=spanfjkgk2Z;jk(x)=2j2(2jxk);j;k2Z:Moreover,thereexiststhewaveletfunction2L2determinedbysuchthatthesetoffunctionsjk(x)=2j2(2jxk);k2Zforxedj