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RegularizedFixed-PointIterationsforNonlinearInverseProblemsS.S.Pereverzyev¤,R.PinnauyandN.SiedowzAugust,2005AbstractInthispaperweintroduceaderivative-free,iterativemethodforsolvingnonlinearill-posedproblemsFx=y,whereinsteadofynoisydatay±withky¡y±k·±aregivenandF:D(F)µX!YisanonlinearoperatorbetweenHilbertspacesXandY.Thismethodisde¯nedbysplittingtheoperatorFintoalinearpartAandanonlinearpartG,suchthatF=A+G.TheniterationsareorganizedasAuk+1=y±¡Guk.Inthecontextofill-posedproblemsweconsiderthesituationwhenAdoesnothaveaboundedinverse,thuseachiterationneedstoberegularized.UndersomeconditionsontheoperatorsAandGwestudythebehavioroftheiterationerror.Weobtainitsstabilitywithrespecttotheiterationnumberkaswellastheoptimalconvergenceratewithrespecttothenoiselevel±,providedthatthesolutionsatis¯esageneralizedsourcecondition.Asanexample,weconsideraninverseproblemofinitialtemperaturereconstructionforanonlinearheatequation,wherethenonlinearityappearsduetoradiatione®ects.Theobtainediterationerrorinthenumericalresultshasthetheoreticallyexpectedbehavior.Thetheoreticalassumptionsareillustratedbyacomputationalexperiment.AMSMSC:65J15,65J20,80A23.Keywords:nonlinearinverseproblem,regularization,aposterioriregularizationparameterchoice,derivative-freeiterativemethod,generalizedsourcecondition,orderoptimalconvergence,nonlinearheatequation,initialtemperature,heatradiation,computationalexperiment.¤Fraunhofer-InstitutfÄurTechno-undWirtschaftsmathematik,Gottlieb-Daimler-Stra¼e,D-67663Kaiserslautern,Germany;e-mail:pereverz@itwm.fhg.deyFachbereichMathematik,TechnischeUniversitÄatKaiserslautern,D-67663Kaiser-slautern,Germany;e-mail:pinnau@mathematik.uni-kl.dezFraunhofer-InstitutfÄurTechno-undWirtschaftsmathematik,Gottlieb-Daimler-Stra¼e,D-67663Kaiserslautern,Germany;e-mail:siedow@itwm.fhg.de11IntroductionLetusconsideranonlinearinverseproblemrepresentedbytheoperatorequationFu=y;(1)withanonlinearoperatorF:D(F)µX!YactingbetweensomeHilbertspacesXandY.Theinnerproductandthecorrespondingnormoneachofthesespaceswillbedenotedby(¢;¢)andk¢k,respectively.Itwillbealwaysclearfromthecontextwhichspaceisconsidered.Weassumethat(1)possessesasolution^u,andthissolutionisuniqueatleastinsomeballaround^u.Weareinterestedinthecasewhen(1)isill-posed,i.e.,thesolutionof(1)doesnotdependcontinuouslyonthedatay.Inparticular,givennoisydatay±,suchthatky¡y±k·±,theequationFu=y±(2)canhavenosolution,andevenifthesolutionexists,itsdistanceto^ucanbearbitrarylarge.Thus,specialmethods,so-calledregularizationmethods,arerequiredforstabilizing(2)[8,14,10,26].Therearebasicallytwoapproachestotheregularizationof(2).Approach1(\regularization¯rst):Insteadoftheill-posednonlinearequation(2)oneconsiderssomefamilyofwell-posednonlinearequationsF®u=y±;(3)withsolutionsfu®gdependingontheregularizationparameter®,andsearchesforsuch®=®(±)thatthenormk^u¡u®ktakesthe(order-)optimalvaluewithrespecttothenoiselevel±.Realiza-tionsofthisapproachareTikhonovregularization[32,9,20,28],Lavrentievregularization[31],andthegeneralregularizationschemeintroducedbyTautenhahn[29].Approach2(\linearization¯rst):HeretheoperatorFisapproximatedbysomelinearoperatorLvFu¼Lvu+bv;(4)whichcandependonthepointv2Xaroundwhichthelinearizationismade.OneofthefrequentlyusedlinearizationschemesinvolvestheFr¶echetderivativeofF.InthiscaseLv=F0v:=F0(v)andbv=Fv¡F0vv.Having(4),oneassociateswith(2)thefollowingiterativeprocedureLukuk+1=y±¡buk;k=0;1;:::(5)Frequently,thelinearizationofnonlinearill-posedequationsalsoleadstoill-posedequations.That'swhyeachiteration(5)needstoberegularized.Thus,insteadof(5)onehastoconsiderL®ukuk+1=y±¡buk;k=0;1;:::;(6)whereL®ukissomeregularizedapproximationofLuk.Butin(5)thenoiseoftherighthandsideisuncontrollable,becauseinadditiontothedatanoisewehavethelinearizationerrorthatisunknown.Thus,withineachiteration(6)itisimpossibletochoosetheregularizationparameter®insuchawaythatthesolutionsof(6)wouldconvergetothesolutionofthelinearizedequationas±!0,becauseforsuchachoicetheknowledgeofthenoiselevelisrequired[3].Thewayoutofthissituationisknownasiterativeregularization.ItwasproposedbyBakushinskii[4]andstudiedin[5,13,12,7].Theideaistochooseasequenceofregularizationparametersf®kgthatsatis¯es1·®k®k+1·C;limk!1®k=0;withsomeconstantC1.Thentheiterationprocedure(6)runswiththissequenceandtheiterationnumberkisusedasaregularizationparameter.Inthispaperweproposeanotherapproachforregularizing(2).Weshow,thatforlinearizationssatisfyingsomeconditionsitispossibletochoosetheregularizationparameterwithineachiterationinsuchawaythatitwillguaranteestabilityoftheiterativeprocess.Namely,theiterationerrork^u¡ukkwilldecreaseuntilsomelevelandthenremainsbelowthislevel.Suchastablebehavior2isunusualforiterativeprocessesusedforill-posednonlinearequations,whereonehastostopiterationssomewheretoavoiderrorexplosion.WeconsideralinearizationoftheoperatorFachievedbyitssplittingintoalinearpartAandanonlinearpartG,i.e.,Fu=Au+Gu:(7)Thissplittingsuggestsa¯xed-pointiterationforsolving(2):Auk+1=y±¡Guk:(8)WeareespeciallyinterestedinthecasewhentheoperatorAdoesnothaveaboundedinverse
本文标题:Regularized Fixed-Point Iterations for Nonlinear I
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