(Hohage)Inverse Problems

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1、lecturenotesonInverseProblemsUniversityofG¨ottingenSummer2002ThorstenHohageContents1Introduction32Algorithmsforthesolutionoflinearinverseproblems133Regularizationmethods214Spectraltheory275Convergenceanalysisoflinearregularizationmethods386Interpretationofsourcecon

2、ditions507Nonlinearoperators558NonlinearTikhonovregularization639Iterativeregularizationmethods69Bibliography7821.IntroductionIn[Kel76]Kellerformulatedthefollowingverygeneraldefinitionofinverseproblems,whichisoftencitedintheliterature:Wecalltwoproblemsinversesofonea

3、notheriftheformulationofeachinvolvesallorpartofthesolutionoftheother.Often,forhistoricalreasons,oneofthetwoproblemshasbeenstudiedextensivelyforsometime,whiletheotherisnewerandnotsowellunderstood.Insuchcases,theformerproblemiscalledthedirectproblem,whilethelatterisc

4、alledtheinverseproblem.Inmanycasesoneofthetwoproblemsisnotwell-posedinthefollowingsense:Definition1.1.(Hadamard)Aproblemiscalledwell-posedif1.thereexistsasolutiontotheproblem(existence),2.thereisatmostonesolutiontotheproblem(uniqueness),3.thesolutiondependscontinuou

5、slyonthedata(stability).Aproblemwhichisnotwell-posediscalledill-posed.Ifoneoftwoproblemswhichareinversetoeachotherisill-posed,wecallittheinverseproblemandtheotheronethedirectproblem.Allinverseproblemswewillconsiderinthefollowingareill-posed.Ifthedataspaceisdefinedas

6、setofsolutionstothedirectproblem,existenceofasolutiontotheinverseproblemisclear.However,asolutionmayfailtoexistifthedataareperturbedbynoise.Thisproblemwillbeaddressedbelow.Uniquenessofasolutiontoaninverseproblemisoftennoteasytoshow.Obviously,itisanimportantissue.If

7、uniquenessisnotguaranteedbythegivendata,theneitheradditionaldatahavetobeobservedorthesetofadmissiblesolutionshastoberestrictedusinga-prioriinformationonthesolution.Inotherwords,aremedyagainstnon-uniquenesscanbeareformulationoftheproblem.AmongthethreeHadamardcriteri

8、a,afailuretomeetthethirdoneismostdelicatetodealwith.Inthiscaseinevitablemeasurementandround-offerrorscanbeamplifiedbyanarbitrarilylargefactorandmak