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1、SIAMREVIEW()1992SocietyforIndustrialandAppliedMathematicsVol.34,No.4,pp.561-580,December1992002ANALYSISOFDISCRETEILL-POSEDPROBLEMSBYMEANSOFTHEL-CURVE*PEltCHltISTIANHANSENtAbstract.Whendiscreteill-posedproblemsareanalyzedandsolvedbyvariousnumericalregu
2、larizationtechniques,averyconvenientwaytodisplayinformationabouttheregularizedsolutionistoplotthenormorseminormofthesolutionversusthenormoftheresidualvector.Inparticular,thegraphassociatedwithTikhonovregularizationplaysacentralrole.Themainpurposeofthi
3、spaperistoadvocatetheuseofthisgraphinthenumericaltreatmentofdiscreteill-posedproblems.Thegraphischaracterizedquantitatively,andseveralimportantrelationsbetweenregularizedsolutionsandthegrapharederived.Itisalsodemonstratedthatseveralmethodsforchoosingt
4、heregularizationparameterarerelatedtolocatingacharacteristicL-shaped"corner"ofthegraph.Keywords,discreteill-posedproblems,leastsquares,generalizedSVD,regularizationAMS(MOS)subjectclassifications.65F20,65F301.Introduction.WesaythatthealgebraicproblemsA
5、xbandmin[IAxarediscreteill-posedproblemsifthematrixAisillconditionedandallitssingularvaluesdecaytozeroinsuchawaythatthereisnoparticulargapinthesingularvaluespectrum.Discreteill-posedproblemsariseinavarietyofapplications:astronomy[5],comput-erizedtomog
6、raphy[32],earlyvision[2],electrocardiography[7],mathematicalphysics[41],andmeteorology[46],tomentionjustafew.TheunderlyingmathematicalproblemisoftenbutnotalwaysalinearFredholmintegralequationofthefirstkind.Thereisavastamountofliteratureonill-posedprob
7、lemsinthesettingofHilbert-spacesandotherinfinite-dimensionalspaces;see,e.g.,[10],[11],[13],[14],[27],[31],[40],[47].Theapproachtakeninthispaperisdifferent:wetakethealgebraicproblemsAxbandminIIAx-bllasourbasis,andweusenumericallinearalgebra--inpartic-u
8、lar,thegeneralizedsingularvaluedecompositionmtoderiveourresults.Introductionstodiscreteill-posedproblemscanbefoundin[4],[5],[29],[33],[35],[43],[44].ThemonographbyHofmann[25]containsawealthofmaterialoninfinite-dimensionalaswellasfinite-dimensi
