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1、阻尼最小二乘法（dampedleastsqures，又称Levenberg–Marquardtalgorithm）theLevenberg–Marquardtalgorithm(LMA)[1]providesanumericalsolutiontotheproblemofminimizingafunction,generallynonlinear,overaspaceofparametersofthefunction.Theseminimizationproblemsariseespeciallyinleasts
2、quarescurvefittingandnonlinearprogramming.TheLMAinterpolatesbetweentheGauss–Newtonalgorithm(GNA就是最小二乘)andthemethodofgradientdescent.TheLMAismorerobustthantheGNA,whichmeansthatinmanycasesitfindsasolutionevenifitstartsveryfaroffthefinalminimum.Forwell-behavedfu
3、nctionsandreasonablestartingparameters,theLMAtendstobeabitslowerthantheGNA.LMAcanalsobeviewedasGauss–Newtonusingatrustregionapproach.However,theLMAfindsonlyalocalminimum,notaglobalminimum.这和最小二乘一样。这是所有线性反演的通病。TheproblemTheprimaryapplicationoftheLevenberg–Marq
4、uardtalgorithmisintheleastsquarescurvefittingproblem:givenasetofmempiricaldatumpairsofindependentanddependentvariables,(xi,yi),optimizetheparametersβofthemodelcurvef(x,β)sothatthesumofthesquaresofthedeviationsbecomesminimal. ThesolutionLikeothernumericminimiz
5、ationalgorithms,theLevenberg–Marquardtalgorithmisaniterativeprocedure.Tostartaminimization,theuserhastoprovideaninitialguessfortheparametervector,β.Incaseswithonlyoneminimum,anuninformedstandardguesslikeβT=(1,1,...,1)willworkfine;incaseswithmultipleminima,the
6、algorithmconvergesonlyiftheinitialguessisalreadysomewhatclosetothefinalsolution.Ineachiterationstep,theparametervector,β,isreplacedbyanewestimate,β+δ.Todetermineδ,thefunctionsareapproximatedbytheirlinearizationswhereisthegradient(row-vectorinthiscase)offwithr
7、especttoβ.Atitsminimum,thesumofsquares,S(β),thegradientofSwithrespecttoδwillbezero.Theabovefirst-orderapproximationofgives.Orinvectornotation,.Takingthederivativewithrespecttoδandsettingtheresulttozerogives:whereJistheJacobianmatrixwhoseithrowequalsJi,andwher
8、eandarevectorswithithcomponentandyi,respectively.Thisisasetoflinearequationswhichcanbesolvedforδ.上面是传统的最小二乘法的线性方程。Levenberg'scontributionistoreplacethisequationbya"dampedversion",whereIis
