Convergence results of a minimax method for findin

ConvergenceResultsofAMinimaxMethodforFindingMultipleCriticalPointsYongxinLiandJianxinZhouyAbstractIn[12],newlocalminimaxtheoremswhichcharacterizeasaddlepointasasolutiontoatwo-levellocalminimaxproblemareestablished.Basedonthelocalcharacter-ization,anumericalminimaxmethodisdesignedforndingmultiplesaddlepoints.ManynumericalexamplesinsemilinearellipticPDEhavebeensuccessfullycarriedouttosolveformultiplesolutions.Oneoftheimportantissuesremainsunsolved,i.e.,theconvergenceofthenumericalminimaxmethod.Inthispaper,werstmodifyStep5inthealgorithmsuchthatitbecomespracticallyeasiertoimplement.Thenweestablishsomeconvergenceresultsforthenumericalminimaxmethodforisolatedornon-isolatedcriticalpoints.Theconvergenceresultsshowthatthealgorithmindeedexceedsthescopeofaminimaxprinciple.Keywords.Multiplecriticalpoints,Localpeakselection,Minimaxalgorithm,ConvergenceAMS(MOS)subjectclassications.58E05,58E30,35A40,35A65Abbreviatedtitles.ConvergenceResultsofAMinimaxMethodIBMT.J.WatsonResearchCenter,YorktownHts,NJ10598.yDepartmentofMathematics,TexasA&MUniversity,CollegeStation,TX77843.SupportedinpartbyNSFGrantDMS96-10076.11IntroductionLetHbeaHilbertspaceandJ:H!RbeaFrechetdierentiablefunctional.DenotebyJ0orrJitsFrechetderivativeandJ00itssecondFrechetderivativeifitexists.Apoint^u2HisacriticalpointofJifJ0(^u)=0asanoperatorJ0:H!H.Anumberc2RiscalledacriticalvalueofJifJ(^u)=cforsomecriticalpoint^u.Foracriticalvaluec,thesetJ−1(c)iscalledacriticallevel.WhenthesecondFrechetderivativeJ00existsatacriticalpoint^u,^uissaidtobenon-degenerateifJ00(^u)isinvertibleasalinearoperatorJ00(^u):H!H,otherwise^uissaidtobedegenerate.Therstcandidatesforacriticalpointarethelocalmaximaandminimatowhichtheclassicalcriticalpointtheorywasdevotedincalculusofvariation.Traditionalnumericalmethodsfocusonndingsuchstablesolutions.Criticalpointsthatarenotlocalextremaareunstableandcalledsaddlepoints,thatis,criticalpointsuofJ,forwhichanyneighborhoodofuinHcontainspointsv;ws.t.J(v)J(u)J(w).Inphysicalsystems,saddlepointsappearasunstableequilibriaortransientexcitedstates.AccordingtotheMorsetheory,theMorseIndex(MI)ofacriticalpoint^uofareal-valuedfunctionalJisthemaximaldimensionofasubspaceofHonwhichtheoperatorJ00(^u)isnegativedenite;thenullityofacriticalpoint^uisthedimensionofthenull-spaceofJ00(^u).Soforanon-degeneratecriticalpoint,ifitsMI=0,thenitisalocalminimizerandastablesolution,andifitsMI0,thenitisamin-maxtypeandunstablesolution.Multiplesolutionswithdierentperformanceandinstabilityexistinmanynonlinearproblemsinnaturalandsocialsciences[20,18,14,22,13].Stabilityisoneofthemaincon-cernsinsystemdesignandcontroltheory.Howeverinmanyapplications,performanceormaneuverabilityismoredesirable,inparticular,insystemdesignorcontrolofemergencyorcombatmachineries.Meanwhileinstablesolutionsmayhavemuchhighermaneuverabilityorperformanceindices.Forprovidingchoiceorbalancebetweenstabilityandmaneuverabilityorperformance,itisimportanttosolveformultiplesolutions.Whencasesarevariational,theybecomemultiplecriticalpointproblems.Soitisimportantforboththeoryandappli-cationstonumericallysolveformultiplecriticalpointsinastableway.Sofar,littleisknown2intheliteraturetodevisesuchafeasiblenumericalalgorithm.Minimaxprincipleisoneofthemostpopularapproachesincriticalpointtheory.However,mostminimaxtheoremsintheliterature(See[1],[16],[17],[18],19,[22]),suchasthemountainpass,variouslinkingandsaddlepointtheorems,requireonetosolveatwo-levelglobaloptimizationproblemandthereforenotforalgorithmimplementation.In[12],motivatedbythenumericalworksofChoi-McKenna[6]andDing-Costa-Chen[11],theMorsetheoryandtheideatodeneasolutionsubmanifold,newlocalminimaxtheoremswhichcharacterizeacriticalpointasasolutiontoatwo-levellocalminimaxproblemareestablished.Basedonthelocalcharacterization,anewnumericalminimaxmethodforndingmultiplecriticalpointsisdevised.ThenumericalmethodisimplementedsuccessfullytosolveaclassofsemilinearellipticPDEonvariousdomainsformultiplesolutions[12].Duetoalimitationtothelengthofthepaperandotherprofoundanalysisinvolved,oneoftheveryimportantissuesremainsunsolvedin[12],i.e.,theconvergenceofthenumericalminimaxmethod,aparamountissueforanynumericalmethod.Theobjectiveofthispaperistoestablishsomeconvergenceresultsforthealgorithm.Theorganizationofthispaperisasfollows.WerstmodifyStep5inthealgorithmbydevelopinganewstepsizeruleinSection2suchthatitbecomespracticallyeasiertoimplement.Somepropertiesofthealgorithmareveriedtherewith.ThenweprovevariousconvergenceresultsforthelocalminimaxalgorithminSection3.OurconvergenceanalysisinSection3showthatouralgorithmcanactuallybeusedtondamoregeneral,non-minimaxtypeofcriticalpoints.InthelastsectionwepresentsmultiplenumericalsolutionstotheHennon'sequationasacomplementtothosedisplayedin[12].Intherestofthissection,weintroducesomenotationsandtheoremsfrom[12]forfutureuse.Inparticular,weproveanewlocalexistencetheorem.ForanysubspaceH0H,letSH0=fvjv2H0;kvk=1gbetheunitsphereinH0.LetLbeaclosedsubspaceinHcalledabasespace,andH=LLL?betheorthogonaldecompositionwhereL?istheorthogonalcomplement