逆问题数学理论导论（英文版）

FollowingKeller[119]wecalltwoproblemsinversetoeachotheriftheformulationofeachofthemrequiresfullorpartialknowledgeoftheother.Bythisdefinition,itisobviouslyarbitrarywhichofthetwoproblemswecallthedirectandwhichwecalltheinverseproblem.Butusually,oneoftheproblemshasbeenstudiedearlierand,perhaps,inmoredetail.Thisoneisusuallycalledthedirectproblem,whereastheotheristheinverseproblem.However,thereisoftenanother,moreimportantdifferencebetweenthesetwoproblems.Hadamard(see[91])introducedtheconceptofawell-posedproblem,originatingfromthephilosophythatthemathematicalmodelofaphysicalproblemhastohavethepropertiesofuniqueness,existence,andstabilityofthesolution.Ifoneofthepropertiesfailstohold,hecalledtheproblemiU-posed.Itturnsoutthatmanyinterestingandimportantinverseproblemsinscienceleadtoill-posedproblems,,whilethecorrespondingdirectproblemsarewell-posed.Often,existenceanduniquenesscanbeforcedbyenlargingorreducingthesolutionspace(thespaceof"models").Forrestoringstability,however,onehastochangethetopologyofthespaces,whichisinmanycasesimpossiblebecauseofthepresenceofmeasurementerrors.Atfirstglance,itseemstobeimpossibletocomputethesolutionofaproblemnumericallyifthesolutionoftheproblemdoesnotdependcontinuouslyonthedata,i.e.,forthecaseofill-posedproblems.Underadditionalaprioriinformationaboutthesolution,suchassmoothnessandboundsonthederivatives,however,itispossibletorestorestabilityandconstructefficientnumericalalgorithms.