Regularization Theory for Ill-posed Problems: Selected Topics

304 Pages·2013·1.879 MB·English

by  Lu S., Pereverzev S.V.

Inverse and Ill-Posed Problems Series 58
ManagingEditor
SergeyI.Kabanikhin,Novosibirsk,Russia;Almaty,Kazakhstan

Shuai Lu, Sergei V. Pereverzev
Regularization Theory
for Ill-posed Problems
Selected Topics
De Gruyter
MathematicsSubjectClassification2010
47A52,65J10,65J20,65J22,65N15,65N20
Authors
Dr.ShuaiLu
FudanUniversity
SchoolofMathematicalSciences
No.220,RoadHandan
200433Shanghai
PRChina
[email protected]
Prof.Dr.SergeiV.Pereverzev
AustrianAcademyofSciences
JohannRadonInstituteforComputationalandAppliedMathematics(RICAM)
Altenbergerstraße69
4040Linz
Austria
[email protected]
ISBN978-3-11-028646-5
e-ISBN978-3-11-028649-6
Set-ISBN978-3-11-028650-2
ISSN1381-4524
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Preface
Thetheoryofinverseproblemshasawidevarietyofapplicationsbecauseanymath-
ematicalmodelneedstobecalibratedbeforeitcanbeused,andsuchacalibrationis
atypicalinverseproblem.
Regularizationtheory,inturn,isthealgorithmicpartofthetheoryofinverseprob-
lems. It provides and analyzes the methods for dealing with ill-posedness, which is
oneofthemainissuesforinverseproblems.
Inspiteofagrowingnumberofmonographsonregularizationtheory(atthetimeof
writing,thelatestpublishedoneis[84]),therearequiteafewtopicswhichhaveonly
recentlybeendevelopedandwhicharenotyetreflectedintheliterature.Thepresent
bookismotivatedbysomeofthese.
Thefirstnoveltyofthisbookisthatitsimultaneouslyanalyzestheill-posedprob-
lems with deterministic and stochastic data noises. Not only does such analysis al-
low uniform theoretical justification of a general regularization scheme for both of
theabove-mentionednoisemodels,italsoprovidesalinktoalargeclassoflearning
theoryalgorithms,whichareessentiallyallofthelinearregularizationschemes.
Notethatthechapteronregularizationalgorithmsinlearningtheoryisanotherfea-
ture which distinguishes this book from existing monographic literature on inverse
problems.
A further novelty of the book is Chapter 3, entitled “Multiparameter regulariza-
tion”.Itisinterestingtoobservethatinexistingpublicationstheperformanceofmulti-
parameter regularization schemes have been variously judged by authors. Some of
them found that multiparameter regularization only marginally improved the one-
parameterversion,whileothersreportedonmostsatisfactorydecisionsgivenbymulti-
parameteralgorithmsincaseswheretheirone-parametercounterpartsfailed.Wehope
thatChapter3willshedlightonthisslightlycontroversialsubject.
Note that in this book the term “multiparameter regularization” is used as a syn-
onym for “multiple penalty regularization”. At the same time, in modern numerical
analysis,theapproximationandregularizationalgorithmsarebecomingmoresophis-
ticatedanddependentonvariousparametersparameterizingeventhespaceswherepe-
nalization,orregularization,isperformed.Theself-tuningofsuchparametersmeans
that a regularization space is automatically adjusted to the considered problem. On
theotherhand,classicalregularizationtheoryrestrictsitselftostudyingthesituation
wherearegularizationspaceisassumedtobegivenapriori.Therefore,tothebestof
ourknowledge,Chapter5ofthepresentbookisthefirstattemptinthemonographic
literature to analyze the adaptive choice of the regularization space. This analysis is
viii Preface
basedontheconceptofmeta-learning,whichisalsofirstintroducedinthecontextof
regularizationtheory.
The meta-learning concept presupposes that the design parameters of algorithms
are selected on the basis of previous experience with similar problems. Therefore,
parameterselectionrulesdevelopedinthiswayareintrinsicallyproblem-oriented.
InChapter5wedemonstrateameta-learning-basedapproachtoregularizationona
problemfromdiabetestechnology,butitwillalsobeshownthatthemainingredients
can be exploited in other applications. At the same time, the material of the chap-
terdescribesoneofthefirstapplications ofregularizationtheoryindiabetes therapy
management,whichisanextremelyimportantmedicalcarearea.
Wehopethatsuchacontextmakesthebookofinterestforawideaudience,within
theinverseproblemscommunityandbeyond.
Thefirstpartofthebookcanalsoberecommendedforuseinlectures.Forexample,
thesectionsoftheintroductorychaptercanbeusedindependentlyingeneralcourses
onnumericalanalysisasexamplesofill-posedproblemsandtheirtreatments.
Atthesametime,Chapter2containsacompactandrathergeneralpresentationof
regularization theory for linear ill-posed problems in the Hilbert space setting under
deterministicandstochasticnoisemodelsandgeneralsourceconditions.Thischapter
canprovidematerialforanadvancedmastercourseonregularizationtheory.Sucha
coursewasgivenattheTechnicalUniversityofKaiserslautern,Germany,andatthe
Stefan Banach International Center in Warsaw, Poland. In Chapter 2 we have really
triedtoadaptthepresentationforthispurpose.Forexample,weavoidedthenumer-
ationoftheformulaeinordertomakethematerialmoreconvenientforpresentation
ontheblackboard.
Thesecondpartofthebookcanbeseenasapresentationofsomefurtherdevelop-
mentsofthebasictheory.Thismaterialisnewinmonographicliteratureonregular-
izationtheoryandcanbeusedinstudents’seminars.
ThebookwaswritteninthestimulatingatmosphereoftheJohannRadonInstitute
forComputationalandAppliedMathematics(RICAM).Thepreliminaryplanforthe
projectwasdiscussedwithitsFoundingDirector,ProfessorHeinzEngl.
Thebookwouldnotbepossiblewithoutourcolleagues.Weareespeciallygrateful
toPeterMathé(WIAS,Berlin),AlexanderGoldenshluger(UniversityofHaifa),Eber-
hardSchock(TechnicalUniversityofKaiserslautern),UlrichTautenhahn(12.01.1951
–10.10.2011),LorenzoRosasco(MIT,Cambridge),BerndHofmann(TechnicalUni-
versityofChemnitz),HuiCao(SunYat-senUniversity),SivananthanSampath(IIT,
Delhi), Valeriya Naumova (RICAM). We are also grateful to Christoph von Friede-
burgandAnjaMöbiusfromDeGruyterforthefinalimpulsetostartwritingthisbook.
SpecialthankstoJinCheng(FudanUniversity,Shanghai),whorecommendedthefirst
authortothesecondoneasaPh.D.student,whichwasthebeginningofthestory.
Finally we gratefully acknowledge financial support from the Austrian Fonds
zur Förderung der wissenschaftlichen Forschung (FWF) (projects P17251-N12
and P20235-N18), Alexander von Humboldt Foundation, National Natural Science
Preface ix
FoundationofChina(KeyProjects91130004and11101093),ShanghaiScienceand
Technology Commission (11ZR1402800 and 11PJ1400800) and the Programme of
IntroducingTalentsofDisciplinetoUniversities(B08018),China.
Shanghai–Linz ShuaiLuandSergeiV.Pereverzev
December2012