Table Of Content

Applied Mathematical Sciences Fioralba Cakoni David Colton A Qualitative Approach to Inverse Scattering Theory Applied Mathematical Sciences Volume 188 FoundingEditors FritzJohn,JosephLaselleandLawrenceSirovich Editors S.S.Antman [email protected] P.J.Holmes [email protected] K.R.Sreenivasan [email protected] Advisors L.Greengard J.Keener R.V.Kohn B.Matkowsky R.Pego C.Peskin A.Singer A.Stevens A.Stuart Forfurthervolumes: http://www.springer.com/series/34 Fioralba Cakoni • David Colton A Qualitative Approach to Inverse Scattering Theory 123 FioralbaCakoni DavidColton DepartmentofMathematicalSciences DepartmentofMathematicalSciences UniversityofDelaware UniversityofDelaware Newark,DE,USA Newark,DE,USA ISSN0066-5452 ISSN2196-968X(electronic) ISBN978-1-4614-8826-2 ISBN978-1-4614-8827-9(eBook) DOI10.1007/978-1-4614-8827-9 SpringerNewYorkHeidelbergDordrechtLondon LibraryofCongressControlNumber:2013950891 MathematicsSubjectClassification(2010):35P25,35R25,35R30,65M30,65R30,78A45 ©SpringerScience+BusinessMediaNewYork2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection withreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeingenteredand executedonacomputersystem,forexclusiveusebythepurchaserofthework.Duplicationofthispub- licationorpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthePublisher’s location,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Permis- sionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violationsareliable toprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpublica- tion,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforanyerrors oromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespecttothe materialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) To the Memory of Natasha Colton 1975–2013 Preface The field of inverse scattering theory has been a particularly active field in applied mathematics for the past 25 years. The aim of research in this field has been to not only detect but also to identify unknown objects through the use of acoustic, electromagnetic, or elastic waves. Although the success of such techniques as ultrasound and x-ray tomography in medical imaging has been truly spectacular, progress has lagged in other areas of application, which are forced to rely on different modalities using limited data in com- plex environments. Indeed, as pointed out in [88] concerning the problem of locating unexploded ordinance, “Target identification is the great unsolved problem. We detect almost everything, we identify nothing.” Until a few years ago, essentially all existing algorithms for target identification were based on either a weak scattering approximation or on the use of nonlinear optimization techniques. A survey of the state of the art for acoustic and electromagnetic wavesas of 1998can be found in [54]. However, as the demands of imaging increased, it became clear that incorrect model assumptions inherent in weak scattering approximations imposed severe lim- itations onwhen reliable reconstructionswerepossible. On the otherhand, it was also realized that for many practical applications nonlinear optimization techniquesrequireaprioriinformationthatisingeneralnotavailable.Hence, in recent years, alternative methods for imaging have been developed that avoidincorrectmodel assumptions but, as opposedto nonlinear optimization techniques, only seek limited information about the scattering object. Such methods come under the general title of qualitative methods in inverse scattering theory. Examples of such an approachare the linear sampling method, [54,107], the factorization method [98,107], the method of singular sources [138,139], the probe method [91,92], and the use of convex scattering sup- ports[74,116],allofwhichseektodetermineanapproximationtotheshapeof the scattering obstacle but in general provide only limited informationabout the material properties of the scatterer. This book is designed to be an introduction to qualitative methods in inverse scattering theory, focusing on the basic ideas of the linear sampling vii viii Preface methodandits closerelative,the factorizationmethod. The obviousquestion is:anintroductionforwhom?Oneofthe problemsinmakingthesenew ideas in inverse scattering theory available to the wider scientific and engineering community is that the research papers in this area make use of mathematics that may be beyond the training of a reader who is not a professional mathematician. This book represents an effort to overcome this problem and to write a monograph that is accessible to anyone having a mathematical background only in advanced calculus and linear algebra. In particular, the necessary material on functional analysis, Sobolev spaces, and the theory of ill-posed problems will be given in the first two chapters. Of course, to do this in a short book such as this one, some proofs will not be given, nor will all theorems be proven in complete generality. In particular, we will use the mapping and discontinuity properties of double- and single-layer potentials with densities in the Sobolev spaces H1/2(∂D) and H−1/2(∂D), respectively, but will notprove any of these results, referring for their proofs to the mono- graphs [111] and [127]. We will furthermore restrict ourselves to a simple model problem, the scattering of time-harmonic electromagneticwaves by an infinite cylinder. This choice means that we can avoid the technical difficul- tiesofthree-dimensionalinversescatteringtheoryfordifferentmodalitiesand insteadrestrictourattentiontothe simplercaseoftwo-dimensionalproblems governedby the Helmholtz equation. For a glimpse ofthe problemsarisingin the three-dimensional “real world,” we refer the reader to [26]. Although, for the foregoing reasons we do not discuss the qualitative approach to the inverse scattering problem for modalities other than electromagnetic waves, the reader should not assume that such approaches do not exist! Indeed, having mastered the material in this book, the reader will be fully preparedto understandthe literatureonqualitativemethods forinverse scattering problems arising in other areas of application, such as acoustics and elasticity. In particular, for qualitative methods in the inverse scattering problem for acoustic waves and underwater sound see [12,133,158,159,160], whereas for elasticity we refer the reader to [5,37,38,73,132,135,150]. We would like to acknowledge the scientific and financial support of the Air Force Office of Scientific Researchandin particularDr. Arje Nachmanof AFOSRandDr.RichardAlbaneseofBrooksAirForceBase.Finally,aspecial thanks to our colleague Peter Monk, who has been a participant with us in developing the qualitative approach to inverse scattering theory and whose advice and insights have been indispensable to our researchefforts. In closing, we note that this book is an updated and expanded version of anearlierbook by the authorsthatoriginallyappearedinthe SpringerSeries on Interactions of Mechanics and Mathematics entitled Qualitative Methods in Inverse Scattering Theory. Newark, Delaware Fioralba Cakoni, David Colton Contents 1 Functional Analysis and Sobolev Spaces ................... 1 1.1 Normed Spaces.......................................... 1 1.2 Bounded Linear Operators................................ 6 1.3 Adjoint Operator........................................ 14 1.4 Sobolev Space Hp[0,2π] .................................. 17 1.5 Sobolev Space Hp(∂D)................................... 23 2 Ill-Posed Problems ........................................ 27 2.1 Regularization Methods .................................. 28 2.2 Singular Value Decomposition............................. 30 2.3 Tikhonov Regularization ................................. 36 3 Scattering by Imperfect Conductors ....................... 45 3.1 Maxwell’s Equations ..................................... 45 3.2 Bessel Functions......................................... 47 3.3 Direct Scattering Problem ................................ 51 4 Inverse Scattering Problems for Imperfect Conductors..... 63 4.1 Far-Field Patterns....................................... 64 4.2 Uniqueness Theorems for Inverse Problem .................. 67 4.3 Linear Sampling Method ................................. 72 4.4 Determination of Surface Impedance ....................... 78 4.5 Limited Aperture Data................................... 81 4.6 Near-Field Data......................................... 83 5 Scattering by Orthotropic Media .......................... 85 5.1 Maxwell Equations for an Orthotropic Medium.............. 85 5.2 Mathematical Formulation of Direct Scattering Problem...... 89 5.3 Variational Methods ..................................... 94 5.4 Solution of Direct Scattering Problem......................105 ix