Inverse Acoustic and Electromagnetic Scattering Theory

418 Pages·2013·2.42 MB·English

by  David Colton

Applied Mathematical Sciences
Volume 93
Editors
S.S.Antman
DepartmentofMathematics
and
InstituteforPhysicalScienceandTechnology
UniversityofMaryland
CollegePark,MD20742-4015
USA
[email protected]
P.Holmes
DepartmentofMechanicalandAerospaceEngineering
PrincetonUniversity
215FineHall
Princeton,NJ08544
[email protected]
K.Sreenivasan
DepartmentofPhysics
andtheCourantInstituteofMathematicalSciences
NewYorkUniversity
70WashingtonSquareSouth
NewYorkCity,NY10012
[email protected]
Advisors
Greengard,L. Keener,J. Keller,J.
Matkowsky,B. Pego,R. Peskin,C.
Stevens,A. Stuart,A.
Forfurthervolumes:
http://www.springer.com/series/34

David Colton•Rainer Kress
Inverse Acoustic
and Electromagnetic
Scattering Theory
Third Edition
123
DavidColton RainerKress
DepartmentMathematicalScience Institutfu¨rNumerischeundAngewandte
UniversityofDelaware Mathematik
Newark,DE Georg-August-Universita¨tGo¨ttingen
USA Go¨ttingen,Germany
ISBN978-1-4614-4941-6 ISBN978-1-4614-4942-3(eBook)
DOI10.1007/978-1-4614-4942-3
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2012943988
Mathematics Subject Classification (2010): 35J05, 35P25, 35R25, 35R30, 45A05, 65M30, 65R20,
65R30,76Q05,78A45
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Preface to the Third Edition
Since the second edition of our bookappearedfourteenyearsago, the field of in-
verse scattering theory has continued to be an active and growing area of applied
mathematics. In this third edition we have tried to bring our book up to date by
includingmanyof the new developmentsin the field that havetaken place during
thisperiod.We againhavemadeno effortto coverall ofthe manynewdirections
in inverse scattering theory but rather have restricted ourselves to a selection of
thosedevelopmentsthatwehaveeitherparticipatedinorareanaturaldevelopment
of material discussed in previous editions. We have also continued to emphasize
simplicity over generality, e.g. smooth domains instead of domains with corners,
isotropicmediaratherthananisotropicmedia,standardboundaryconditionsrather
thanmoregeneralizedones,etc.Bysodoing,wehopethatourbookwillcontinue
toserveasabasicintroductiontothefieldofinversescatteringtheory.
Inordertobringourbookuptodate,considerablechangeshavebeenmadetothe
secondedition.Inparticular,newsectionshavebeenaddedonthelinearsampling
andfactorizationmethodsforsolvingtheinversescatteringproblemaswellasex-
pandedtreatmentsofiterationmethodsanduniquenesstheoremsfortheinverseob-
stacleproblem.Theseadditionshavealsorequiredustoexpandourpresentationon
bothtransmission eigenvaluesandboundaryintegralequationsin Sobolevspaces.
Thesechangesinturnsuggestamoreintegratedviewofinversescatteringtheory.In
particular,whatwaspreviouslyreferredto as the Colton–Monkand Kirsch–Kress
methods,respectively,arenowviewedastwoexamplesofwhatarecalleddecompo-
sitionmethods.Fromthispointofviewthetechniquesofiteration,decomposition
andsamplingformanaturaltrilogyofmethodsforsolvinginversescatteringprob-
lems.Althoughafewresultsfromthesecondeditionhavebeenremovedduetothe
factthatwenowconsiderthemtobeobsolete,forhistoricalreasonswehavetended
todososparingly.
Wehopethatthisneweditionofourbookwillcontinuetoservereaderswhoare
already in the field of inverse scattering theory as well as to attract newcomersto
thisbeautifulareaofappliedmathematics.
Newark,Delaware DavidColton
Go¨ttingen,Germany RainerKress
vii

Preface to the Second Edition
Inthefiveyearssincethefirsteditionofthisbookappeared,thefieldofinversescat-
teringtheoryhascontinuedtogrowandflourish.Hence,whentheopportunityfora
secondeditionpresenteditself,wewerepleasedtohavethepossibilityofupdating
ourmonographtotakeintoaccountrecentdevelopmentsinthearea.Asinthefirst
edition,wehavebeenmotivatedbyourownviewofinversescatteringandhavenot
attemptedto includeall ofthe manynewdirectionsin the field.However,we feel
thatthisneweditionrepresentsastateoftheartoverviewofthebasicelementsof
themathematicaltheoryofacousticandelectromagneticinversescattering.
In addition to making minor corrections and additional comments in the text
andupdatingthereferences,wehaveaddednewsectionsonNewton’smethodfor
solving the inverse obstacle problem (Section 5.3), the spectral theory of the far
fieldoperator(Section8.4),aproofoftheuniquenessofthesolutiontotheinverse
mediumproblemfor acoustic waves (Section 10.2)and a methodfor determining
the support of an inhomogeneousmedium from far field data by solving a linear
integralequationofthefirstkind(Section10.7).
We hope that this second edition will attract new readers to the beautiful and
intriguingfieldofinversescattering.
Newark,Delaware DavidColton
Go¨ttingen,Germany RainerKress
ix