An introduction to inverse problems with applications PDF
Preview An introduction to inverse problems with applications
An Introduction to Inverse Problems with Applications Francisco Duarte Moura Neto and Antônio José da Silva Neto An Introduction to Inverse Problems with Applications ABC Authors FranciscoDuarteMouraNeto AntônioJosédaSilvaNeto Dept.deModelagemComputational Dept.deEngenhariaMecânicaeEnergia UniversidadedoEstadodoRiodeJaneiro UniversidadedoEstadodoRiodeJaneiro RuaBonfim25,parteUERJ RuaBonfim25,parteUERJ NovaFriburgo,RJ28625-570 NovaFriburgo,RJ28625-570 Brazil Brazil E-mail:[email protected] E-mail:[email protected] ISBN978-3-642-32556-4 e-ISBN978-3-642-32557-1 DOI10.1007/978-3-642-32557-1 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2012945297 (cid:2)c Springer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer. PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations areliabletoprosecutionundertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpub- lication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforany errorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespect tothematerialcontainedherein. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) This book is dedicated to myloveMaysa,ourbelovedchildrenDanielandCecilia, andmydearparentsMariaLuizandFrancisco(inmemoriam) fdmn mybelovedones,Gilsineida,LucasandLuísa, andtomydearparentsAntônio(inmemoriam)andJarleide ajsn Foreword For the benefit of the general readership, it might be a good idea to first explain thedifferencebetween“inverseproblems”versus“analysisproblems”usingsimple generaltermsasdefinedbyProfessorShiroKuboofOsakaUniversityinhisbook publishedintheearly1990s. Analysis problems are well-posed problems concerned with finding distribu- tion(s) of certain variable(s) in a domain of a given size and shape that can be multiplyconnectedand time-dependent.Results of the analysis problemsalso de- pend on boundary and/or initial conditions given at every point of a boundary of the domain. Properties of the media filling the domain must also be given in ad- ditiontostrengthsandlocations/distributionofanypossiblesources/sinks.Finally, equation(s)governingthefieldvariable(s)mustbegiven.Ifallofthesepiecesofin- formationaregiven,thentheproblemoffindingthefielddistributionofvariable(s) isawell-posedproblemthatcanbesolvedusingappropriatenumericalintegration algorithms. However, if at least one of these pieces of information is missing, such under- specified problems become inverse problems of determining the missing piece of informationinadditiontosimultaneouslysolvingtheoriginalanalysisproblem.To enableourselvesto accomplishthis complextask, we mustbegivenan additional pieceofinformation(typically,apartofthesolutionofthecorrespondinganalysis problem)whichmakestheinverseproblemanover-specifiedorill-posedproblem. Forexample,whengivensizeandshapeofanisotropicplateandeitherNeumann orDirichletboundaryconditionsateverypointalongtheedgesoftheplate,steady state heat conduction in the plate will be governed by the LaPlace’s equation for temperature. This would be a classical well-posed analysis problem. However, if boundary conditions are not given on one boundary of this plate, the problem of findingtemperaturedistributionintheplatebecomesunder-specifiedandcannotbe solved.ThisproblemwillbecomesolvableifweprovidebothDirichletandNeuman boundaryconditionssimultaneouslyonatleastsomepartsoftheplate’sboundaries whichwillmakethisanover-specifiedorill-posedinverseproblemofdetermining themissingboundaryconditionsandsimultaneouslydeterminingthedistributionof temperaturethroughouttheplate. Inverseproblemshavetraditionallybeenconsideredmathematicallychallenging problems and have consequently been studied predominantly by mathematicians. Since therearemanypracticalinverseproblemsina varietyofdisciplinesthatre- quire mathematicaltools for their solution, it is scientists and engineersthat have been developing many of these methods recently out of necessity to obtain prac- tical results. Consequently, an initially wide gap between scientists and engineers VIII Foreword versus applied mathematicians has been steadily narrowing as both communities haverealizedthattheyhavemanythingstolearnfromeachother. This book is a welcome and unique publication that uses a common language to blendthe rigourof the applied mathematicsworld and the reality of a research scientist’s or an engineer’sworld.Thus, it should appealto everyonewho has the basicknowledgeofdifferentialequationsandatleastarudimentaryunderstanding ofbasicmathematicalmodelsusedinfieldtheoryandgeneralcontinuummechanics and transport processes. Specifically, applied mathematicians will be able to find here physicalrelevance for some of their theoretical work and learn to appreciate theimportanceofdevelopingunderstandable,easy-to-use,easytoadaptandreliable algorithmsforthesolutionofdifferentclassesofinverseproblems.Atthesametime, researchscientistsandengineerswillbeable to learnfromthisbookthatsome of the methods and formulations that they have been using in the past are prone to problemsofnon-uniquenessoftheresultsandthataccuracyofmanyofthepractical methodscouldeasilybecomearealissuewhensolvinginverseproblems. Actually,thisbookcouldbeusednotonlyasavaluablereferencebook,butalso asatextbookforstudentsinthefieldsofappliedmathematics,engineeringandexact sciences.Besidesitssimplelanguage,thisbookiseasytocomprehendalsobecause it contains a number of illustrative examplesand exercises demonstratingeach of themajorconceptsandalgorithms. For example, basic concepts of regularization of ill-posed problems are very nicelyexplainedanddemonstratedsothatevenacompletenovicetothisfieldcan understandandapplythem.Formulationsandapplicationsinimageprocessingand thermalfieldspresentedinthisbookhavedirectpracticalapplicationsandaddsig- nificantlytothemorecompleteunderstandingofthegeneralproblematicsofinverse problems governedby elliptic and parabolic partial differential equations. Inverse scatteringproblemshavenotbeencoveredinthisbookasthisfieldcaneasilyfilla separatebook. Many formulations for the solution of inverse problems used to be very disci- plinespecificandevenproblemspecific.Thus,despitetheirmathematicalelegance andsolutionefficiencyandaccuracy,mostoftheclassicalinverseproblemssolution methodshadseverelimitationsconcerningtheirfieldsofapplicability.Furthermore, mostofthesemethodsusedtobehighlymathematical,thusrequiringhighlymath- ematicaleducationonthepartofusers. Since industry requires fast and simple algorithms for the solution of a wide varietyofinverseproblems,thisimpliesagrowingneedforusersthatdonothave averyhighdegreeofmathematicaleducation.Consequently,manyofthecurrently used general algorithms for the solution of inverse problems eventually result in somesortofafunctionalthatneedstobeminimized.Thishasbeenrecognizedby the authors of this book which have therefore included some of the most popular minimizationalgorithmsinthistext. Hence,thisbookprovidesaclosedlooponhowtoformulateaninverseproblem, how to choose an appropriate algorithm for its solution, and how to perform the solutionprocedure. Foreword IX Irecommendthisbookhighlytothosethatarelearningaboutinverseproblems aswellastothosethatthinkthattheyknoweverythingaboutsuchproblems.Both entities will be pleasantly surprised with the ease that concepts, formulationsand solutionalgorithmsareexplainedinthisbook. GeorgeS.Dulikravich Miami,Florida June2011 Preface Archimedes,isthiscrownmadeofgold? KingHieroIIofSyracusea,circa250BCE. (...)tofindashapeofabellbymeansofthesounds whichitiscapableofsendingout. SirA.Schusterb,1882. Canoneheartheshapeofadrum? MarcKacc,1966. aHieroII(308BCE-215BCE). bSirA.Schuster(1851-1934). cMarcKac(1914-1984). Oninverseproblems Perhapsthemostfamousinverseproblemforthemathematicalcommunityis:Can oneheartheshapeofadrum?[40,65].Thatis:Isoneabletofigureouttheshapeofa drumbasedonthesounditemits?Thecorrespondingdirectproblemistodetermine thesoundemittedbyadrumofknownshape.Thesolutiontothedirectproblemis longknown,butthesolutiontotheinverseproblemeludedthescientificcommunity foralongtime.Itwasfoundtobenegative:therearetwodrums,differentinshape, thatemitthe same sound,see [36]. Severalothermathematicalaspectsconcerning theresolutionofinverseproblemshavebeeninvestigatedinrecentyears. Parallelto that,a largenumberof significantinverseproblemmethodologyap- plications were developed in engineering,medicine, geophysicsand astrophysics, aswellasinseveralotherbranchesofscienceandtechnology. Why? Because inverse problems is an interdisciplinary area that matches the mathematical model of a problem to its experimental data. Or, given a bunch of numbers, data, in a data drivenresearch, looks for a mathematical model. It is an interfacebetweentheoryandpractice! Aboutthisbook Thegeneralpurposeofthisbookistointroducecertainkeyideasoninverseprob- lemsanddiscusssomemeaningfulapplications.Withthisapproach,wehopetobe abletostimulate thereaderto studyinverseproblemsandtouse theminpractical situations. XII Preface The bookis divided,thoughnotin sequence,in two main parts, one of a more mathematical nature, and the other more linked to applications. It adopts an ele- mentaryapproachtothemathematicalanalysisofinverseproblemsanddevelopsa generalmethodologyforthesolutionofrealinverseproblems.Further,itdiscusses aseriesofapplicationsofthismethodology,rangingfromimageprocessingapplied to medicine, to problems of radiation applied to tomography, and onto problems of conductive heat transfer used in the design of thermal devices. The choice of applicationsreflecttheacquaintanceoftheauthors. Inordertomakethebookofamanageablesizeandsuitableforalargeraudience, we optedto make the presentationofmathematicalconceptsin the contextoflin- ear,finitedimensional,inverseproblems.Inthissetting,keyissuescanbehandled withmathematicalcare,inarather“pedestrian”andeasywaybecausetheproblem islinearandfinitedimensional,requiringonlyacquaintancewithbasicideasfrom linear algebra. Geometrical ideas, which are a key to generalization, are empha- sized. Someoftheapplicationsconsidered,however,involveproblemswhicharenei- ther finite dimensional nor linear. The treatment then is numerical, and the ideas from the first part of the book are used as guidelines, through extrapolation.This is possible, in part, because, to simplify, this book deals, severaltimes, with least squaresmethods.Althoughthesubjectsinthisbookareintricate,thechapterscan be read, somewhat, independentlyof one another.This is because of the intended redundancy,employedforpedagogicalreasons,like in an oldteaching’stradition: attention,associationandrepetition.Tomakeiteasiertoperusethebook,adescrip- tionofthebook’scontent,chapterbychapter,isgivenonpages4-6. The pre-requisitesto read this bookare calculus of severalvariablesand linear algebra.Nonetheless,afewconceptsandresultsfromlinearalgebraandcalculusare reviewedintheappendix,inordertomakethebookreasonablyself-contained.Even thoughknowledgeof differentialequationsis necessary to understandsome parts, basicconceptsonthissubjectarenotsuppliedorreviewed.Knowledgeofnumerical methodsmightbeusefulforreadingcertainsections.We includedexercisesatthe endofeachchapter,manyofthemguided,tomakeiteasierforreaderstograsp,and extendtheconceptspresented. We believe that this book can be read, with some interest and to their profit, by upper division undergraduate students and beginning graduate students in ap- plied mathematics, physics, engineering,and biology.Since it also includes some thoughts on mathematical modeling, which are in the back of the minds of re- searchers,butarenotusuallyspelledout,thisbookmayinterestthemtoo. Someremarks Practicalmatters:weuseemphasisedexpressionstosignalagroupofwordswitha specificmeaningwhentheyarebeingdefinedeitherexplicitorimplicitly.Theend ofanexample,aproof,oranargumentisindicatedbyasmallblacksquare. Theauthorswillbehappytoreceiveanycommentsand/orsuggestionsonthisbook.
The list of books you might like
