Iterative Methods for Approximate Solution of Inverse Problems

297 Pages·2004·1.96 MB·English

by  A. B. Bakushinsky

Iterative Methods for Approximate Solution
of Inverse Problems
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume577
Iterative Methods for
Approximate Solution of
Inverse Problems
by
A.B. Bakushinsky
Russian Academy of Science, Moscow,
Russia
and
M.Yu. Kokurin
Mary State University, Yoshkar-Ola,
Russia
AC.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 1-4020-3121-1 (HB)
ISBN 1-4020-3122-X (e-book)
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Contents
Dedication v
Acknowledgments ix
Introduction xi
1. IRREGULAREQUATIONSASILL–POSEDPROBLEMS 1
1.1 Preliminaries 1
1.2 Irregular Equations and Contemporary Theory of Ill–Posed
Problems 10
2. REGULARIZATIONMETHODSFORLINEAREQUATIONS 17
2.1 GeneralSchemeofConstructingAffineRegularizationAlgorithms
forLinearEquationsinHilbertSpace 18
2.2 GeneralSchemeofConstructingRegularizationAlgorithmsin
BanachSpace 36
2.3 NecessityofSourcewiseRepresentationforRateofConvergence
EstimatesinBanachSpace 52
3. PARAMETRIC APPROXIMATIONS OF SOLUTIONS TO
NONLINEAROPERATOREQUATIONS 89
3.1 ClassicalLinearizationSchemes 90
3.2 ParametricApproximationsofIrregularEquations 92
3.3 ParametricApproximationsinInverseScatteringProblem 98
4. ITERATIVEPROCESSESONTHEBASISOF
PARAMETRICAPPROXIMATIONS 107
4.1 GeneralizedGauss–NewtonTypeMethodsforNonlinearIrregular
EquationsinHilbertSpace 108
viii ITERATIVEMETHODSFORINVERSEPROBLEMS
4.2 NecessityofSourcewiseRepresentationforRateofConvergence
EstimatesofGauss–NewtonTypeMethods 119
4.3 GeneralizedNewton–KantorovichTypeMethodsforNonlinear
IrregularEquationsinBanachSpace 126
4.4 NecessityofSourcewiseRepresentationforRateofConvergence
EstimatesofNewton–KantorovichTypeMethods 134
4.5 ContinuousMethodsforIrregularOperatorEquationsinHilbert
andBanachSpaces 140
5. STABLEITERATIVEPROCESSES 159
5.1 Stable Gradient Projection Methods with Adaptive Choice of
Projectors 160
5.2 ProjectionMethodwithaPrioriChoiceofProjectors 175
5.3 ProjectionMethodforFindingQuasisolutions 178
5.4 StableMethodsontheBasisofParametricApproximations 183
5.5 StableContinuousApproximationsandAttractorsofDynamical
SystemsinHilbertSpace 189
5.6 Iteratively Regularized Gradient Method and Its Continuous
Version 197
5.7 OnConstructionofStableIterativeMethodsforSmoothIrregular
EquationsandEquationswithDiscontinuousOperators 208
6. APPLICATIONSOFITERATIVEMETHODS 225
6.1 ReconstructionofBoundedHomogeneousInclusion 226
6.2 ReconstructionofSeparatingSurfaceofHomogeneousMedia
byMeasurementsofGravitationalAttractionForce 232
6.3 AcousticInverseMediumProblem 244
6.4 InverseAcousticObstacleScattering. FarFieldObservation 256
6.5 InverseAcousticObstacleScattering. NearFieldObservation 262
7. NOTES 275
References 279
Index 289
Acknowledgments
We expresssinceregratitudetoour pupilsandcolleagues fornumerous
interestingandhelpfuldiscussionsofthepresentedresults.
Introduction
Theconceptofaninverseproblem(IP)havegainedwidespreadacceptance
inmodernappliedmathematics,althoughitisunlikelythatanyrigorousformal
definitionofthisconceptexists. Mostcommonly,byIPismeantaproblemof
determining various quantitative characteristics of a medium such as density,
sound speed, electric permittivity and conductivity, shape of a solid body by
observations over physical (e.g., sound, electromagnetic, heat, gravitational)
fields in the medium [3, 4, 26, 27, 30, 33, 42, 67, 72, 81, 101, 108, 116, 120,
124, 125]. The fields may be of natural appearance or specially induced, sta-
tionaryordependingontime. Animportantandfrequentlyaddressedclassof
IPs arises when it is required to determine some physical characteristics of a
boundedpartofthemediumwhileoutofthispartthecharacteristicsareknown.
To be more specific, let a homogeneous medium contain a bounded inhomo-
geneous inclusion with unknown characteristics that are beyond the reach of
immediate measurements; properties of the homogeneous background there-
witharegiven. Itisrequiredtodeterminecharacteristicsoftheinclusionusing
resultsofmeasurementsofphysicalfieldsinthecompositemediumformedby
the background and the inclusion. As this takes place, the measurements are
usuallymadeinaregionofobservationlocatedfarfromtheinhomogeneityto
be recovered. As a rule, a formal mathematical model of IP under consider-
ation can be derived with relative ease. This is most often done with the use
of a suitable model of the field propagation phenomenon for inhomogeneous
media. In this context, the problem of determination of field values in a re-
gionofobservationbygivencharacteristicsofthemediumiscalledthedirect
problem. Inmanyinstances,directproblemscanbereducedtoclassicalpartial
differentialequationsofsecondordersupplementedwithnecessaryinitialand
boundaryconditions. Ellipticequationsusuallyariseinmathematicalmodelsof
thegravitationalfield, stationarytemperaturefieldsandtime–harmonicsound
fields,equationsofparabolictypedescribephenomenaofheatconduction,hy-
perbolicequationsgovernpropagationofnonstationaryscalarwavefields,the
xii
system of Maxwell’s equations describes electromagnetic fields, etc. Since
properties of the homogeneous medium outside the inclusion are fixed, any
solution method for the direct problem naturally defines a mapping that takes
eachsetofadmissiblephysicalorgeometriccharacteristicsoftheinclusionto
thefieldvaluesontheregionofobservation. Thereforetheprocessofsolving
IPformallyisequivalenttoexactorapproximatedeterminationofapreimage
ofgivenfieldvaluesunderthismapping. AsignificantfeatureofappliedIPsis
thatfieldsaretypicallymeasuredapproximatelyandhencetruefieldvaluesare
notinourdisposal. Wecanhopeonlytogetsomeestimatesforthemeasured
physicalvaluestogetherwithcorrespondingboundsofthemeasurementerrors.
A strict formulation of an operator equation connected with IP involves as an
obligatory part a detailed description of functional spaces of feasible images
and admissible preimages of the problem’s operator. These spaces are often
calledtheobservationspaceandthesolutionspacerespectively;theytypically
arisetobeinfinite–dimensionalHilbertorBanachspaces. Afundamentalfact
to regularization theory is that observation and solution spaces can’t be taken
quite arbitrarily. In practice, their choice is usually strictly determined by an
available technique of registering field characteristics and by qualitative re-
quirementsoninterpretationsofthemeasurements. Asolutiontosuchoperator
equationgenerallycan’tbeexpressedinanexplicitformsothattheuseofap-
proximatemethodsofnumericalanalysisistheonlywaytogetthesolutionor
its approximation. Unfortunately, equations associated with IPs usually don’t
satisfytheregularitycondition,whichistypicalforconstructionandanalysisof
classicalsolutionmethodsforsmoothoperatorequations. Theregularityofan
equationwithadifferentiableoperatormeansthatthederivativeofthisoperator
must have a continuous inverse or pseudo–inverse mapping. If this condition
isviolated,thenwesaythattheequationisirregular.
Systematicstudiesofirregularscalarequationsandfinite–dimensionalsys-
tems of nonlinear equations were originated by Schr¨oder’s work [129]. By
now, theory of numerical methods for nonlinear finite–dimensional systems
has accumulated a wide variety of results on solving irregular problems (see,
e.g., [47, 64, 69] and references therein). Certain of approaches developed
within the context of this theory were extended to abstract operator equations
ininfinite–dimensionalspaces. Mostofcurrentpapersonsolvingfiniteirreg-
ular systems involve special conditions on a nonlinearity of an equation; we
pointtotheverypopularconditionof2–regularity([69])anditsmodifications
andtovariousconditionsontheJacobianatthesolution. Letusemphasizethat
analogsofsuchconditionsintheinfinite–dimensionalcaserequirethatranges
ofoperators’derivativesbeclosedsubspacesofanobservationspace. Unfortu-
nately,attemptstoapplythementionedapproachestoequationsoftypicalIPs
have not met with success so far. The matter is likely that the conditions on
derivatives’degenerationcitedabovearenotcharacteristicforequationsasso-