统计和计算逆问题

图书标准信息

• 作者 [芬]凯皮奥（Kaipio J.） 著
• 出版社 世界图书出版公司
• 出版时间 2015-01
• 版次 1
• ISBN 9787510086311
• 定价 39.00元
• 装帧 平装
• 开本 24开
• 纸张 胶版纸
• 页数 339页
• 正文语种 英语

【内容简介】

　　Thisbookisaimedatpostgraduatestudentsinappliedmathematicsaswellasatengineeringandphysicsstudentswithafirmbackgroundinmathematics.Thefirstfourchapterscanbeusedasthematerialforafirstcourseoninverseproblemswithafocusoncomputationalandstatisticalaspects.Ontheotherhand,Chapters3and4,whichdiscussstatisticalandnonstationaryinversionmethods,canbeusedbystudentsalreadyhavingknowldegeofclassicalinversionmethods.
　　Thereisrichliterature,includingnumeroustextbooks,ontheclassicalaspectsofinverseproblems.Fromthenumericalpointofview,thesebooksconcentrateonproblemsinwhichthemeasurementerrorsareeitherverysmallorin,whichtheerrorpropertiesareknownexactly.Inrealworldproblems,however,theerrorsareseldomverysmallandtheirpropertiesinthedeterministicsensearenotwellknown.Forexample,inclassicalliteraturetheerrornormisusuallyassumedtobeaknownrealnumber.Inreality,theerrornormisarandomvariablewhosemeanmightbeknown.

【目录】

Preface
1InverseProblemsandInterpretationofMeasurements
1.1IntroductoryExamples
1.2InverseCrimes

2ClassicalRegularizationMethods
2.1Introduction：FredholmEquation
2.2TruncatedSingularValueDecomposition
2.3TikhonovRegularization
2.3.1GeneralizationsoftheTikhonovRegularization
2.4RegularizationbyTruncatedIterativeMethods
2.4.1Landweber-FridmanIteration
2.4.2KaczmarzIterationandART
2.4.3KrylovSubspaceMethods
2.5NotesandComments

3StatisticalInversionTheory
3.1InverseProblemsandBayes'Formula
3.1.1Estimators
3.2ConstructionoftheLikelihoodFunction
3.2.1AdditiveNoise
3.2.2OtherExplicitNoiseModels
3.2.3CountingProcessData
3.3PriorModels
3.3.1GaussianPriors
3.3.2ImpulsePriorDensities
3.3.3Discontinuities
3.3.4MarkovRandomFields
3.3.5Sample-basedDensities
3.4GaussianDensities
3.4.1GaussianSmoothnessPriors
3.5InterpretingthePosteriorDistribution
3.6MarkovChainMonteCarloMethods
3.6.1TheBasicIdea
3.6.2Metropolis-HastingsConstructionoftheKernel
3.6.3GibbsSampler
3.6.4Convergence
3.7HierarcicalModels
3.8NotesandComments

4NonstationaryInverseProblems
4.1BayesianFiltering
4.1.1ANonstationaryInverseProblem
4.1.2EvolutionandObservationModels
4.2KalmanFilters
4.2.1LinearGaussianProblems
4.2.2ExtendedKalmanFilters
4.3ParticleFilters
4.4SpatialPriors
4.5Fixed-lagandFixed-intervalSmoothing
4.6Higher-orderMarkovModels
4.7NotesandComments

5ClassicalMethodsRevisited
5.1EstimationTheory
5.1.1MaximumLikelihoodEstimation
5.1.2EstimatorsInducedbyBayesCosts
5.1.3EstimationErrorwithAffineEstimators
5.2TestCases
5.2.1PriorDistributions
5.2.2ObservationOperators
5.2.3TheAdditiveNoiseModels
5.2.4TestProblems
5.3Sample-BasedErrorAnalysis
5.4TruncatedSingularValueDecomposition
5.5ConjugateGradient.Iteration
5.6TikhonovRegularization
5.6.1PriorStructureandRegularizationLevel
5.6.2MisspeeificationoftheGaussianObservationErrorModel
5.6.3AdditiveCauchyErrors
5.7DiseretizationandPriorModels
5.8StatisticalModelReduction，ApproximationErrorsandInverseCrimes
5.8.1AnExample：FullAngleTomographyandCGNE
5.9NotesandComments

6ModelProblems
6.1X-rayTomography
6.1.1RadonTransform
6.1.2DiscreteModel
6.2InverseSourceProblems
6.2.1Quasi-staticMaxwell'sEquations
6.2.2ElectricInverseSourceProblems
6.2.3MagneticInverseSourceProblems
6.3ImpedanceTomography
6.4OpticalTomography
6.4.1TheRadiationTransferEquation
6.4.2DiffusionApproximation
6.4.3Time-harmonicMeasurement
6.5NotesandComments

7CaseStudies
7.1ImageDeblurringandRecoveryofAnomalies
7.1.1TheModelProblem
7.1.2ReducedandApproximationErrorModels
7.1.3SamplingthePosteriorDistribution
7.1.4EffectsofModellingErrors
7.2LimitedAngleTomography：DentalX-rayImaging
7.2.1TheLayerEstimation
7.2.2MAPEstimates
7.2.3Sampling：GibbsSampler
7.3BiomagneticInverseProblem：SourceLocalization
7.3.1ReconstructionwithGaussianWhiteNoisePriorModel
7.3.2ReconstructionofDipoleStrengthswiththee1-priorModel
7.4DynamicMEGbyBayesFiltering
7.4.1ASingleDipoleModel
7.4.2MoreRealisticGeometry
7.4.3MultipleDipoleModels
7.5ElectricalImpedanceTomography：OptimalCurrentPatterns
7.5.1APosterioriSynthesizedCurrentPatterns
7.5.2OptimizationCriterion
7.5.3NumericalExamples
7.6ElectricalImpedanceTomography：HandlingApproximationErrors
7.6.1MeshesandProjectors
7.6.2ThePriorDistributionandthePriorModel
7.6.3TheEnhancedErrorModel
7.6.4TheMAPEstimates
7.7ElectricalImpedanceProcessTomography
7.7.1TheEvolutionModel
7.7.2TheObservationModelandtheComputationalScheme
7.7.3TheFixed-lagStateEstimate
7.7.4EstimationoftheFlowProfile
7.8OpticalTomographyinAnisotropicMedia
7.8.1TheAnisotropyModel
7.8.2LinearizedModel
7.9OpticalTomography：BoundaryRecovery
7.9.1TheGeneralEllipticCase
7.9.2ApplicationtoOpticalDiffusionTomography
7.10NotesandComments

AAppendix：LinearAlgebraandFunctionalAnalysis
A.1LinearAlgebra
A.2FunctionalAnalysis
A.3SobolevSpaces
BAppendix2：BasicsonProbability
B.1BasicConcepts
B.2ConditionalProbabilities
References
Index