统计和计算逆问题

内容简介

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　　This book is aimed at postgraduate students in applied mathematics as well as at engineering and physics students with a firm background in mathematics. The first four chapters can be used as the material for a first course on inverse problems with a focus on computational and statistical aspects. On the other hand, Chapters 3 and 4, which discuss statistical and nonstationary inversion methods, can be used by students already having knowldege of classical inversion methods.　　There is rich literature, including numerous textbooks, on the classical aspects of inverse problems. From the numerical point of view, these books concentrate on problems in which the measurement errors are either very small or in,which the error properties are known exactly. In real world problems, however, the errors are seldom very small and their properties in the deterministic sense are not well known. For example, in classical literature the error norm is usually assumed to be a known real number. In reality, the error norm is a random variable whose mean might be known.

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目录

Preface1 Inverse Problems and Interpretation of Measurements1.1 Introductory Examples1.2 Inverse Crimes2 Classical Regularization Methods2.1 Introduction： Fredholm Equation2.2 Truncated Singular Value Decomposition2.3 Tikhonov Regularization2.3.1 Generalizations of the Tikhonov Regularization2.4 Regularization by Truncated Iterative Methods2.4.1 Landweber-Fridman Iteration2.4.2 Kaczmarz Iteration and ART2.4.3 Krylov Subspace Methods2.5 Notes and Comments3 Statistical Inversion Theory3.1 Inverse Problems and Bayes’ Formula3.1.1 Estimators3.2 Construction of the Likelihood Function3.2.1 Additive Noise3.2.2 Other Explicit Noise Models3.2.3 Counting Process Data3.3 Prior Models3.3.1 Gaussian Priors3.3.2 Impulse Prior Densities3.3.3 Discontinuities3.3.4 Markov Random Fields3.3.5 Sample-based Densities3.4 Gaussian Densities3.4.1 Gaussian Smoothness Priors3.5 Interpreting the Posterior Distribution3.6 Markov Chain Monte Carlo Methods3.6.1 The Basic Idea3.6.2 Metropolis-Hastings Construction of the Kernel3.6.3 Gibbs Sampler3.6.4 Convergence3.7 Hierarcical Models3.8 Notes and Comments4 Nonstationary Inverse Problems4.1 Bayesian Filtering4.1.1 A Nonstationary Inverse Problem4.1.2 Evolution and Observation Models4.2 Kalman Filters4.2.1 Linear Gaussian Problems4.2.2 Extended Kalman Filters4.3 Particle Filters4.4 Spatial Priors4.5 Fixed-lag and Fixed-interval Smoothing4.6 Higher-order Markov Models4.7 Notes and Comments5 Classical Methods Revisited5.1 Estimation Theory5.1.1 Maximum Likelihood Estimation5.1.2 Estimators Induced by Bayes Costs5.1.3 Estimation Error with Affine Estimators5.2 Test Cases5.2.1 Prior Distributions5.2.2 Observation Operators5.2.3 The Additive Noise Models5.2.4 Test Problems5.3 Sample-Based Error Analysis5.4 Truncated Singular Value Decomposition5.5 Conjugate Gradient.Iteration5.6 Tikhonov Regularization5.6.1 Prior Structure and Regularization Level5.6.2 Misspeeification of the Gaussian Observation Error Model5.6.3 Additive Cauchy Errors5.7 Diseretization and Prior Models5.8 Statistical Model Reduction， Approximation Errors and Inverse Crimes5.8.1 An Example： Full Angle Tomography and CGNE5.9 Notes and Comments6 Model Problems6.1 X-ray Tomography6.1.1 Radon Transform6.1.2 Discrete Model6.2 Inverse Source Problems6.2.1 Quasi-static Maxwell’s Equations6.2.2 Electric Inverse Source Problems6.2.3 Magnetic Inverse Source Problems6.3 Impedance Tomography6.4 Optical Tomography6.4.1 The Radiation Transfer Equation6.4.2 Diffusion Approximation6.4.3 Time-harmonic Measurement6.5 Notes and Comments7 Case Studies7.1 Image Deblurring and Recovery of Anomalies7.1.1 The Model Problem7.1.2 Reduced and Approximation Error Models7.1.3 Sampling the Posterior Distribution7.1.4 Effects of Modelling Errors7.2 Limited Angle Tomography： Dental X-ray Imaging7.2.1 The Layer Estimation7.2.2 MAP Estimates7.2.3 Sampling： Gibbs Sampler7.3 Biomagnetic Inverse Problem： Source Localization7.3.1 Reconstruction with Gaussian White Noise Prior Model7.3.2 Reconstruction of Dipole Strengths with the e1-prior Model7.4 Dynamic MEG by Bayes Filtering7.4.1 A Single Dipole Model7.4.2 More Realistic Geometry7.4.3 Multiple Dipole Models7.5 Electrical Impedance Tomography： Optimal Current Patterns7.5.1 A Posteriori Synthesized Current Patterns7.5.2 Optimization Criterion7.5.3 Numerical Examples7.6 Electrical Impedance Tomography： Handling Approximation Errors7.6.1 Meshes and Projectors7.6.2 The Prior Distribution and the Prior Model7.6.3 The Enhanced Error Model7.6.4 The MAP Estimates7.7 Electrical Impedance Process Tomography7.7.1 The Evolution Model7.7.2 The Observation Model and the Computational Scheme7.7.3 The Fixed-lag State Estimate7.7.4 Estimation of the Flow Profile7.8 Optical Tomography in Anisotropic Media7.8.1 The Anisotropy Model7.8.2 Linearized Model7.9 Optical Tomography： Boundary Recovery7.9.1 The General Elliptic Case7.9.2 Application to Optical Diffusion Tomography7.10 Notes and CommentsA Appendix： Linear Algebra and Functional AnalysisA.1 Linear AlgebraA.2 Functional AnalysisA.3 Sobolev SpacesB Appendix 2： Basics on ProbabilityB.1 Basic ConceptsB.2 Conditional ProbabilitiesReferencesIndex

封面

书名:统计和计算逆问题

作者:凯皮奥

页数:339

定价:¥39.0

出版社:世界图书出版公司

出版日期:2015-01-01

ISBN:9787510086311

PDF电子书大小:144MB 高清扫描完整版