| Preface 1 Inverse Problems and Interpretation of Measurements 1.1 Introductory Examples 1.2 Inverse Crimes 2 Classical Regularization Methods 2.1 Introduction: Fredholm Equation 2.2 Truncated Singular Value Decomposition 2.3 Tikhonov Regularization 2.3.1 Generalizations of the Tikhonov Regularization 2.4 Regularization by Truncated Iterative Methods 2.4.1 Landweber-Fridman Iteration 2.4.2 Kaczmarz Iteration and ART 2.4.3 Krylov Subspace Methods 2.5 Notes and Comments 3 Statistical Inversion Theory 3.1 Inverse Problems and Bayes' Formula 3.1.1 Estimators 3.2 Construction of the Likelihood Function 3.2.1 Additive Noise 3.2.2 Other Explicit Noise Models 3.2.3 Counting Process Data 3.3 Prior Models 3.3.1 Gaussian Priors 3.3.2 Impulse Prior Densities 3.3.3 Discontinuities 3.3.4 Markov Random Fields 3.3.5 Sample-based Densities 3.4 Gaussian Densities 3.4.1 Gaussian Smoothness Priors 3.5 Interpreting the Posterior Distribution 3.6 Markov Chain Monte Carlo Methods 3.6.1 The Basic Idea 3.6.2 Metropoli-Hastings Constluetion of the Kernel 3.6.3 Gibbs Samples" 3.6.4 Convergence 3.7 Hierarcieal Models 3.8 Notes and Comments 4 Nonstationary Inverse Problems 4.1 Bayesian Filtering 4.1.1 A Nonstationary Inverse Problem 4.1.2 Evolution and Observation Models 4.2 Kalman Filters 4.2.1 Linear Gaussian Problems 4.2.2 Extended Kalman Filters 4.3 Particle Filters 4.4 Spatial Priors 4.5 Fixed-lag and Fixed-interval Smoothing 4.6 Higher-order Markov Models 4.7 Notes and Comments 5 Classical Methods Revisited 5.1 Estimation Theory 5.1.1 Maximum Likelihood Estimation 5.1.2 Estimators Induced by Bayes Costs 5.1.3 Estimation Error with Affine Estimators 5.2 Test Cases 5.2.1 Prior Distributions 5.2.2 Observation Operators 5.2.3 The Additive Noise Models 5.2.4 Test Problems 5.3 Sample-Based Error Analysis 5.4 Truncated Singular Value Decomposition 5.5 Conjugate Gradient Iteration 5.6 Tikhonov Regularization 5.6.1 Prior Structure and Regularization Level 5.6.2 Misspeeifieation of the Gaussian Observation Error Model 5.6.3 Additive Cauchy Errors 5.7 Discretization and Prior Models 5.8 Statistical Model Reduction, Approximation Errors and Inverse Crimes 5.8.1 An Example: Full Angle Tomography and CGNE... …… 6 Model problems 7 Case studies A Linear algebra and functional analysis B Basics on probability References Index |
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