Statistical and computational inverse problems统计的和计算的反演问题
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作者： Jari Kaipio 著
出 版 社：
出版时间： 2004-12-1字数：版次： 1页数： 339印刷时间： 2004/12/01开本： 16开印次： 1纸张： 胶版纸I S B N ： 9780387220734包装： 精装内容简介
The book develops the statistical approach to inverse problems with an emphasis on modeling and computations. The framework is the Bayesian paradigm, where all variables are modeled as random variables, the randomness reflecting the degree of belief of their values, and the solution of the inverse problem is expressed in terms of probability densities. The book discusses in detail the construction of prior models, the measurement noise modeling and Bayesian estimation. Markov Chain Monte Carlo-methods as well as optimization methods are employed to explore the probability distributions. The results and techniques are clarified with classroom examples that are often non-trivial but easy to follow. Besides the simple examples, the book contains previously unpublished research material, where the statistical approach is developed further to treat such problems as discretization errors, and statistical model reduction. Furthermore, the techniques are then applied to a number of real world applications such as limited angle tomography, image deblurring, electrical impedance tomography and biomagnetic inverse problems. The book is intended to researchers and advanced students in applied mathematics, computational physics and engineering. The first part of the book can be used as a text book on advanced inverse problems courses.
The authors Jari Kaipio and Erkki Somersalo are Professors in the Applied Physics Department of the University of Kuopio, Finland and the Mathematics Department at the Helsinki University of Technology, Finland, respectively.
目录
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...
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6 Model problems
7 Case studies
A Linear algebra and functional analysis
B Basics on probability
References
Index
