《Computational Methods for Inverse Problems》(Curtis R. Vogel) provides the reader with a basic understanding of both the underlying mathematics and the computational methods used to solve inverse problems. It also addresses specialized topics like image reconstruction, parameter identification, total variation methods, nonnegativity constraints, and regularization parameter selection methods.

Inverse problems arise in a number of important practical applications,ranging from biomedical imaging to seismic prospecting.《Computational Methods for Inverse Problems》(Curtis R. Vogel) provides the reader with a basic understanding of both the underlying mathematics and the computational methods used to solve inverse problems.《Computational Methods for Inverse Problems》 also addresses specialized topics like image reconstruction, parameter identification, total variation methods, nonnegativity constraints, and regularization parameter selection methods.

Because inverse problems typically involve the estimation of certain quantities based on indirect measurements, the estimation process is often ill-posed. Regularization methods, which have been developed to deal with this ill-posedness, are carefully explained in the early chapters of Computational Methods for Inverse Problems. The book also integrates mathematical and statistical theory with applications and practical computational methods, including topics like maximum likelihood estimation and Bayesian estimation.

Several web-based resources are available to make this monograph interactive, including a collection of MATLAB m-files used to generate many of the examples and figures. These resources enable readers to conduct their own computational experiments in order to gain insight. They also provide templates for the implementation of regularization methods and numerical solution techniques for other inverse problems. Moreover, they include some realistic test problems to be used to develop and test various numerical methods.

Computational Methods for Inverse Problems is intended for graduate students and researchers in applied mathematics, engineering, and the physical sciences who may encounter inverse problems in their work.

Foreword

Preface

1 Introduction

1.1 An Illustrative Example

1.2 Regularization by Filtering

1.2.1 A Deterministic Error Analysis

1.2.2 Rates of Convergence

1.2.3 A Posteriori Regularization Parameter Selection

1.3 Variational Regularization Methods

1.4 Iterative Regularization Methods

Exercises

2 Analytical Tools

2.1 Ill-Posedness and Regularization

2.1.1 Compact Operators, Singular Systems, and the SVD

2.1.2 Least Squares Solutions and the Pseudo-Inverse .

2.2 Regularization Theory

2.3 Optimization Theory

2.4 Generalized Tikhonov Regularization

2.4.1 Penalty Functionals

2.4.2 Data Discrepancy Functionals

2.4.3 Some Analysis

Exercises

3 Numerical Optimization Tools

3.1 The Steepest Descent Method

3.2 The Conjugate Gradient Method

3.2.1 Preconditioning

3.2.2 Nonlinear CG Method

3.3 Newton's Method

3.3.1 Trust Region Globalization of Newton's Method

3.3.2 The BFGS Method

3.4 Inexact Line Search

Exercises

6 Statistical Estimation Theory

4.1 Preliminary Definitions and Notation

4.2 Maximum Likelihood Estimation

4.3 Bayesian Estimation

4.4 Linear Least Squares Estimation

4.4. l Best Linear Unbiased Estimation

4.4.2 Minimum Variance Linear Estimation

4.5 The EM Algorithm

4.5.1 An Illustrative Example

Exercises

5 Image Deblurring

5.1 A Mathematical Model for Image Blurring

5.1.1 A Two-Dimensional Test Problem

5.2 Computational Methods for Toeplitz Systems

5.2.1 Discrete Fourier Transform and Convolution

5.2.2 The FFT Algorithm

5.2.3 Toeplitz and Circulant Matrices

5.2.4 Best Circulant Approximation

5.2.5 Block Toeplitz and Block Circulant Matrices

5.3 Fourier-Based Deblurring Methods

5.3.1 Direct Fourier Inversion

5.3.2 CG for Block Toeplitz Systems

5.3.3 Block Circulant Preconditioners

5.3.4 A Comparison of Block Circulant Preconditioners

5.4 Multilevel Techniques

Exercises

6 Parameter Identification

6.1 An Abstract Framework

6.1.1 Gradient Computations

6.1.2 Adjoint, or Costate, Methods

6.1.3 Hessian Computations

6.1.4 Gauss-Newton Hessian Approximation

6.2 A One-Dimensional Example

6.3 A Convergence Result

Exercises

7 Regularization Parameter Selection Methods

7.1 The Unbiased Predictive Risk Estimator Method

7.1.1 Implementation of the UPRE Method~.

7.1.2 Randomized Trace Estimation

7.1.3 A Numerical Illustration of Trace Estimation

7.1.4 Nonlinear Variants of UPRE

7.2 Generalized Cross Validation

7.2.1 A Numerical Comparison of UPRE and GCV

7.3 The Discrepancy Principle

7.3.1 Implementation of the Discrepancy Principle

7.4 The L-Curve Method

7.4.1 A Numerical Illustration of the L-Curve Method

7.5 Other Regularization Parameter Selection Methods

7.6 Analysis of Regularization Parameter Selection Methods

7.6. l Model Assumptions and Preliminary Results

7.6.2 Estimation and Predictive Errors for TSVD

7.6.3 Estimation and Predictive Errors for Tikhonov Regularization

7.6.4 Analysis of the Discrepancy Principle

7.6.5 Analysis of GCV

7.6.6 Analysis of the L-Curve Method

7.7 A Comparison of Methods

Exercises

8 Total Variation Regularization

8.1 Motivation

8.2 Numerical Methods for Total Variation

8.2.1 A One-Dimensional Discretization

8.2.2 A Two-Dimensional Discretization

8.2.3 Steepest Descent and Newton's Method for Total Variation

8.2.4 Lagged Diffusivity Fixed Point Iteration

8.2.5 A Primal-Dual Newton Method

8.2.6 Other Methods

8.3 Numerical Comparisons

8.3.1 Results for a One-Dimensional Test Problem

8.3.2 Two-Dimensional Test Results

8.4 Mathematical Analysis of Total Variation

8.4.1 Approximations to the TV Functional

Exercises

9 Nonnegativity Constraints

9.1 An Illustrative Example

9.2 Theory of Constrained Optimization

9.2. l Nonnegativity Constraints

9.3 Numerical Methods for Nonnegatively Constrained Minimization

9.3.1 The Gradient Projection Method

9.3.2 A Projected Newton Method

9.3.3 A Gradient Projection-Reduced Newton Method

9.3.4 A Gradient Projection-CG Method

9.3.5 Other Methods

9.4 Numerical Test Results

9.4.1 Results for One-Dimensional Test Problems

9.4.2 Results for a Two-Dimensional Test Problem

9.5 Iterative Nonnegative Regularization Methods

9.5.1 Richardson-Lucy Iteration

9.5.2 A Modified Steepest Descent Algorithm

Exercises

Bibliography

Index