| 《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. |
| Curtis R. Vogel is a Professor in the Department of Mathematical Sciences at Montana State University. His research interests include numerical analysis,mathematical modeling, optimization, inverse and ill-posed problems, and scientific computing. He has written many refereed articles and reports on these topics. |
| 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 |
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