Computational Methods for Inverse Problems

ISBN-10: 0898715504

ISBN-13: 9780898715507

Edition: N/A

Authors: Curtis R. Vogel

List price: $56.00
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Book details

List price: $56.00
Publisher: Society for Industrial and Applied Mathematics
Binding: Paperback
Pages: 199
Size: 7.00" wide x 10.00" long x 0.55" tall
Weight: 1.144
Language: English

Foreword
Preface
Introduction
An Illustrative Example
Regularization by Filtering
A Deterministic Error Analysis
Rates of Convergence
A Posteriori Regularization Parameter Selection
Variational Regularization Methods
Iterative Regularization Methods
Exercises
Analytical Tools
Ill-Posedness and Regularization
Compact Operators, Singular Systems, and the SVD
Least Squares Solutions and the Pseudo-Inverse
Regularization Theory
Optimization Theory
Generalized Tikhonov Regularization
Penalty Functionals
Data Discrepancy Functionals
Some Analysis
Exercises
Numerical Optimization Tools
The Steepest Descent Method
The Conjugate Gradient Method
Preconditioning
Nonlinear CG Method
Newton's Method
Trust Region Globalization of Newton's Method
The BFGS Method
Inexact Line Search
Exercises
Statistical Estimation Theory
Preliminary Definitions and Notation
Maximum Likelihood Estimation
Bayesian Estimation
Linear Least Squares Estimation
Best Linear Unbiased Estimation
Minimum Variance Linear Estimation
The EM Algorithm
An Illustrative Example
Exercises
Image Deblurring
A Mathematical Model for Image Blurring
A Two-Dimensional Test Problem
Computational Methods for Toeplitz Systems
Discrete Fourier Transform and Convolution
The FFT Algorithm
Toeplitz and Circulant Matrices
Best Circulant Approximation
Block Toeplitz and Block Circulant Matrices
Fourier-Based Deblurring Methods
Direct Fourier Inversion
CG for Block Toeplitz Systems
Block Circulant Preconditioners
A Comparison of Block Circulant Preconditioners
Multilevel Techniques
Exercises
Parameter Identification
An Abstract Framework
Gradient Computations
Adjoint, or Costate, Methods
Hessian Computations
Gauss--Newton Hessian Approximation
A One-Dimensional Example
A Convergence Result
Exercises
Regularization Parameter Selection Methods
The Unbiased Predictive Risk Estimator Method
Implementation of the UPRE Method
Randomized Trace Estimation
A Numerical Illustration of Trace Estimation
Nonlinear Variants of UPRE
Generalized Cross Validation
A Numerical Comparison of UPRE and GCV
The Discrepancy Principle
Implementation of the Discrepancy Principle
The L-Curve Method
A Numerical Illustration of the L-Curve Method
Other Regularization Parameter Selection Methods
Analysis of Regularization Parameter Selection Methods
Model Assumptions and Preliminary Results
Estimation and Predictive Errors for TSVD
Estimation and Predictive Errors for Tikhonov Regularization
Analysis of the Discrepancy Principle
Analysis of GCV
Analysis of the L-Curve Method
A Comparison of Methods
Exercises
Total Variation Regularization
Motivation
Numerical Methods for Total Variation
A One-Dimensional Discretization
A Two-Dimensional Discretization
Steepest Descent and Newton's Method for Total Variation
Lagged Diffusivity Fixed Point Iteration
A Primal-Dual Newton Method
Other Methods
Numerical Comparisons
Results for a One-Dimensional Test Problem
Two-Dimensional Test Results
Mathematical Analysis of Total Variation
Approximations to the TV Functional
Exercises
Nonnegativity Constraints
An Illustrative Example
Theory of Constrained Optimization
Nonnegativity Constraints
Numerical Methods for Nonnegatively Constrained Minimization
The Gradient Projection Method
A Projected Newton Method
A Gradient Projection-Reduced Newton Method
A Gradient Projection-CG Method
Other Methods
Numerical Test Results
Results for One-Dimensional Test Problems
Results for a Two-Dimensional Test Problem
Iterative Nonnegative Regularization Methods
Richardson--Lucy Iteration
A Modified Steepest Descent Algorithm
Exercises
Bibliography
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