Computational methods for inverse problems /

Computational methods for inverse problems /

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Bibliographic Details
Author / Creator:Vogel, Curtis R.
Imprint:Philadelphia : SIAM, c2002.
Description:xvi, 183 p. : ill. ; 26 cm.
Language:English
Series:Frontiers in applied mathematics ; 23
Subject:
Inverse problems (Differential equations) -- Numerical solutions.
Inverse problems (Differential equations) -- Numerical solutions.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/7408461
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ISBN:0898715504
9780898715507
Notes:Includes bibliography and index.
Table of Contents:
  • 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
  • 4. Statistical Estimation Theory
  • 4.1. Preliminary Definitions and Notation
  • 4.2. Maximum Likelihood Estimation
  • 4.3. Bayesian Estimation
  • 4.4. Linear Least Squares Estimation
  • 4.4.1. 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
  • 7.6.1. 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
  • 9.2.1. 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
Purchased by the Francis H. Dowley Endowed Library Fund bookplate

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