反问题的计算方法

《反问题的计算方法(英文影印版)》

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

《反问题的计算方法(英文影印版)》

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 a, lgorithm

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. i 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

9　nonnegativity constraints

exercises

bibliography

inverse problems arise in a number of
important practical applications, ranging from biomedical imaging
to seismic prospecting. this book 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.
　　 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.