反问题的计算方法（英文版）
出版时间：2011年版
内容简介
inverse problems arise in a number ofimportant practical applications， ranging from biomedical imagingto seismic prospecting. this book provides the reader with a basicunderstanding of both the underlying mathematics and thecomputational methods used to solve inverse problems. it alsoaddresses specialized topics like image reconstruction， parameteridentification， total variation methods， nonnegativity constraints，and regularization parameter selection methods. because inverse problems typically involve the estimation ofcertain quantities based on indirect measurements， the estimationprocess is often ill-posed. regularization methods， which have beendeveloped to deal with this ill-posedness， are carefully explainedin the early chapters of computational methods for inverseproblems. the book also integrates mathematical and statisticaltheory with applications and practical computational methods，including topics like maximum likelihood estimation and bayesianestimation. several web-based resources are available to make this monographinteractive， including a collection of matlab m-files used togenerate many of the examples and figures. these resources enablereaders to conduct their own computational experiments in order togain insight. they also provide templates for the implementation ofregularization methods and numerical solution techniques for otherinverse problems. moreover， they include some realistic testproblems to be used to develop and test various numericalmethods. computational methods for inverse problems is intended forgraduate students and researchers in applied mathematics，engineering， and the physical sciences who may encounter inverseproblems in their work.
目录
《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 tikhonovregularization
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
