（计算反演问题中的优化与正则化方法及其应用）(国内英文版)Optimization and Regularization for Computational Inverse Problems and Applications

作者： 王彦飞，（俄）亚哥拉，杨长春主编

出 版 社： 高等教育出版社

出版时间： 2010-5-1字数： 620000版次： 1页数： 350印刷时间： 2010-5-1开本： 16开印次： 1纸张： 胶版纸I S B N ： 9787040285154包装： 精装

Optimization and Regularization for Computational Inverse Problems and Applications focuses on advances in inversion theory and recent developments with practical applications, particularly emphasizing the combination of optimization and regularization for solving inverse problems. This book covers both the methods, including standard regularization theory, Fejer processes for linear and nonlinear problems, the balancing principle, extrapolated regularization, nonstandard regularization, nonlinear gradient method, the nonmonotone gradient method, subspace method and Lie group method; and the practical applications, such as the reconstruction problem for inverse scattering, molecular spectra data processing, quantitative remote sensing inversion, seismic inversion using the Lie group method, and the gravitational lensing problem.

Scientists, researchers and engineers, as well as graduate students engaged in applied mathematics, engineering, geophysics, medical science, image processing, remote sensing and atmospheric science will benefit from this book.

Part I Introduction

1 Inverse Problems, Optimization and Regularization: A

Multi-Disciplinary Subject

Yanfei Wang and Changchun Yang

1.1 Introduction

1.2 Examples about mathematical inverse problems

1.3 Examples in applied science and engineering

1.4 Basic theory

1.5 Scientific computing

1.6 Conclusion

Referertces

Part II Regularization Theory and Recent Developments

2 Ill-Posed Problems and Methods for Their Numerical Solution

Anatoly G. Yagola

2.1 Well-posed and ill-posed problems

2.2 Definition of the regularizing algorithm

2.3 Ill-posed problems on compact sets

2.4 Ill-posed problems with sourcewise represented solutions

2.5 Variational approach for constructing regularizing algorithms

2.6 Nonlinear ill-posed problems

2.7 Iterative and other methods

References

3 Inverse Problems with A Priori Information

Vladimir V. Vasin

3.1 Introduction

3.2 Formulation of the problem with a priori information

3.3 The main classes of mappings of the Fejer type and their properties

3.4 Convergence theorems of the method of successive approximations for the pseudo-contractive operators

3.5 Examples of operators of the Fejer type

3.6 Fejer processes for nonlinear equations

3.7 Applied problems with a priori information and methods for solution

3.7.1 Atomic structure characterization

3.7.2 Radiolocation of the ionosphere

3.7.3 Image reconstruction

3.7.4 Thermal sounding of the atmosphere

3.7.5 Testing a wellbore/reservoir

3.8 Conclusions

References

4 Regularization of Naturally Linearized Parameter

Identification Problems and the Application of the Balancing

Principle

Hui Cao and Sergei Pereverzyev

4.1 Introduction

4.2 Discretized Tikhonov regularization and estimation of accuracy

4.2.1 Generalized source condition

4.2.2 Discretized Tikhonov regularization

4.2.3 Operator monotone index functions

4.2.4 Estimation of the accuracy

4.3 Parameter identification in elliptic equation

4.3.1 Natural linearization

4.3.2 Data smoothing and noise level analysis

4.3.3 Estimation of the accuracy

4.3.4 Balancing principle

4.3.5 Numerical examples

4.4 Parameter identification in parabolic equation

4.4.1 Natural linearization for recovering b(x) = a(u(T, x))

4.4.2 Regularized identification of the diffusion coefficient a(u)

4.4.3 Extended balancing principle

4.4.4 Numerical examples

References

5 Extrapolation Techniques of Tikhonov Regularization

Tingyan Xiao, Yuan Zhao and Guozhong Su

5.1 Introduction

5.2 Notations and preliminaries

5.3 Extrapolated regularization based on vector-valued function approximation

5.3.1 Extrapolated scheme based on Lagrange interpolation

5.3.2 Extrapolated scheme based on Hermitian interpolation

5.3.3 Extrapolation scheme based on rational interpolation

5.4 Extrapolated regularization based on improvement of regularizing qualification

5.5 The choice of parameters in the extrapolated regularizing approximation

5.6 Numerical experiments

5.7 Conclusion

References

6 Modified Regularization Scheme with Application in Reconstructing Neumann-Dirichlet Mapping

Pingli Xie and Jin Cheng

6.1 Introduction

6.2 Regularization method

6.3 Computational aspect

6.4 Numerical simulation results for the modified regularization

6.5 The Neumann-Dirichlet mapping for elliptic equation of second order

6.6 The numerical results of the Neumann-Dirichlet mapping

6.7 Conclusion

References

Part III Nonstandard Regularization and Advanced Optimization Theory and Methods

7 Gradient Methods for Large Scale Convex Quadratic Functions

Yaxiang Yuan

7.1 Introduction

7.2 A generalized convergence result

7.3 Short BB steps

7.4 Numerical results

7.5 Discussion and conclusion

References

8 Convergence Analysis of Nonlinear Conjugate Gradient

Methods

Yuhong Dai

8.1 Introduction

8.2 Some preliminaries

8.3 A sufficient and necessary condition on βk

8.3.1 Proposition of the condition

8.3.2 Sufficiency of (8.3.5)

8.3.3 Necessity of (8.3.5)

8.4 Applications of the condition (8.3.5)

8.4.1 Property (#)

8.4.2 Applications to some known conjugate gradient methods

8.4.3 Application to a new conjugate gradient method

8.5 Discussion

References

9 Full Space and Subspace Methods for Large Scale Image

Restoration

Yanfei Wang, Shiqian Ma and Qinghua Ma

9.1 Introduction

9.2 Image restoration without regularization

9.3 Image restoration with regularization

9.4 Optimization methods for solving the smoothing regularized functional

9.4.1 Minimization of the convex quadratic programming problem with projection

9.4.2 Limited memory BFGS method with projection

9.4.3 Subspace trust region methods

9.5 Matrix-Vector Multiplication (MVM)

9.5.1 MVM: FFT-based method

9.5.2 MVM with sparse matrix

9.6 Numerical experiments

9.7 Conclusions

References

Part IV Numerical Inversion in Geoscience and Quantitative Remote Sensing

10 Some Reconstruction Methods for Inverse Scattering Problems

Jijun Liu and Haibing Wang

10.1 Introduction

10.2 Iterative methods and decomposition methods

10.2.1 Iterative methods

10.2.2 Decomposition methods

10.2.3 Hybrid method

10.3 Singular source methods

10.3.1 Probe method

10.3.2 Singular sources method

10.3.3 Linear sampling method

10.3.4 Factorization method

10.3.5 Range test method

10.3.6 No response test method

10.4 Numerical schemes

References

11 Inverse Problems of Molecular Spectra Data Processing

Gulnara Kuramshina

11.1 Introduction

11.2 Inverse vibrational problem

11.3 The mathematical formulation of the inverse vibrational problem

11.4 Regularizing algorithms for solving the inverse vibrational problem

11.5 Model of scaled molecular force field

11.6 General inverse problem of structural chemistry

11.7 Intermolecular potential

11.8 Examples of calculations

11.8.1 Calculation of methane intermolecular potential

11.8.2 Prediction of vibrational spectrum of fullerene C240

References

12 Numerical Inversion Methods in Geoscience and Quantitative

Remote Sensing

Yanfei Wang and Xiaowen Li

12.1 Introduction

12.2 Examples of quantitative remote sensing inverse problems: land

surface parameter retrieval problem

12.3 Formulation of the forward and inverse problem

12.4 What causes ill-posedness

12.5 Tikhonov variational regularization

12.5.1 Choices of the scale operator D

12.5.2 Regularization parameter selection methods

12.6 Solution methods

12.6.1 Gradient-type methods

12.6.2 Newton-type methods

12.7 Numerical examples

12.8 Conclusions

References

13 Pseudo-Differential Operator and Inverse Scattering of

Multidimensional Wave Equation

Hong Liu, Li He

13.1 Introduction

13.2 Notations of operators and symbols

13.3 Description in symbol domain

13.4 Lie algebra integral expressions

13.5 Wave equation on the ray coordinates

13.6 Symbol expression of one-way wave operator equations

13.7 Lie algebra expression of travel time

13.8 Lie algebra integral expression of prediction operator

13.9 Spectral factorization expressions of reflection data

13.10 Conclusions

References

14 Tikhonov Regularization for Gravitational Lensing Research.

Boris Artamonov, Ekaterina Koptelova, Elena Shimanovskaya and

Anatoly G. Yagola

14.1 Introduction

14.2 Regularized deconvolution of images with point sources and smooth background

14.2.1 Formulation of the problem

14.2.2 Tikhonov regularization approach

14.2.3 A priori information

14.3 Application of the Tikhonov regularization approach to quasar profile reconstruction

14.3.1 Brief introduction to microlensing

14.3.2 Formulation of the problem

14.3.3 Implementation of the Tikhonov regularization approach

14.3.4 Numerical results of the Q2237 profile reconstruction

14.4 Conclusions

References

Index