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
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