計算反演問題中的優化與正則化方法及其應用...

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 ...

Part I Introduction1 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 ReferertcesPart II Regularization Theory and Recent Developments2 Ill-Posed Problems and Methods for Their Numerical Solution Anatoly G. Yagola 2.1 Well-posed and ill-posed prob...