Front Matter
 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 Scienti_c computing
  1.6 Conclusion
  References
 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 De_nition 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 Fej_er 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 Fej_er type
  3.6 Fej_er processes for nonlinear equations
  3.7 Applied problems with a priori information and methods forsolution
   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 Identi_cation 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 identi_cation 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 identi_cation in parabolic equation
   4.4.1 Natural linearization for recovering b(x) = a(u(T; x))
   4.4.2 Regularized identi_cation of the di_usion coe_cient 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 quali_cation
  5.5 The choice of parameters in the extrapolated regularizing approximation
  5.6 Numerical experiments
  5.7 Conclusion
  References
 6 Modi_ed 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 modi_ed 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 su_cient and necessary condition on _k
   8.3.1 Proposition of the condition
   8.3.2 Su_ciency 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 _eld
  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-Di_erential 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 reection 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 pro_le 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 pro_le reconstruction
  14.4 Conclusions
  References
 Index
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