The book covers many directions in the modern theory of inverse and illposed problems： mathematical physics， optimal inverse design， inverse scattering， inverse vibration， biomedical imaging， oceanography， seismic imaging and remote sensing； methods including standard regularization，parallel computing for multidimensional problems， Nystr6m method，numerical differentiation， analytic continuation， perturbation regularization，filtering， optimization and sparse solving methods are fully addressed.
- 目录
- Front Matter
- I Introduction
- 1 S. I. Kabanikhin Inverse Problems of Mathematical Physics
- 1.1 Introduction
- 1.2 Examples of Inverse and Ill-posed Problems
- 1.3 Well-posed and Ill-posed Problems
- 1.4 The Tikhonov Theorem
- 1.5 The Ivanov Theorem: Quasi-solution
- 1.6 The Lavrentiev’sMethod
- 1.7 The Tikhonov RegularizationMethod
- References
- II Recent Advances in Regularization Theory and Methods
- 2 D. V. Lukyanenko and A. G. Yagola Using Parallel Computing for Solving Multidimensional Ill-posed Problems
- 2.1 Introduction
- 2.2 Using Parallel Computing
- 2.2.1 Main idea of parallel computing
- 2.2.2 Parallel computing limitations
- 2.3 Parallelization of Multidimensional Ill-posed Problem
- 2.3.1 Formulation of the problem and method of solution
- 2.3.2 Finite-difference approximation of the functional and its gradient
- 2.3.3 Parallelization of the minimization problem
- 2.4 Some Examples of Calculations
- 2.5 Conclusions
- References
- 3 M. T. Nair Regularization of Fredholm Integral Equations of the First Kind using Nystr Approximation
- 3.1 Introduction
- 3.2 NystrMethod for Regularized Equations
- 3.2.1 Nystr approximation of integral operators
- 3.2.2 Approximation of regularized equation
- 3.2.3 Solvability of approximate regularized equation
- 3.2.4 Method of numerical solution
- 3.3 Error Estimates
- 3.3.1 Some preparatory results
- 3.3.2 Error estimate with respect to _ · _2
- 3.3.3 Error estimate with respect to _ · _∞
- 3.3.4 A modified method
- 3.4 Conclusion
- References
- 4 T. Y. Xiao, H. Zhang and L. L. Hao Regularization of Numerical Differentiation: Methods and Applications
- 4.1 Introduction
- 4.2 Regularizing Schemes
- 4.2.1 Basic settings
- 4.2.2 Regularized difference method (RDM)
- 4.2.3 Smoother-Based regularization (SBR)
- 4.2.4 Mollifier regularization method (MRM)
- 4.2.5 Tikhonov’s variational regularization (TiVR)
- 4.2.6 Lavrentiev regularization method (LRM)
- 4.2.7 Discrete regularizationmethod (DRM)
- 4.2.8 Semi-Discrete Tikhonov regularization (SDTR)
- 4.2.9 Total variation regularization (TVR)
- 4.3 Numerical Comparisons
- 4.4 Applied Examples
- 4.4.1 Simple applied problems
- 4.4.2 The inverse heat conduct problems (IHCP)
- 4.4.3 The parameter estimation in new product diffusion model
- 4.4.4 Parameter identification of sturm-liouville operator
- 4.4.5 The numerical inversion of Abel transform
- 4.4.6 The linear viscoelastic stress analysis
- 4.5 Discussion and Conclusion
- References
- 5 C. L. Fu, H. Cheng and Y. J. Ma Numerical Analytic Continuation and Regularization
- 5.1 Introduction
- 5.2 Description of the Problems in Strip Domain and Some Assumptions
- 5.2.1 Description of the problems
- 5.2.2 Some assumptions
- 5.2.3 The ill-posedness analysis for the Problems 5.2.1 and 5.2.2
- 5.2.4 The basic idea of the regularization for Problems 5.2.1 and 5.2.2
- 5.3 Some RegularizationMethods
- 5.3.1 Some methods for solving Problem 5.2.1
- 5.3.2 Some methods for solving Problem 5.2.2
- 5.4 Numerical Tests
- References
- 6 G. S. Li An Optimal Perturbation Regularization Algorithm for Function Reconstruction and Its Applications
- 6.1 Introduction
- 6.2 The Optimal Perturbation Regularization Algorithm
- 6.3 Numerical Simulations
- 6.3.1 Inversion of time-dependent reaction coefficient
- 6.3.2 Inversion of space-dependent reaction coefficient
- 6.3.3 Inversion of state-dependent source term
- 6.3.4 Inversion of space-dependent diffusion coefficient
- 6.4 Applications
- 6.4.1 Determining magnitude of pollution source
- 6.4.2 Data reconstruction in an undisturbed soil-column experiment
- 6.5 Conclusions
- References
- 7 L. V. Zotov and V. L. Panteleev Filtering and Inverse Problems Solving
- 7.1 Introduction
- 7.2 SLAE Compatibility
- 7.3 Conditionality
- 7.4 Pseudosolutions
- 7.5 Singular Value Decomposition
- 7.6 Geometry of Pseudosolution
- 7.7 Inverse Problems for the Discrete Models of Observations
- 7.8 TheModel in Spectral Domain
- 7.9 Regularization of Ill-posed Systems
- 7.10 General Remarks, the Dilemma of Bias and Dispersion
- 7.11 Models, Based on the Integral Equations
- 7.12 Panteleev Corrective Filtering
- 7.13 Philips-Tikhonov Regularization
- References
- III Optimal Inverse Design and Optimization Methods
- 8 G. S. Dulikravich and I. N. Egorov Inverse Design of Alloys’ Chemistry for Specified Thermo-Mechanical Properties by using Multi-objective Optimization
- 8.1 Introduction
- 8.2 Multi-Objective Constrained Optimization and Response Surfaces
- 8.3 Summary of IOSO Algorithm
- 8.4 Mathematical Formulations of Objectives and Constraints
- 8.5 Determining Names of Alloying Elements and Their Concentrations for Specified Properties of Alloys
- 8.6 Inverse Design of Bulk Metallic Glasses
- 8.7 Open Problems
- 8.8 Conclusions
- References
- 9 Z. H. Xiang Two Approaches to Reduce the Parameter Identification Errors
- 9.1 Introduction
- 9.2 The Optimal Sensor Placement Design
- 9.2.1 The well-posedness analysis of the parameter identification procedure
- 9.2.2 The algorithm for optimal sensor placement design
- 9.2.3 The integrated optimal sensor placement and parameter identification algorithm
- 9.2.4 Examples
- 9.3 The Regularization Method with the Adaptive Updating of Apriori Information
- 9.3.1 Modified extended Bayesian method for parameter identification
- 9.3.2 The well-posedness analysis of modified extended Bayesianmethod
- 9.3.3 Examples
- 9.4 Conclusion
- References
- 10 Y. H. Dai A General Convergence Result for the BFGS Method
- 10.1 Introduction
- 10.2 The BFGS Algorithm
- 10.3 A General Convergence Result for the BFGS Algorithm
- 10.4 Conclusion and Discussions
- References
- IV Recent Advances in Inverse Scattering
- 11 X. D. Liu and B. Zhang Uniqueness Results for Inverse Scattering Problems
- 11.1 Introduction
- 11.2 Uniqueness for Inhomogeneity n
- 11.3 Uniqueness for Smooth Obstacles
- 11.4 Uniqueness for Polygon or Polyhedra
- 11.5 Uniqueness for Balls or Discs
- 11.6 Uniqueness for Surfaces or Curves
- 11.7 Uniqueness Results in a LayeredMedium
- 11.8 Open Problems
- References
- 12 G. Bao and P. J. Li Shape Reconstruction of Inverse Medium Scattering for the Helmholtz Equation
- 12.1 Introduction
- 12.2 Analysis of the scatteringmap
- 12.3 Inversemedium scattering
- 12.3.1 Shape reconstruction
- 12.3.2 Born approximation
- 12.3.3 Recursive linearization
- 12.4 Numerical experiments
- 12.5 Concluding remarks
- References
- V Inverse Vibration, Data Processing and Imaging
- 13 G. M. Kuramshina, I. V. Kochikov and A. V. Stepanova Numerical Aspects of the Calculation of Molecular Force Fields from Experimental Data
- 13.1 Introduction
- 13.2 Molecular Force FieldModels
- 13.3 Formulation of Inverse Vibration Problem
- 13.4 Constraints on the Values of Force Constants Based on Quantum Mechanical Calculations
- 13.5 Generalized Inverse Structural Problem
- 13.6 Computer Implementation
- 13.7 Applications
- References
- 14 J. J. Liu and H. L. Xu Some Mathematical Problems in Biomedical Imaging
- 14.1 Introduction
- 14.2 MathematicalModels
- 14.2.1 Forward problem
- 14.2.2 Inverse problem
- 14.3 Harmonic Bz Algorithm
- 14.3.1 Algorithmdescription
- 14.3.2 Convergence analysis
- 14.3.3 The stable computation of ΔBz
- 14.4 Integral EquationsMethod
- 14.4.1 Algorithmdescription
- 14.4.2 Regularization and discretization
- 14.5 Numerical Experiments
- References
- VI Numerical Inversion in Geosciences
- 15 S. I. Kabanikhin and M. A. Shishlenin Numerical Methods for Solving Inverse Hyperbolic Problems
- 15.1 Introduction
- 15.2 Gel’fand-Levitan-KreinMethod
- 15.2.1 The two-dimensional analogy of Gel’fand-Levitan-Krein equation
- 15.2.2 N-approximation of Gel’fand-Levitan-Krein equation
- 15.2.3 Numerical results and remarks
- 15.3 Linearized Multidimensional Inverse Problem for the Wave Equation
- 15.3.1 Problemformulation
- 15.3.2 Linearization
- 15.4 Modified Landweber Iteration
- 15.4.1 Statement of the problem
- 15.4.2 Landweber iteration
- 15.4.3 Modification of algorithm
- 15.4.4 Numerical results
- References
- 16 H. B. Song, X. H. Huang, L. M. Pinheiro, Y. Song, C. Z. Dong and Y.Bai Inversion Studies in Seismic Oceanography
- 16.1 Introduction of Seismic Oceanography
- 16.2 Thermohaline Structure Inversion
- 16.2.1 Inversion method for temperature and salinity
- 16.2.2 Inversion experiment of synthetic seismic data
- 16.2.3 Inversion experiment of GO data (Huang et al., 2011)
- 16.3 Discussion and Conclusion
- References
- 17 L. J. Gelius Image Resolution Beyond the Classical Limit
- 17.1 Introduction
- 17.2 Aperture and Resolution Functions
- 17.3 Deconvolution Approach to Improved Resolution
- 17.4 MUSIC Pseudo-Spectrum Approach to Improved Resolution
- 17.5 Concluding Remarks
- References
- 18 Y. F. Wang, Z. H. Li and C. C. Yang Seismic Migration and Inversion
- 18.1 Introduction
- 18.2 MigrationMethods: A Brief Review
- 18.2.1 Kirchhoffmigration.
- 18.2.2 Wave field extrapolation
- 18.2.3 Finite difference migration in ω − X domain
- 18.2.4 Phase shift migration.
- 18.2.5 Stoltmigration
- 18.2.6 Reverse timemigration
- 18.2.7 Gaussian beam migration
- 18.2.8 Interferometricmigration
- 18.2.9 Ray tracing
- 18.3 SeismicMigration and Inversion
- 18.3.1 The forwardmodel
- 18.3.2 Migration deconvolution
- 18.3.3 Regularization model
- 18.3.4 Solving methods based on optimization
- 18.3.5 Preconditioning
- 18.3.6 Preconditioners
- 18.4 Illustrative Examples
- 18.4.1 Regularized migration inversion for point diffraction scatterers
- 18.4.2 Comparison with the interferometric migration
- 18.5 Conclusion
- References
- 19 Y. F. Wang, J. J. Cao, T. Sun and C. C. Yang Seismic Wavefields Interpolation Based on Sparse Regularization and Compressive Sensing
- 19.1 Introduction
- 19.2 Sparse Transforms
- 19.2.1 Fourier, wavelet, Radon and ridgelet transforms
- 19.2.2 The curvelet transform
- 19.3 Sparse RegularizingModeling
- 19.3.1 Minimization in l0 space
- 19.3.2 Minimization in l1 space
- 19.3.3 Minimization in lp-lq space
- 19.4 Brief Review of Previous Methods in Mathematics
- 19.5 Sparse OptimizationMethods
- 19.5.1 l0 quasi-normapproximationmethod
- 19.5.2 l1-normapproximationmethod
- 19.5.3 Linear programmingmethod
- 19.5.4 Alternating directionmethod
- 19.5.5 l1-norm constrained trust region method
- 19.6 Sampling
- 19.7 Numerical Experiments
- 19.7.1 Reconstruction of shot gathers
- 19.7.2 Field data
- 19.8 Conclusion
- References
- 20 H. Yang Some Researches on Quantitative Remote Sensing Inversion
- 20.1 Introduction
- 20.2 Models
- 20.3 A PrioriKnowledge
- 20.4 Optimization Algorithms
- 20.5 Multi-stage Inversion Strategy
- 20.6 Conclusion
- References
- Index
