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.
- 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