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
