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应用反问题中的计算方法(英文版)


作者:
王彦飞, Anatoly G. Yagola, 杨长春
定价:
139.00元
ISBN:
978-7-04-034499-8
版面字数:
640.000千字
开本:
16开
全书页数:
530页
装帧形式:
精装
重点项目:
暂无
出版时间:
2012-10-16
读者对象:
学术著作
一级分类:
自然科学
二级分类:
数学与统计
三级分类:
计算数学

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

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