Preface -- Editor's Preface -- I Introduction -- 1 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's Method -- 1.7 The Tikhonov Regularization Method -- References -- II Recent Advances in Regularization Theory and Methods -- 2 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 Regularization of Fredholm Integral Equations of the First Kind using Nyström Approximation -- 3.1 Introduction -- 3.2 Nyström Method for Regularized Equations -- 3.2.1 Nyström 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

4.2.5 Tikhonov's variational regularization (TiVR) -- 4.2.6 Lavrentiev regularization method (LRM) -- 4.2.7 Discrete regularization method (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 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 Regularization Methods -- 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 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 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 The Model 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 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 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 A-priori Information -- 9.3.1 Modified extended Bayesian method for parameter identification -- 9.3.2 The well-posedness analysis of modified extended Bayesian method -- 9.3.3 Examples -- 9.4 Conclusion -- References -- 10 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 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 Layered Medium -- 11.8 Open Problems -- References -- 12 Shape Reconstruction of Inverse Medium Scattering for the Helmholtz Equation -- 12.1 Introduction -- 12.2 Analysis of the scattering map -- 12.3 Inverse medium 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 Numerical Aspects of the Calculation of Molecular Force Fields from Experimental Data -- 13.1 Introduction -- 13.2 Molecular Force Field Models -- 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 Some Mathematical Problems in Biomedical Imaging -- 14.1 Introduction -- 14.2 Mathematical Models -- 14.2.1 Forward problem -- 14.2.2 Inverse problem -- 14.3 Harmonic Bz Algorithm -- 14.3.1 Algorithm description -- 14.3.2 Convergence analysis -- 14.3.3 The stable computation of ΔΒΖ -- 14.4 Integral Equations Method -- 14.4.1 Algorithm description -- 14.4.2 Regularization and discretization -- 14.5 Numerical Experiments -- References -- VI Numerical Inversion in Geosciences.

15 Numerical Methods for Solving Inverse Hyperbolic Problems -- 15.1 Introduction -- 15.2 Gel'fand-Levitan-Krein Method -- 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 Problem formulation -- 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 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 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 Seismic Migration and Inversion -- 18.1 Introduction -- 18.2 Migration Methods: A Brief Review -- 18.2.1 Kirchhoff migration -- 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 Stolt migration -- 18.2.6 Reverse time migration -- 18.2.7 Gaussian beam migration -- 18.2.8 Interferometric migration -- 18.2.9 Ray tracing -- 18.3 Seismic Migration and Inversion -- 18.3.1 The forward model -- 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.