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Ⅰ Introduction1

1 Inverse Problems of Mathematical Physics&S.I.Kabanikhin3

1.1 Introduction3

1.2 Examples of Inverse and Ill-posed Problems12

1.3 Well-posed and Ill-posed Problems24

1.4 The Tikhonov Theorem26

1.5 The Ivanov Theorem:Quasi-solution29

1.6 The Lavrentiev's Method33

1.7 The Tikhonov Regularization Method35

References44

Ⅱ Recent Advances in Regularization Theory and Methods47

2 Using Parallel Computing for Solving Multidimensional Ill-posed Problems&D.V.Lukyanenko and A.G.Yagola49

2.1 Introduction49

2.2 Using Parallel Computing51

2.2.1 Main idea of parallel computing51

2.2.2 Parallel computing limitations52

2.3 Parallelization of Multidimensional Ill-posed Problem53

2.3.1 Formulation of the problem and method of solution53

2.3.2 Finite-difference approximation of the functional and its gradient56

2.3.3 Parallelization of the minimization problem58

2.4 Some Examples of Calculations61

2.5 Conclusions63

References63

3 Regularization of Fredholm Integral Equations of the First Kind using Nystr？m Approximation&M.T Nair65

3.1 Introduction65

3.2 Nystr？m Method for Regularized Equations68

3.2.1 Nystr？m approximation of integral operators68

3.2.2 Approximation of regularized equation69

3.2.3 Solvability of approximate regularized equation70

3.2.4 Method of numerical solution73

3.3 Error Estimates74

3.3.1 Some preparatory results74

3.3.2 Error estimate with respect to ‖·‖277

3.3.3 Error estimate with respect to ‖·‖∞77

3.3.4 A modified method78

3.4 Conclusion80

References81

4 Regularization of Numerical Differentiation：Methods and Applications&T.Y.Xiao,H.Zhang and L.L.Hao83

4.1 Introduction83

4.2 Regularizing Schemes87

4.2.1 Basic settings87

4.2.2 Regularized difference method(RDM)88

4.2.3 Smoother-Based regularization(SBR)89

4.2.4 Mollifier regularization method(MRM)90

4.2.5 Tikhonov's variational regularization(TiVR)92

4.2.6 Lavrentiev regularization method(LRM)93

4.2.7 Discrete regularization method(DRM)94

4.2.8 Semi-Discrete Tikhonov regularization(SDTR)96

4.2.9 Total variation regularization(TVR)99

4.3 Numerical Comparisons102

4.4 Applied Examples105

4.4.1 Simple applied problems106

4.4.2 The inverse heat conduct problems(IHCP)107

4.4.3 The parameter estimation in new product diffusion model108

4.4.4 Parameter identification of sturm-liouville operator110

4.4.5 The numerical inversion of Abel transform112

4.4.6 The linear viscoelastic stress analysis114

4.5 Discussion and Conclusion115

References117

5 Numerical Analytic Continuation and Regularization&C.L.Fu,H.Cheng and Y.J.Ma121

5.1 Introduction121

5.2 Description of the Problems in Strip Domain and Some Assumptions124

5.2.1 Description of the problems124

5.2.2 Some assumptions125

5.2.3 The ill-posedness analysis for the Problems 5.2.1 and 5.2.2 125

5.2.4 The basic idea of the regularization for Problems 5.2.1 and 5.2.2 126

5.3 Some Regularization Methods126

5.3.1 Some methods for solving Problem 5.2.1 126

5.3.2 Some methods for solving Problem 5.2.2 133

5.4 Numerical Tests135

References140

6 An Optimal Perturbation Regularization Algorithm for Function Reconstruction and Its Applications143

6.1 Introduction143

6.2 The Optimal Perturbation Regularization Algorithm144

6.3 Numerical Simulations147

6.3.1 Inversion of time-dependent reaction coefficient147

6.3.2 Inversion of space-dependent reaction coefficient149

6.3.3 Inversion of state-dependent source term151

6.3.4 Inversion of space-dependent diffusion coefficient157

6.4 Applications159

6.4.1 Determining magnitude of pollution source159

6.4.2 Data reconstruction in an undisturbed soil-column experiment162

6.5 Conclusions165

References166

7 Filtering and Inverse Problems Solving&L.V.Zotov and V.L.Panteleev169

7.1 Introduction169

7.2 SLAE Compatibility170

7.3 Conditionality171

7.4 Pseudosolutions173

7.5 Singular Value Decomposition175

7.6 Geometry of Pseudosolution177

7.7 Inverse Problems for the Discrete Models of Observations178

7.8 The Model in Spectral Domain180

7.9 Regularization of Ill-posed Systems181

7.10 General Remarks,the Dilemma of Bias and Dispersion181

7.11 Models,Based on the Integral Equations184

7.12 Panteleev Corrective Filtering185

7.13 Philips-Tikhonov Regularization186

References194

Ⅲ Optimal Inverse Design and Optimization Methods195

8 Inverse Design of Alloys'Chemistry for Specified Thermo-Mechanical Properties by using Multi-ob jective Optimization197

8.1 Introduction198

8.2 Multi-Objective Constrained Optimization and Response Surfaces199

8.3 Summary of IOSO Algorithm201

8.4 Mathematical Formulations of Objectives and Constraints203

8.5 Determining Names of Alloying Elements and Their Concentra-tions for Specifed Properties of Alloys212

8.6 Inverse Design of Bulk Metallic Glasses214

8.7 Open Problems215

8.8 Conclusions218

References219

9 Two Approaches to Reduce the Parameter Identification Errors&Z.H.Xiang221

9.1 Introduction221

9.2 The Optimal Sensor Placement Design223

9.2.1 The well-posedness analysis of the parameter identifica-tion procedure223

9.2.2 The algorithm for optimal sensor placement design226

9.2.3 The integrated optimal sensor placement and parameter identification algorithm229

9.2.4 Examples229

9.3 The Regularization Method with the Adaptive Updating of A-priori Information233

9.3.1 Modified extended Bayesian method for parameter identification234

9.3.2 The well-posedness analysis of modified extended Bayesian method234

9.3.3 Examples236

9.4 Conclusion238

References238

10 A General Convergence Result for the BFGS Method&Y.H.Dai241

10.1 Introduction241

10.2 The BFGS Algorithm243

10.3 A General Convergence Result for the BFGS Algorithm244

10.4 Conclusion and Discussions246

References247

Ⅳ Recent Advances in Inverse Scattering249

11 Uniqueness Results for Inverse Scattering Problems&X.D.Liu and B.Zhang251

11.1 Introduction251

11.2 Uniqueness for Inhomogeneity n256

11.3 Uniqueness for Smooth Obstacles256

11.4 Uniqueness for Polygon or Polyhedra262

11.5 Uniqueness for Balls or Discs263

11.6 Uniqueness for Surfaces or Curves265

11.7 Uniqueness Results in a Layered Medium265

11.8 Open Problems272

References276

12 Shape Reconstruction of Inverse Medium Scattering for the Helmholtz Equation&G.Bao and P.J.Li283

12.1 Introduction283

12.2 Analysis of the scattering map285

12.3 Inverse medium scattering290

12.3.1 Shape reconstruction291

12.3.2 Born approximation292

12.3.3 Recursive linearization294

12.4 Numerical experiments298

12.5 Concluding remarks303

References303

Ⅴ Inverse Vibration,Data Processing and Imaging307

13 Numerical Aspects of the Calculation of Molecular Force Fields from Experimental Data&G.M.Kuramshina,I.V.Kochikov and A.V.Stepanova309

13.1 Introduction309

13.2 Molecular Force Field Models311

13.3 Formulation of Inverse Vibration Problem312

13.4 Constraints on the Values of Force Constants Based on Quantum Mechanical Calculations314

13.5 Generalized Inverse Structural Problem319

13.6 Computer Implementation321

13.7 Applications323

References327

14 Some Mathematical Problems in Biomedical Imaging&J.J.Liu and H.L.Xu331

14.1 Introduction331

14.2 Mathematical Models334

14.2.1 Forward problem334

14.2.2 Inverse problem336

14.3 Harmonic Bz Algorithm339

14.3.1 Algorithm description340

14.3.2 Convergence analysis342

14.3.3 The stable computation of △Bz344

14.4 Integral Equations Method348

14.4.1 Algorithm description348

14.4.2 Regularization and discretization352

14.5 Numerical Experiments354

References362

Ⅵ Numerical Inversion in Geosciences367

15 Numerical Methods for Solving Inverse Hyperbolic Problems&S.I.Kabanikhin and M.A.Shishlenin369

15.1 Introduction369

15.2 Gel'fand-Levitan-Krein Method370

15.2.1 The two-dimensional analogy of Gel'fand-Levitan-Krein equation374

15.2.2 N-approximation of Gel'fand-Levitan-Krein equation377

15.2.3 Numerical results and remarks379

15.3 Linearized Multidimensional Inverse Problem for the Wave Equation379

15.3.1 Problem formulation381

15.3.2 Linearization382

15.4 Modified Landweber Iteration384

15.4.1 Statement of the problem385

15.4.2 Landweber iteration387

15.4.3 Modification of algorithm388

15.4.4 Numerical results389

References390

16 Inversion Studies in Seismic Oceanography&H.B.Song,X.H.Huang,L.M.Pinheiro,Y.Song,C.Z.Dong and Y.Bai395

16.1 Introduction of Seismic Oceanography395

16.2 Thermohaline Structure Inversion398

16.2.1 Inversion method for temperature and salinity398

16.2.2 Inversion experiment of synthetic seismic data399

16.2.3 Inversion experiment of GO data(Huang et a1.,2011)402

16.3 Discussion and Conclusion406

References408

17 Image Resolution Beyond the Classical Limit&L.,Gelius411

17.1 Introduction411

17.2 Aperture and Resolution Functions412

17.3 Deconvolution Approach to Improved Resolution417

17.4 MUSIC Pseudo-Spectrum Approach to Improved Resolution424

17.5 Concluding Remarks434

References436

18 Seismic Migration and Inversion&Y.F.Wang,Z.H.Li and C.C.Yang439

18.1 Introduction439

18.2 Migration Methods:A Brief Review440

18.2.1 Kirchhoff migration440

18.2.2 Wave field extrapolation441

18.2.3 Finite difference migration in ω-X domain442

18.2.4 Phase shift migration443

18.2.5 Stolt migration443

18.2.6 Reverse time migration446

18.2.7 Gaussian beam migration447

18.2.8 Interferometric migration447

18.2.9 Ray tracing449

18.3 Seismic Migration and Inversion452

18.3.1 The forward model454

18.3.2 Migration deconvolution456

18.3.3 Regularization model457

18.3.4 Solving methods based on optimization458

18.3.5 Preconditioning462

18.3.6 Preconditioners464

18.4 Illustrative Examples465

18.4.1 Regularized migration inversion for point diffraction scatterers465

18.4.2 Comparison with the interferometric migration468

18.5 Conclusion468

References471

19 Seismic Wavefields Interpolation Based on Sparse Regularization and Compressive Sensing&Y.F.Wang,J.J.Cao,T.Sun and C.C.Yang475

19.1 Introduction475

19.2 Sparse Transforms477

19.2.1 Fourier,wavelet,Radon and ridgelet transforms477

19.2.2 The curvelet transform480

19.3 Sparse Regularizing Modeling481

19.3.1 Minimization in l0 space481

19.3.2 Minimization in l1 space481

19.3.3 Minimization in lp-lq space482

19.4 Brief Review of Previous Methods in Mathematics482

19.5 Sparse Optimization Methods485

19.5.1 l0 quasi-norm approximation method485

19.5.2 l1-norm approximation method487

19.5.3 Linear programming method489

19.5.4 Alternating direction method491

19.5.5 l1-norm constrained trust region method493

19.6 Sampling496

19.7 Numerical Experiments497

19.7.1 Reconstruction of shot gathers497

19.7.2 Field data498

19.8 Conclusion503

References503

20 Some Researches on Quantitative Remote Sensing Inversion&H.Yang509

20.1 Introduction509

20.2 Models511

20 3 A Priori Knowledge514

20.4 Optimization Algorithms516

20.5 Multi-stage Inversion Strategy520

20.6 Conclusion524

References525

Index529