应用反问题中的计算方法PDF电子书下载

Ⅰ Introduction 1

1 Inverse Problems of Mathematical Physics&S.I.Kabanikhin 3

1.1 Introduction 3

1.2 Examples of Inverse and Ill-posed Problems 12

1.3 Well-posed and Ill-posed Problems 24

1.4 The Tikhonov Theorem 26

1.5 The Ivanov Theorem:Quasi-solution 29

1.6 The Lavrentiev's Method 33

1.7 The Tikhonov Regularization Method 35

References 44

Ⅱ Recent Advances in Regularization Theory and Methods 47

2 Using Parallel Computing for Solving Multidimensional Ill-posed Problems&D.V.Lukyanenko and A.G.Yagola 49

2.1 Introduction 49

2.2 Using Parallel Computing 51

2.2.1 Main idea of parallel computing 51

2.2.2 Parallel computing limitations 52

2.3 Parallelization of Multidimensional Ill-posed Problem 53

2.3.1 Formulation of the problem and method of solution 53

2.3.2 Finite-difference approximation of the functional and its gradient 56

2.3.3 Parallelization of the minimization problem 58

2.4 Some Examples of Calculations 61

2.5 Conclusions 63

References 63

3 Regularization of Fredholm Integral Equations of the First Kind using Nystr？m Approximation&M.T Nair 65

3.1 Introduction 65

3.2 Nystr？m Method for Regularized Equations 68

3.2.1 Nystr？m approximation of integral operators 68

3.2.2 Approximation of regularized equation 69

3.2.3 Solvability of approximate regularized equation 70

3.2.4 Method of numerical solution 73

3.3 Error Estimates 74

3.3.1 Some preparatory results 74

3.3.2 Error estimate with respect to ‖·‖2 77

3.3.3 Error estimate with respect to ‖·‖∞ 77

3.3.4 A modified method 78

3.4 Conclusion 80

References 81

4 Regularization of Numerical Differentiation：Methods and Applications&T.Y.Xiao,H.Zhang and L.L.Hao 83

4.1 Introduction 83

4.2 Regularizing Schemes 87

4.2.1 Basic settings 87

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 Comparisons 102

4.4 Applied Examples 105

4.4.1 Simple applied problems 106

4.4.2 The inverse heat conduct problems(IHCP) 107

4.4.3 The parameter estimation in new product diffusion model 108

4.4.4 Parameter identification of sturm-liouville operator 110

4.4.5 The numerical inversion of Abel transform 112

4.4.6 The linear viscoelastic stress analysis 114

4.5 Discussion and Conclusion 115

References 117

5 Numerical Analytic Continuation and Regularization&C.L.Fu,H.Cheng and Y.J.Ma 121

5.1 Introduction 121

5.2 Description of the Problems in Strip Domain and Some Assumptions 124

5.2.1 Description of the problems 124

5.2.2 Some assumptions 125

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 Methods 126

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 Tests 135

References 140

6 An Optimal Perturbation Regularization Algorithm for Function Reconstruction and Its Applications 143

6.1 Introduction 143

6.2 The Optimal Perturbation Regularization Algorithm 144

6.3 Numerical Simulations 147

6.3.1 Inversion of time-dependent reaction coefficient 147

6.3.2 Inversion of space-dependent reaction coefficient 149

6.3.3 Inversion of state-dependent source term 151

6.3.4 Inversion of space-dependent diffusion coefficient 157

6.4 Applications 159

6.4.1 Determining magnitude of pollution source 159

6.4.2 Data reconstruction in an undisturbed soil-column experiment 162

6.5 Conclusions 165

References 166

7 Filtering and Inverse Problems Solving&L.V.Zotov and V.L.Panteleev 169

7.1 Introduction 169

7.2 SLAE Compatibility 170

7.3 Conditionality 171

7.4 Pseudosolutions 173

7.5 Singular Value Decomposition 175

7.6 Geometry of Pseudosolution 177

7.7 Inverse Problems for the Discrete Models of Observations 178

7.8 The Model in Spectral Domain 180

7.9 Regularization of Ill-posed Systems 181

7.10 General Remarks,the Dilemma of Bias and Dispersion 181

7.11 Models,Based on the Integral Equations 184

7.12 Panteleev Corrective Filtering 185

7.13 Philips-Tikhonov Regularization 186

References 194

Ⅲ Optimal Inverse Design and Optimization Methods 195

8 Inverse Design of Alloys'Chemistry for Specified Thermo-Mechanical Properties by using Multi-ob jective Optimization 197

8.1 Introduction 198

8.2 Multi-Objective Constrained Optimization and Response Surfaces 199

8.3 Summary of IOSO Algorithm 201

8.4 Mathematical Formulations of Objectives and Constraints 203

8.5 Determining Names of Alloying Elements and Their Concentra-tions for Specifed Properties of Alloys 212

8.6 Inverse Design of Bulk Metallic Glasses 214

8.7 Open Problems 215

8.8 Conclusions 218

References 219

9 Two Approaches to Reduce the Parameter Identification Errors&Z.H.Xiang 221

9.1 Introduction 221

9.2 The Optimal Sensor Placement Design 223

9.2.1 The well-posedness analysis of the parameter identifica-tion procedure 223

9.2.2 The algorithm for optimal sensor placement design 226

9.2.3 The integrated optimal sensor placement and parameter identification algorithm 229

9.2.4 Examples 229

9.3 The Regularization Method with the Adaptive Updating of A-priori Information 233

9.3.1 Modified extended Bayesian method for parameter identification 234

9.3.2 The well-posedness analysis of modified extended Bayesian method 234

9.3.3 Examples 236

9.4 Conclusion 238

References 238

10 A General Convergence Result for the BFGS Method&Y.H.Dai 241

10.1 Introduction 241

10.2 The BFGS Algorithm 243

10.3 A General Convergence Result for the BFGS Algorithm 244

10.4 Conclusion and Discussions 246

References 247

Ⅳ Recent Advances in Inverse Scattering 249

11 Uniqueness Results for Inverse Scattering Problems&X.D.Liu and B.Zhang 251

11.1 Introduction 251

11.2 Uniqueness for Inhomogeneity n 256

11.3 Uniqueness for Smooth Obstacles 256

11.4 Uniqueness for Polygon or Polyhedra 262

11.5 Uniqueness for Balls or Discs 263

11.6 Uniqueness for Surfaces or Curves 265

11.7 Uniqueness Results in a Layered Medium 265

11.8 Open Problems 272

References 276

12 Shape Reconstruction of Inverse Medium Scattering for the Helmholtz Equation&G.Bao and P.J.Li 283

12.1 Introduction 283

12.2 Analysis of the scattering map 285

12.3 Inverse medium scattering 290

12.3.1 Shape reconstruction 291

12.3.2 Born approximation 292

12.3.3 Recursive linearization 294

12.4 Numerical experiments 298

12.5 Concluding remarks 303

References 303

Ⅴ Inverse Vibration,Data Processing and Imaging 307

13 Numerical Aspects of the Calculation of Molecular Force Fields from Experimental Data&G.M.Kuramshina,I.V.Kochikov and A.V.Stepanova 309

13.1 Introduction 309

13.2 Molecular Force Field Models 311

13.3 Formulation of Inverse Vibration Problem 312

13.4 Constraints on the Values of Force Constants Based on Quantum Mechanical Calculations 314

13.5 Generalized Inverse Structural Problem 319

13.6 Computer Implementation 321

13.7 Applications 323

References 327

14 Some Mathematical Problems in Biomedical Imaging&J.J.Liu and H.L.Xu 331

14.1 Introduction 331

14.2 Mathematical Models 334

14.2.1 Forward problem 334

14.2.2 Inverse problem 336

14.3 Harmonic Bz Algorithm 339

14.3.1 Algorithm description 340

14.3.2 Convergence analysis 342

14.3.3 The stable computation of △Bz 344

14.4 Integral Equations Method 348

14.4.1 Algorithm description 348

14.4.2 Regularization and discretization 352

14.5 Numerical Experiments 354

References 362

Ⅵ Numerical Inversion in Geosciences 367

15 Numerical Methods for Solving Inverse Hyperbolic Problems&S.I.Kabanikhin and M.A.Shishlenin 369

15.1 Introduction 369

15.2 Gel'fand-Levitan-Krein Method 370

15.2.1 The two-dimensional analogy of Gel'fand-Levitan-Krein equation 374

15.2.2 N-approximation of Gel'fand-Levitan-Krein equation 377

15.2.3 Numerical results and remarks 379

15.3 Linearized Multidimensional Inverse Problem for the Wave Equation 379

15.3.1 Problem formulation 381

15.3.2 Linearization 382

15.4 Modified Landweber Iteration 384

15.4.1 Statement of the problem 385

15.4.2 Landweber iteration 387

15.4.3 Modification of algorithm 388

15.4.4 Numerical results 389

References 390

16 Inversion Studies in Seismic Oceanography&H.B.Song,X.H.Huang,L.M.Pinheiro,Y.Song,C.Z.Dong and Y.Bai 395

16.1 Introduction of Seismic Oceanography 395

16.2 Thermohaline Structure Inversion 398

16.2.1 Inversion method for temperature and salinity 398

16.2.2 Inversion experiment of synthetic seismic data 399

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

16.3 Discussion and Conclusion 406

References 408

17 Image Resolution Beyond the Classical Limit&L.,Gelius 411

17.1 Introduction 411

17.2 Aperture and Resolution Functions 412

17.3 Deconvolution Approach to Improved Resolution 417

17.4 MUSIC Pseudo-Spectrum Approach to Improved Resolution 424

17.5 Concluding Remarks 434

References 436

18 Seismic Migration and Inversion&Y.F.Wang,Z.H.Li and C.C.Yang 439

18.1 Introduction 439

18.2 Migration Methods:A Brief Review 440

18.2.1 Kirchhoff migration 440

18.2.2 Wave field extrapolation 441

18.2.3 Finite difference migration in ω-X domain 442

18.2.4 Phase shift migration 443

18.2.5 Stolt migration 443

18.2.6 Reverse time migration 446

18.2.7 Gaussian beam migration 447

18.2.8 Interferometric migration 447

18.2.9 Ray tracing 449

18.3 Seismic Migration and Inversion 452

18.3.1 The forward model 454

18.3.2 Migration deconvolution 456

18.3.3 Regularization model 457

18.3.4 Solving methods based on optimization 458

18.3.5 Preconditioning 462

18.3.6 Preconditioners 464

18.4 Illustrative Examples 465

18.4.1 Regularized migration inversion for point diffraction scatterers 465

18.4.2 Comparison with the interferometric migration 468

18.5 Conclusion 468

References 471

19 Seismic Wavefields Interpolation Based on Sparse Regularization and Compressive Sensing&Y.F.Wang,J.J.Cao,T.Sun and C.C.Yang 475

19.1 Introduction 475

19.2 Sparse Transforms 477

19.2.1 Fourier,wavelet,Radon and ridgelet transforms 477

19.2.2 The curvelet transform 480

19.3 Sparse Regularizing Modeling 481

19.3.1 Minimization in l0 space 481

19.3.2 Minimization in l1 space 481

19.3.3 Minimization in lp-lq space 482

19.4 Brief Review of Previous Methods in Mathematics 482

19.5 Sparse Optimization Methods 485

19.5.1 l0 quasi-norm approximation method 485

19.5.2 l1-norm approximation method 487

19.5.3 Linear programming method 489

19.5.4 Alternating direction method 491

19.5.5 l1-norm constrained trust region method 493

19.6 Sampling 496

19.7 Numerical Experiments 497

19.7.1 Reconstruction of shot gathers 497

19.7.2 Field data 498

19.8 Conclusion 503

References 503

20 Some Researches on Quantitative Remote Sensing Inversion&H.Yang 509

20.1 Introduction 509

20.2 Models 511

20 3 A Priori Knowledge 514

20.4 Optimization Algorithms 516

20.5 Multi-stage Inversion Strategy 520

20.6 Conclusion 524

References 525

Index 529