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