Computational Methods for Applied Inverse Problems (eBook)

Yanfei Wang, Anatoly G. Yagola, Changchun Yang (Herausgeber)

eBook Download: PDF

2012
550 Seiten
De Gruyter (Verlag)
9783110259056 (ISBN)

Infos

This monograph reports recent advances of inversion theory and recent developments with practical applications in frontiers of sciences, especially inverse design and novel computational methods for inverse problems. Readers who do research in applied mathematics, engineering, geophysics, biomedicine, image processing, remote sensing, and environmental science will benefit from the contents since the book incorporates a background of using statistical and non-statistical methods, e.g., regularization and optimization techniques for solving practical inverse problems.

 

Yanfei Wang, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China; Anatoly G. Yagola, Lomonosov Moscow State University, Russia; Changchun Yang, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China.

Autor

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Yanfei Wang, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China; Anatoly G. Yagola, Lomonosov Moscow State University, Russia; Changchun Yang, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing, China.

Inhalt

Preface 5
Editor’s Preface 7
I Introduction 21
1 Inverse Problems of Mathematical Physics 23
1.1 Introduction 23
1.2 Examples of Inverse and Ill-posed Problems 32
1.3 Well-posed and Ill-posed Problems 44
1.4 The Tikhonov Theorem 46
1.5 The Ivanov Theorem: Quasi-solution 49
1.6 The Lavrentiev’s Method 53
1.7 The Tikhonov Regularization Method 55
References 64
II Recent Advances in Regularization Theory and Methods 67
2 Using Parallel Computing for Solving Multidimensional Ill-posed Problems 69
2.1 Introduction 69
2.2 Using Parallel Computing 71
2.2.1 Main idea of parallel computing 71
2.2.2 Parallel computing limitations 72
2.3 Parallelization of Multidimensional Ill-posed Problem 73
2.3.1 Formulation of the problem and method of solution 73
2.3.2 Finite-difference approximation of the functional and its gradient 76
2.3.3 Parallelization of the minimization problem 78
2.4 Some Examples of Calculations 81
2.5 Conclusions 83
References 83
3 Regularization of Fredholm Integral Equations of the First Kind using Nyström Approximation 85
3.1 Introduction 85
3.2 Nyström Method for Regularized Equations 88
3.2.1 Nyström approximation of integral operators 88
3.2.2 Approximation of regularized equation 89
3.2.3 Solvability of approximate regularized equation 90
3.2.4 Method of numerical solution 93
3.3 Error Estimates 94
3.3.1 Some preparatory results 94
3.3.2 Error estimate with respect to || · ||2 97
3.3.3 Error estimate with respect to || · ||8 97
3.3.4 A modified method 98
3.4 Conclusion 100
References 101
4 Regularization of Numerical Differentiation: Methods and Applications 103
4.1 Introduction 103
4.2 Regularizing Schemes 107
4.2.1 Basic settings 107
4.2.2 Regularized difference method (RDM) 108
4.2.3 Smoother-Based regularization (SBR) 109
4.2.4 Mollifier regularization method (MRM) 110
4.2.5 Tikhonov’s variational regularization (TiVR) 112
4.2.6 Lavrentiev regularization method (LRM) 113
4.2.7 Discrete regularization method (DRM) 114
4.2.8 Semi-Discrete Tikhonov regularization (SDTR) 116
4.2.9 Total variation regularization (TVR) 119
4.3 Numerical Comparisons 122
4.4 Applied Examples 125
4.4.1 Simple applied problems 126
4.4.2 The inverse heat conduct problems (IHCP) 127
4.4.3 The parameter estimation in new product diffusion model 128
4.4.4 Parameter identification of sturm-liouville operator 130
4.4.5 The numerical inversion of Abel transform 132
4.4.6 The linear viscoelastic stress analysis 134
4.5 Discussion and Conclusion 135
References 137
5 Numerical Analytic Continuation and Regularization 141
5.1 Introduction 141
5.2 Description of the Problems in Strip Domain and Some Assumptions 144
5.2.1 Description of the problems 144
5.2.2 Some assumptions 145
5.2.3 The ill-posedness analysis for the Problems 5.2.1 and 5.2.2 145
5.2.4 The basic idea of the regularization for Problems 5.2.1 and 5.2.2 146
5.3 Some Regularization Methods 146
5.3.1 Some methods for solving Problem 5.2.1 146
5.3.2 Some methods for solving Problem 5.2.2 153
5.4 Numerical Tests 155
References 160
6 An Optimal Perturbation Regularization Algorithm for Function Reconstruction and Its Applications 163
6.1 Introduction 163
6.2 The Optimal Perturbation Regularization Algorithm 164
6.3 Numerical Simulations 167
6.3.1 Inversion of time-dependent reaction coefficient 167
6.3.2 Inversion of space-dependent reaction coefficient 169
6.3.3 Inversion of state-dependent source term 171
6.3.4 Inversion of space-dependent diffusion coefficient 177
6.4 Applications 179
6.4.1 Determining magnitude of pollution source 179
6.4.2 Data reconstruction in an undisturbed soil-column experiment 182
6.5 Conclusions 185
References 186
7 Filtering and Inverse Problems Solving 189
7.1 Introduction 189
7.2 SLAE Compatibility 190
7.3 Conditionality 191
7.4 Pseudosolutions 193
7.5 Singular Value Decomposition 195
7.6 Geometry of Pseudosolution 197
7.7 Inverse Problems for the Discrete Models of Observations 198
7.8 The Model in Spectral Domain 200
7.9 Regularization of Ill-posed Systems 201
7.10 General Remarks, the Dilemma of Bias and Dispersion 201
7.11 Models, Based on the Integral Equations 204
7.12 Panteleev Corrective Filtering 205
7.13 Philips-Tikhonov Regularization 206
References 214
III Optimal Inverse Design and Optimization Methods 215
8 Inverse Design of Alloys’ Chemistry for Specified Thermo-Mechanical Properties by using Multi-objective Optimization 217
8.1 Introduction 218
8.2 Multi-Objective Constrained Optimization and Response Surfaces 219
8.3 Summary of IOSO Algorithm 221
8.4 Mathematical Formulations of Objectives and Constraints 223
8.5 Determining Names of Alloying Elements and Their Concentrations for Specified Properties of Alloys 232
8.6 Inverse Design of Bulk Metallic Glasses 234
8.7 Open Problems 235
8.8 Conclusions 238
References 239
9 Two Approaches to Reduce the Parameter Identification Errors 241
9.1 Introduction 241
9.2 The Optimal Sensor Placement Design 243
9.2.1 The well-posedness analysis of the parameter identification procedure 243
9.2.2 The algorithm for optimal sensor placement design 246
9.2.3 The integrated optimal sensor placement and parameter identification algorithm 249
9.2.4 Examples 249
9.3 The Regularization Method with the Adaptive Updating of A-priori Information 253
9.3.1 Modified extended Bayesian method for parameter identification 254
9.3.2 The well-posedness analysis of modified extended Bayesian method 254
9.3.3 Examples 256
9.4 Conclusion 258
References 258
10 A General Convergence Result for the BFGS Method 261
10.1 Introduction 261
10.2 The BFGS Algorithm 263
10.3 A General Convergence Result for the BFGS Algorithm 264
10.4 Conclusion and Discussions 266
References 267
IV Recent Advances in Inverse Scattering 269
11 Uniqueness Results for Inverse Scattering Problems 271
11.1 Introduction 271
11.2 Uniqueness for Inhomogeneity n 276
11.3 Uniqueness for Smooth Obstacles 276
11.4 Uniqueness for Polygon or Polyhedra 282
11.5 Uniqueness for Balls or Discs 283
11.6 Uniqueness for Surfaces or Curves 285
11.7 Uniqueness Results in a Layered Medium 285
11.8 Open Problems 292
References 296
12 Shape Reconstruction of Inverse Medium Scattering for the Helmholtz Equation 303
12.1 Introduction 303
12.2 Analysis of the scattering map 305
12.3 Inverse medium scattering 310
12.3.1 Shape reconstruction 311
12.3.2 Born approximation 312
12.3.3 Recursive linearization 314
12.4 Numerical experiments 318
12.5 Concluding remarks 323
References 323
V Inverse Vibration, Data Processing and Imaging 327
13 Numerical Aspects of the Calculation of Molecular Force Fields from Experimental Data 329
13.1 Introduction 329
13.2 Molecular Force Field Models 331
13.3 Formulation of Inverse Vibration Problem 332
13.4 Constraints on the Values of Force Constants Based on Quantum Mechanical Calculations 334
13.5 Generalized Inverse Structural Problem 339
13.6 Computer Implementation 341
13.7 Applications 343
References 347
14 Some Mathematical Problems in Biomedical Imaging 351
14.1 Introduction 351
14.2 Mathematical Models 354
14.2.1 Forward problem 354
14.2.2 Inverse problem 356
14.3 Harmonic Bz Algorithm 359
14.3.1 Algorithm description 360
14.3.2 Convergence analysis 362
14.3.3 The stable computation of ... 364
14.4 Integral Equations Method 368
14.4.1 Algorithm description 368
14.4.2 Regularization and discretization 372
14.5 Numerical Experiments 374
References 382
VI Numerical Inversion in Geosciences 387
15 Numerical Methods for Solving Inverse Hyperbolic Problems 389
15.1 Introduction 389
15.2 Gel’fand-Levitan-Krein Method 390
15.2.1 The two-dimensional analogy of Gel'fand-Levitan-Krein equation 394
15.2.2 N-approximation of Gel'fand-Levitan-Krein equation 397
15.2.3 Numerical results and remarks 399
15.3 Linearized Multidimensional Inverse Problem for the Wave Equation 399
15.3.1 Problem formulation 401
15.3.2 Linearization 402
15.4 Modified Landweber Iteration 404
15.4.1 Statement of the problem 405
15.4.2 Landweber iteration 407
15.4.3 Modification of algorithm 408
15.4.4 Numerical results 409
References 410
16 Inversion Studies in Seismic Oceanography 415
16.1 Introduction of Seismic Oceanography 415
16.2 Thermohaline Structure Inversion 418
16.2.1 Inversion method for temperature and salinity 418
16.2.2 Inversion experiment of synthetic seismic data 419
16.2.3 Inversion experiment of GO data (Huang et al., 2011) 422
16.3 Discussion and Conclusion 426
References 428
17 Image Resolution Beyond the Classical Limit 431
17.1 Introduction 431
17.2 Aperture and Resolution Functions 432
17.3 Deconvolution Approach to Improved Resolution 437
17.4 MUSIC Pseudo-Spectrum Approach to Improved Resolution 444
17.5 Concluding Remarks 454
References 456
18 Seismic Migration and Inversion 459
18.1 Introduction 459
18.2 Migration Methods: A Brief Review 460
18.2.1 Kirchhoff migration 460
18.2.2 Wave field extrapolation 461
18.2.3 Finite difference migration in . — X domain 462
18.2.4 Phase shift migration 463
18.2.5 Stolt migration 463
18.2.6 Reverse time migration 466
18.2.7 Gaussian beam migration 467
18.2.8 Interferometric migration 467
18.2.9 Ray tracing 469
18.3 Seismic Migration and Inversion 472
18.3.1 The forward model 474
18.3.2 Migration deconvolution 476
18.3.3 Regularization model 477
18.3.4 Solving methods based on optimization 478
18.3.5 Preconditioning 482
18.3.6 Preconditioners 484
18.4 Illustrative Examples 485
18.4.1 Regularized migration inversion for point diffraction scatterers 485
18.4.2 Comparison with the interferometric migration 488
18.5 Conclusion 488
References 491
19 Seismic Wavefields Interpolation Based on Sparse Regularization and Compressive Sensing 495
19.1 Introduction 495
19.2 Sparse Transforms 497
19.2.1 Fourier, wavelet, Radon and ridgelet transforms 497
19.2.2 The curvelet transform 500
19.3 Sparse Regularizing Modeling 501
19.3.1 Minimization in l0 space 501
19.3.2 Minimization in l1 space 501
19.3.3 Minimization in lp-lq space 502
19.4 Brief Review of Previous Methods in Mathematics 502
19.5 Sparse Optimization Methods 505
19.5.1 lo quasi-norm approximation method 505
19.5.2 l1-norm approximation method 507
19.5.3 Linear programming method 509
19.5.4 Alternating direction method 511
19.5.5 l1-norm constrained trust region method 513
19.6 Sampling 516
19.7 Numerical Experiments 517
19.7.1 Reconstruction of shot gathers 517
19.7.2 Field data 518
19.8 Conclusion 523
References 523
20 Some Researches on Quantitative Remote Sensing Inversion 529
20.1 Introduction 529
20.2 Models 531
20.3 A Priori Knowledge 534
20.4 Optimization Algorithms 536
20.5 Multi-stage Inversion Strategy 540
20.6 Conclusion 544
References 545
Index 549

Produkt-Details

• Erscheint lt. Verlag: 30.10.2012
• Reihe/Serie: Inverse and Ill-Posed Problems Series
• Reihe/Serie: Inverse and Ill-Posed Problems Series
• Reihe/Serie: ISSN
• Reihe/Serie: ISSN
• Co-Autor: Higher Education Press, Y. Bai, G. Bao, J. J. Cao, H. Cheng, Y. H. Dai, C. Z. Dong, G. S. Dulikravich, I. N. Egorov, C. L. Fu, L. J. Gelius, L. L. Hao, X. H. Huang, S. I. Kabanikhin, I. V. Kochikov, G. M. Kuramshina, G. S. Li, P. J. Li, Z. H. Li, J. J. Liu, X. D. Liu, Y. J. Ma, M. T. Nair, V. L. Panteleev, L. M. Pinheiro, M. A. Shishlenin, H. B. Song, Y. Song, A. V. Stepanova, T. Sun, Y. F. Wang, Z. H. Xiang, T. Y. Xiao, H. L. Xu, A. G. Yagola, C. C. Yang, H. Yang, B. Zang, H. Zhang, L. V. Zotov
• Zusatzinfo: 173 b/w ill., 41 b/w tbl.
• Verlagsort: Berlin/Boston
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geografie / Kartografie
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Geologie
• Themenwelt: Naturwissenschaften ► Geowissenschaften ► Hydrologie / Ozeanografie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie
• Themenwelt: Technik
• Schlagworte: Computational Method • Functional Analysis • Geography • Geophysics • Image Processing • inverse problem • Oceanography • operator theory • Optimization • Regularization • Remote Sensing
• ISBN-13: 9783110259056 / 9783110259056