Wang / Yagola / Yang | Computational Methods for Applied Inverse Problems | E-Book | sack.de
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E-Book, Englisch, 550 Seiten

Reihe: ISSN

Wang / Yagola / Yang Computational Methods for Applied Inverse Problems


E-Book, Englisch, 550 Seiten

Reihe: ISSN

ISBN: 978-3-11-025905-6
Verlag: De Gruyter
Format: PDF
 Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

Nowadays inverse problems and applications in science and engineering represent an extremely active research field. The subjects are related to mathematics, physics, geophysics, geochemistry, oceanography, geography and remote sensing, astronomy, biomedicine, and other areas of applications.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. The practical applications include inverse scattering, chemistry, molecular spectra data processing, quantitative remote sensing inversion, seismic imaging, oceanography, and astronomical imaging.The book serves as a reference book and 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.
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Zielgruppe

Applied Mathematicians, Geoscientists, Professionals Working in Image Processing; Academic Libraries

Autoren/Hrsg.

Wang, Yanfei
Yagola, Anatoly G.
Yang, Changchun

Fachgebiete

Weitere Infos & Material

1;Preface;5
2;Editor’s Preface;7
3;I Introduction;21
3.1;1 Inverse Problems of Mathematical Physics;23
3.1.1;1.1 Introduction;23
3.1.2;1.2 Examples of Inverse and Ill-posed Problems;32
3.1.3;1.3 Well-posed and Ill-posed Problems;44
3.1.4;1.4 The Tikhonov Theorem;46
3.1.5;1.5 The Ivanov Theorem: Quasi-solution;49
3.1.6;1.6 The Lavrentiev’s Method;53
3.1.7;1.7 The Tikhonov Regularization Method;55
3.1.8;References;64
4;II Recent Advances in Regularization Theory and Methods;67
4.1;2 Using Parallel Computing for Solving Multidimensional Ill-posed Problems;69
4.1.1;2.1 Introduction;69
4.1.2;2.2 Using Parallel Computing;71
4.1.2.1;2.2.1 Main idea of parallel computing;71
4.1.2.2;2.2.2 Parallel computing limitations;72
4.1.3;2.3 Parallelization of Multidimensional Ill-posed Problem;73
4.1.3.1;2.3.1 Formulation of the problem and method of solution;73
4.1.3.2;2.3.2 Finite-difference approximation of the functional and its gradient;76
4.1.3.3;2.3.3 Parallelization of the minimization problem;78
4.1.4;2.4 Some Examples of Calculations;81
4.1.5;2.5 Conclusions;83
4.1.6;References;83
4.2;3 Regularization of Fredholm Integral Equations of the First Kind using Nyström Approximation;85
4.2.1;3.1 Introduction;85
4.2.2;3.2 Nyström Method for Regularized Equations;88
4.2.2.1;3.2.1 Nyström approximation of integral operators;88
4.2.2.2;3.2.2 Approximation of regularized equation;89
4.2.2.3;3.2.3 Solvability of approximate regularized equation;90
4.2.2.4;3.2.4 Method of numerical solution;93
4.2.3;3.3 Error Estimates;94
4.2.3.1;3.3.1 Some preparatory results;94
4.2.3.2;3.3.2 Error estimate with respect to || · ||2;97
4.2.3.3;3.3.3 Error estimate with respect to || · ||8;97
4.2.3.4;3.3.4 A modified method;98
4.2.4;3.4 Conclusion;100
4.2.5;References;101
4.3;4 Regularization of Numerical Differentiation: Methods and Applications;103
4.3.1;4.1 Introduction;103
4.3.2;4.2 Regularizing Schemes;107
4.3.2.1;4.2.1 Basic settings;107
4.3.2.2;4.2.2 Regularized difference method (RDM);108
4.3.2.3;4.2.3 Smoother-Based regularization (SBR);109
4.3.2.4;4.2.4 Mollifier regularization method (MRM);110
4.3.2.5;4.2.5 Tikhonov’s variational regularization (TiVR);112
4.3.2.6;4.2.6 Lavrentiev regularization method (LRM);113
4.3.2.7;4.2.7 Discrete regularization method (DRM);114
4.3.2.8;4.2.8 Semi-Discrete Tikhonov regularization (SDTR);116
4.3.2.9;4.2.9 Total variation regularization (TVR);119
4.3.3;4.3 Numerical Comparisons;122
4.3.4;4.4 Applied Examples;125
4.3.4.1;4.4.1 Simple applied problems;126
4.3.4.2;4.4.2 The inverse heat conduct problems (IHCP);127
4.3.4.3;4.4.3 The parameter estimation in new product diffusion model;128
4.3.4.4;4.4.4 Parameter identification of sturm-liouville operator;130
4.3.4.5;4.4.5 The numerical inversion of Abel transform;132
4.3.4.6;4.4.6 The linear viscoelastic stress analysis;134
4.3.5;4.5 Discussion and Conclusion;135
4.3.6;References;137
4.4;5 Numerical Analytic Continuation and Regularization;141
4.4.1;5.1 Introduction;141
4.4.2;5.2 Description of the Problems in Strip Domain and Some Assumptions;144
4.4.2.1;5.2.1 Description of the problems;144
4.4.2.2;5.2.2 Some assumptions;145
4.4.2.3;5.2.3 The ill-posedness analysis for the Problems 5.2.1 and 5.2.2;145
4.4.2.4;5.2.4 The basic idea of the regularization for Problems 5.2.1 and 5.2.2;146
4.4.3;5.3 Some Regularization Methods;146
4.4.3.1;5.3.1 Some methods for solving Problem 5.2.1;146
4.4.3.2;5.3.2 Some methods for solving Problem 5.2.2;153
4.4.4;5.4 Numerical Tests;155
4.4.5;References;160
4.5;6 An Optimal Perturbation Regularization Algorithm for Function Reconstruction and Its Applications;163
4.5.1;6.1 Introduction;163
4.5.2;6.2 The Optimal Perturbation Regularization Algorithm;164
4.5.3;6.3 Numerical Simulations;167
4.5.3.1;6.3.1 Inversion of time-dependent reaction coefficient;167
4.5.3.2;6.3.2 Inversion of space-dependent reaction coefficient;169
4.5.3.3;6.3.3 Inversion of state-dependent source term;171
4.5.3.4;6.3.4 Inversion of space-dependent diffusion coefficient;177
4.5.4;6.4 Applications;179
4.5.4.1;6.4.1 Determining magnitude of pollution source;179
4.5.4.2;6.4.2 Data reconstruction in an undisturbed soil-column experiment;182
4.5.5;6.5 Conclusions;185
4.5.6;References;186
4.6;7 Filtering and Inverse Problems Solving;189
4.6.1;7.1 Introduction;189
4.6.2;7.2 SLAE Compatibility;190
4.6.3;7.3 Conditionality;191
4.6.4;7.4 Pseudosolutions;193
4.6.5;7.5 Singular Value Decomposition;195
4.6.6;7.6 Geometry of Pseudosolution;197
4.6.7;7.7 Inverse Problems for the Discrete Models of Observations;198
4.6.8;7.8 The Model in Spectral Domain;200
4.6.9;7.9 Regularization of Ill-posed Systems;201
4.6.10;7.10 General Remarks, the Dilemma of Bias and Dispersion;201
4.6.11;7.11 Models, Based on the Integral Equations;204
4.6.12;7.12 Panteleev Corrective Filtering;205
4.6.13;7.13 Philips-Tikhonov Regularization;206
4.6.14;References;214
5;III Optimal Inverse Design and Optimization Methods;215
5.1;8 Inverse Design of Alloys’ Chemistry for Specified Thermo-Mechanical Properties by using Multi-objective Optimization;217
5.1.1;8.1 Introduction;218
5.1.2;8.2 Multi-Objective Constrained Optimization and Response Surfaces;219
5.1.3;8.3 Summary of IOSO Algorithm;221
5.1.4;8.4 Mathematical Formulations of Objectives and Constraints;223
5.1.5;8.5 Determining Names of Alloying Elements and Their Concentrations for Specified Properties of Alloys;232
5.1.6;8.6 Inverse Design of Bulk Metallic Glasses;234
5.1.7;8.7 Open Problems;235
5.1.8;8.8 Conclusions;238
5.1.9;References;239
5.2;9 Two Approaches to Reduce the Parameter Identification Errors;241
5.2.1;9.1 Introduction;241
5.2.2;9.2 The Optimal Sensor Placement Design;243
5.2.2.1;9.2.1 The well-posedness analysis of the parameter identification procedure;243
5.2.2.2;9.2.2 The algorithm for optimal sensor placement design;246
5.2.2.3;9.2.3 The integrated optimal sensor placement and parameter identification algorithm;249
5.2.2.4;9.2.4 Examples;249
5.2.3;9.3 The Regularization Method with the Adaptive Updating of A-priori Information;253
5.2.3.1;9.3.1 Modified extended Bayesian method for parameter identification;254
5.2.3.2;9.3.2 The well-posedness analysis of modified extended Bayesian method;254
5.2.3.3;9.3.3 Examples;256
5.2.4;9.4 Conclusion;258
5.2.5;References;258
5.3;10 A General Convergence Result for the BFGS Method;261
5.3.1;10.1 Introduction;261
5.3.2;10.2 The BFGS Algorithm;263
5.3.3;10.3 A General Convergence Result for the BFGS Algorithm;264
5.3.4;10.4 Conclusion and Discussions;266
5.3.5;References;267
6;IV Recent Advances in Inverse Scattering;269
6.1;11 Uniqueness Results for Inverse Scattering Problems;271
6.1.1;11.1 Introduction;271
6.1.2;11.2 Uniqueness for Inhomogeneity n;276
6.1.3;11.3 Uniqueness for Smooth Obstacles;276
6.1.4;11.4 Uniqueness for Polygon or Polyhedra;282
6.1.5;11.5 Uniqueness for Balls or Discs;283
6.1.6;11.6 Uniqueness for Surfaces or Curves;285
6.1.7;11.7 Uniqueness Results in a Layered Medium;285
6.1.8;11.8 Open Problems;292
6.1.9;References;296
6.2;12 Shape Reconstruction of Inverse Medium Scattering for the Helmholtz Equation;303
6.2.1;12.1 Introduction;303
6.2.2;12.2 Analysis of the scattering map;305
6.2.3;12.3 Inverse medium scattering;310
6.2.3.1;12.3.1 Shape reconstruction;311
6.2.3.2;12.3.2 Born approximation;312
6.2.3.3;12.3.3 Recursive linearization;314
6.2.4;12.4 Numerical experiments;318
6.2.5;12.5 Concluding remarks;323
6.2.6;References;323
7;V Inverse Vibration, Data Processing and Imaging;327
7.1;13 Numerical Aspects of the Calculation of Molecular Force Fields from Experimental Data;329
7.1.1;13.1 Introduction;329
7.1.2;13.2 Molecular Force Field Models;331
7.1.3;13.3 Formulation of Inverse Vibration Problem;332
7.1.4;13.4 Constraints on the Values of Force Constants Based on Quantum Mechanical Calculations;334
7.1.5;13.5 Generalized Inverse Structural Problem;339
7.1.6;13.6 Computer Implementation;341
7.1.7;13.7 Applications;343
7.1.8;References;347
7.2;14 Some Mathematical Problems in Biomedical Imaging;351
7.2.1;14.1 Introduction;351
7.2.2;14.2 Mathematical Models;354
7.2.2.1;14.2.1 Forward problem;354
7.2.2.2;14.2.2 Inverse problem;356
7.2.3;14.3 Harmonic Bz Algorithm;359
7.2.3.1;14.3.1 Algorithm description;360
7.2.3.2;14.3.2 Convergence analysis;362
7.2.3.3;14.3.3 The stable computation of ...;364
7.2.4;14.4 Integral Equations Method;368
7.2.4.1;14.4.1 Algorithm description;368
7.2.4.2;14.4.2 Regularization and discretization;372
7.2.5;14.5 Numerical Experiments;374
7.2.6;References;382
8;VI Numerical Inversion in Geosciences;387
8.1;15 Numerical Methods for Solving Inverse Hyperbolic Problems;389
8.1.1;15.1 Introduction;389
8.1.2;15.2 Gel’fand-Levitan-Krein Method;390
8.1.2.1;15.2.1 The two-dimensional analogy of Gel'fand-Levitan-Krein equation;394
8.1.2.2;15.2.2 N-approximation of Gel'fand-Levitan-Krein equation;397
8.1.2.3;15.2.3 Numerical results and remarks;399
8.1.3;15.3 Linearized Multidimensional Inverse Problem for the Wave Equation;399
8.1.3.1;15.3.1 Problem formulation;401
8.1.3.2;15.3.2 Linearization;402
8.1.4;15.4 Modified Landweber Iteration;404
8.1.4.1;15.4.1 Statement of the problem;405
8.1.4.2;15.4.2 Landweber iteration;407
8.1.4.3;15.4.3 Modification of algorithm;408
8.1.4.4;15.4.4 Numerical results;409
8.1.5;References;410
8.2;16 Inversion Studies in Seismic Oceanography;415
8.2.1;16.1 Introduction of Seismic Oceanography;415
8.2.2;16.2 Thermohaline Structure Inversion;418
8.2.2.1;16.2.1 Inversion method for temperature and salinity;418
8.2.2.2;16.2.2 Inversion experiment of synthetic seismic data;419
8.2.2.3;16.2.3 Inversion experiment of GO data (Huang et al., 2011);422
8.2.3;16.3 Discussion and Conclusion;426
8.2.4;References;428
8.3;17 Image Resolution Beyond the Classical Limit;431
8.3.1;17.1 Introduction;431
8.3.2;17.2 Aperture and Resolution Functions;432
8.3.3;17.3 Deconvolution Approach to Improved Resolution;437
8.3.4;17.4 MUSIC Pseudo-Spectrum Approach to Improved Resolution;444
8.3.5;17.5 Concluding Remarks;454
8.3.6;References;456
8.4;18 Seismic Migration and Inversion;459
8.4.1;18.1 Introduction;459
8.4.2;18.2 Migration Methods: A Brief Review;460
8.4.2.1;18.2.1 Kirchhoff migration;460
8.4.2.2;18.2.2 Wave field extrapolation;461
8.4.2.3;18.2.3 Finite difference migration in . — X domain;462
8.4.2.4;18.2.4 Phase shift migration;463
8.4.2.5;18.2.5 Stolt migration;463
8.4.2.6;18.2.6 Reverse time migration;466
8.4.2.7;18.2.7 Gaussian beam migration;467
8.4.2.8;18.2.8 Interferometric migration;467
8.4.2.9;18.2.9 Ray tracing;469
8.4.3;18.3 Seismic Migration and Inversion;472
8.4.3.1;18.3.1 The forward model;474
8.4.3.2;18.3.2 Migration deconvolution;476
8.4.3.3;18.3.3 Regularization model;477
8.4.3.4;18.3.4 Solving methods based on optimization;478
8.4.3.5;18.3.5 Preconditioning;482
8.4.3.6;18.3.6 Preconditioners;484
8.4.4;18.4 Illustrative Examples;485
8.4.4.1;18.4.1 Regularized migration inversion for point diffraction scatterers;485
8.4.4.2;18.4.2 Comparison with the interferometric migration;488
8.4.5;18.5 Conclusion;488
8.4.6;References;491
8.5;19 Seismic Wavefields Interpolation Based on Sparse Regularization and Compressive Sensing;495
8.5.1;19.1 Introduction;495
8.5.2;19.2 Sparse Transforms;497
8.5.2.1;19.2.1 Fourier, wavelet, Radon and ridgelet transforms;497
8.5.2.2;19.2.2 The curvelet transform;500
8.5.3;19.3 Sparse Regularizing Modeling;501
8.5.3.1;19.3.1 Minimization in l0 space;501
8.5.3.2;19.3.2 Minimization in l1 space;501
8.5.3.3;19.3.3 Minimization in lp-lq space;502
8.5.4;19.4 Brief Review of Previous Methods in Mathematics;502
8.5.5;19.5 Sparse Optimization Methods;505
8.5.5.1;19.5.1 lo quasi-norm approximation method;505
8.5.5.2;19.5.2 l1-norm approximation method;507
8.5.5.3;19.5.3 Linear programming method;509
8.5.5.4;19.5.4 Alternating direction method;511
8.5.5.5;19.5.5 l1-norm constrained trust region method;513
8.5.6;19.6 Sampling;516
8.5.7;19.7 Numerical Experiments;517
8.5.7.1;19.7.1 Reconstruction of shot gathers;517
8.5.7.2;19.7.2 Field data;518
8.5.8;19.8 Conclusion;523
8.5.9;References;523
8.6;20 Some Researches on Quantitative Remote Sensing Inversion;529
8.6.1;20.1 Introduction;529
8.6.2;20.2 Models;531
8.6.3;20.3 A Priori Knowledge;534
8.6.4;20.4 Optimization Algorithms;536
8.6.5;20.5 Multi-stage Inversion Strategy;540
8.6.6;20.6 Conclusion;544
8.6.7;References;545
9;Index;549

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.

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