Computational Methods for Applied Inverse Problems

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