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E-raamat: Computational Methods for Applied Inverse Problems, Volume 56 [De Gruyter e-raamatud]

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Computational Methods for Applied Inverse Problems, Volume 56Suurem pilt
Teised raamatud teemal:
Wiley OnlineRohkem infot De Gruyter e-raamatute kohta
Raamatu kodulehekülg: http://dx.doi.org/10.1515/9783110259056
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.
Preface v
Editor's Preface vii
I Introduction 1(46)
1 Inverse Problems of Mathematical Physics
3(44)
S.I. Kabanikhin
1.1 Introduction
3(9)
1.2 Examples of Inverse and Ill posed Problems
12(12)
1.3 Well posed and Ill posed Problems
24(2)
1.4 The Tikhonov Theorem
26(3)
1.5 The Ivanov Theorem: Quasi solution
29(4)
1.6 The Lavrentiev's Method
33(2)
1.7 The Tikhonov Regularization Method
35(9)
References
44(3)
II Recent Advances in Regularization Theory and Methods 47(148)
2 Using Parallel Computing for Solving Multidimensional Ill posed Problems
49(16)
D.V. Lukyanenko
A.G. Yagola
2.1 Introduction
49(2)
2.2 Using Parallel Computing
51(2)
2.2.1 Main idea of parallel computing
51(1)
2.2.2 Parallel computing limitations
52(1)
2.3 Parallelization of Multidimensional Ill posed Problem
53(8)
2.3.1 Formulation of the problem and method of solution
53(3)
2.3.2 Finite difference approximation of the functional and its gradient
56(2)
2.3.3 Parallelization of the minimization problem
58(3)
2.4 Some Examples of Calculations
61(2)
2.5 Conclusions
63(1)
References
63(2)
3 Regularization of Fredholm Integral Equations of the First Kind using Nystrom Approximation
65(18)
M.T. Nair
3.1 Introduction
65(3)
3.2 Nystrom Method for Regularized Equations
68(6)
3.2.1 Nystrom approximation of integral operators
68(1)
3.2.2 Approximation of regularized equation
69(1)
3.2.3 Solvability of approximate regularized equation
70(3)
3.2.4 Method of numerical solution
73(1)
3.3 Error Estimates
74(6)
3.3.1 Some preparatory results
74(3)
3.3.2 Error estimate with respect to ||·||2
77(1)
3.3.3 Error estimate with respect to ||·||infinity
77(1)
3.3.4 A modified method
78(2)
3.4 Conclusion
80(1)
References
81(2)
4 Regularization of Numerical Differentiation: Methods and Applications
83(38)
T.Y. Xiao
H. Zhang
L.L. Hao
4.1 Introduction
83(4)
4.2 Regularizing Schemes
87(15)
4.2.1 Basic settings
87(1)
4.2.2 Regularized difference method (RDM)
88(1)
4.2.3 Smoother Based regularization (SBR)
89(1)
4.2.4 Mollifier regularization method (MRM)
90(2)
4.2.5 Tikhonov's variational regularization (TiVR)
92(1)
4.2.6 Lavrentiev regularization method (LRM)
93(1)
4.2.7 Discrete regularization method (DRM)
94(2)
4.2.8 Semi Discrete Tikhonov regularization (SDTR)
96(3)
4.2.9 Total variation regularization (TVR)
99(3)
4.3 Numerical Comparisons
102(3)
4.4 Applied Examples
105(10)
4.4.1 Simple applied problems
106(1)
4.4.2 The inverse heat conduct problems (IHCP)
107(1)
4.4.3 The parameter estimation in new product diffusion model
108(2)
4.4.4 Parameter identification of sturm liouville operator
110(2)
4.4.5 The numerical inversion of Abel transform
112(2)
4.4.6 The linear viscoclastic stress analysis
114(1)
4.5 Discussion and Conclusion
115(2)
References
117(4)
5 Numerical Analytic Continuation and Regularization
121(22)
C.L. Fu
H. Cheng
Y.J. Ma
5.1 Introduction
121(3)
5.2 Description of the Problems in Strip Domain and Some Assumptions
124(2)
5.2.1 Description of the problems
124(1)
5.2.2 Some assumptions
125(1)
5.2.3 The ill posedness analysis for the Problems 5.2.1 and 5.2.2
125(1)
5.2.4 The basic idea of the regularization for Problems 5.2.1 and 5.2.2
126(1)
5.3 Some Regularization Methods
126(9)
5.3.1 Some methods for solving Problem 5.2.1
126(7)
5.3.2 Some methods for solving Problem 5.2.2
133(2)
5.4 Numerical Tests
135(5)
References
140(3)
6 An Optimal Perturbation Regularization Algorithm for Function Reconstruction and Its Applications
143(26)
G.S. Li
6.1 Introduction
143(1)
6.2 The Optimal Perturbation Regularization Algorithm
144(3)
6.3 Numerical Simulations
147(12)
6.3.1 Inversion of time dependent reaction coefficient
147(2)
6.3.2 Inversion of space dependent reaction coefficient
149(2)
6.3.3 Inversion of state dependent source term
151(6)
6.3.4 Inversion of space dependent diffusion coefficient
157(2)
6.4 Applications
159(6)
6.4.1 Determining magnitude of pollution source
159(3)
6.4.2 Data reconstruction in an undisturbed soil column experiment
162(3)
6.5 Conclusions
165(1)
References
166(3)
7 Filtering and Inverse Problems Solving
169(26)
L.V. Zotov
V.L. Panteleev
7.1 Introduction
169(1)
7.2 SLAE Compatibility
170(1)
7.3 Conditionality
171(2)
7.4 Pseudosolutions
173(2)
7.5 Singular Value Decomposition
175(2)
7.6 Geometry of Pseudosolution
177(1)
7.7 Inverse Problems for the Discrete Models of Observations
178(2)
7.8 The Model in Spectral Domain
180(1)
7.9 Regularization of Ill posed Systems
181(1)
7.10 General Remarks, the Dilemma of Bias and Dispersion
181(3)
7.11 Models, Based on the Integral Equations
184(1)
7.12 Panteleev Corrective Filtering
185(1)
7.13 Philips Tikhonov Regularization
186(8)
References
194(1)
III Optimal Inverse Design and Optimization Methods 195(54)
8 Inverse Design of Alloys' Chemistry for Specified Thermo Mechanical Properties by using Multi objective Optimization
197(24)
G.S. Dulikravich
I.N. Egorov
8.1 Introduction
198(1)
8.2 Multi Objective Constrained Optimization and Response Surfaces
199(2)
8.3 Summary of IOSO Algorithm
201(2)
8.4 Mathematical Formulations of Objectives and Constraints
203(9)
8.5 Determining Names of Alloying Elements and Their Concentrations for Specified Properties of Alloys
212(2)
8.6 Inverse Design of Bulk Metallic Glasses
214(1)
8.7 Open Problems
215(3)
8.8 Conclusions
218(1)
References
219(2)
9 Two Approaches to Reduce the Parameter Identification Errors
221(20)
Z.H. Xiang
9.1 Introduction
221(2)
9.2 The Optimal Sensor Placement Design
223(10)
9.2.1 The well posedness analysis of the parameter identification procedure
223(3)
9.2.2 The algorithm for optimal sensor placement design
226(3)
9.2.3 The integrated optimal sensor placement and parameter identification algorithm
229(1)
9.2.4 Examples
229(4)
9.3 The Regularization Method with the Adaptive Updating of A priori Information
233(5)
9.3.1 Modified extended Bayesian method for parameter identification
234(1)
9.3.2 The well posedness analysis of modified extended Bayesian method
234(2)
9.3.3 Examples
236(2)
9.4 Conclusion
238(1)
References
238(3)
10 A General Convergence Result for the BFGS Method
241(8)
Y.H. Dai
10.1 Introduction
241(2)
10.2 The BFGS Algorithm
243(1)
10.3 A General Convergence Result for the BFGS Algorithm
244(2)
10.4 Conclusion and Discussions
246(1)
References
247(2)
IV Recent Advances in Inverse Scattering 249(58)
11 Uniqueness Results for Inverse Scattering Problems
251(32)
X.D. Liu
B. Zhang
11.1 Introduction
251(5)
11.2 Uniqueness for Inhomogeneity n
256(1)
11.3 Uniqueness for Smooth Obstacles
256(6)
11.4 Uniqueness for Polygon or Polyhedra
262(1)
11.5 Uniqueness for Balls or Discs
263(2)
11.6 Uniqueness for Surfaces or Curves
265(1)
11.7 Uniqueness Results in a Layered Medium
265(7)
11.8 Open Problems
272(4)
References
276(7)
12 Shape Reconstruction of Inverse Medium Scattering for the Helmholtz Equation
283(24)
G. Bao
P.J. Li
12.1 Introduction
283(2)
12.2 Analysis of the scattering map
285(5)
12.3 Inverse medium scattering
290(8)
12.3.1 Shape reconstruction
291(1)
12.3.2 Born approximation
292(2)
12.3.3 Recursive linearization
294(4)
12.4 Numerical experiments
298(5)
12.5 Concluding remarks
303(1)
References
303(4)
V Inverse Vibration, Data Processing and Imaging 307(60)
13 Numerical Aspects of the Calculation of Molecular Force Fields from Experimental Data
309(22)
G.M. Kuramshina
I.V. Kochikov
A.V. Stepanova
13.1 Introduction
309(2)
13.2 Molecular Force Field Models
311(1)
13.3 Formulation of Inverse Vibration Problem
312(2)
13.4 Constraints on the Values of Force Constants Based on Quantum Mechanical Calculations
314(5)
13.5 Generalized Inverse Structural Problem
319(2)
13.6 Computer Implementation
321(2)
13.7 Applications
323(4)
References
327(4)
14 Some Mathematical Problems in Biomedical Imaging
331(36)
J.J. Liu
H.L. Xu
14.1 Introduction
331(3)
14.2 Mathematical Models
334(5)
14.2.1 Forward problem
334(2)
14.2.2 Inverse problem
336(3)
14.3 Harmonic Bz Algorithm
339(9)
14.3.1 Algorithm description
340(2)
14.3.2 Convergence analysis
342(2)
14.3.3 The stable computation of ΔBz
344(4)
14.4 Integral Equations Method
348(6)
14.4.1 Algorithm description
348(4)
14.4.2 Regularization and discretization
352(2)
14.5 Numerical Experiments
354(8)
References
362(5)
VI Numerical Inversion in Geosciences 367(162)
15 Numerical Methods for Solving Inverse Hyperbolic Problems
369(26)
S.I. Kabanikhin
M.A. Shishlenin
15.1 Introduction
369(1)
15.2 Gel'fand Levitan Krein Method
370(9)
15.2.1 The two dimensional analogy of Gel'fand Levitan Krein equation
374(3)
15.2.2 N-approximation of Gel'fand Levitan Krein equation
377(2)
15.2.3 Numerical results and remarks
379(1)
15.3 Linearized Multidimensional Inverse Problem for the Wave Equation
379(5)
15.3.1 Problem formulation
381(1)
15.3.2 Linearization
382(2)
15.4 Modified Landweber Iteration
384(6)
15.4.1 Statement of the problem
385(2)
15.4.2 Land Weber Iteration
387(1)
15.4.3 Modification of algorithm
388(1)
15.4.4 Numerical results
389(1)
References
390(5)
16 Inversion Studies in Seismic Oceanography
395(16)
H.B. Song
X.H. Huang
L.M. Pinheiro
Y. Song
C.Z. Dong
Y. Bai
16.1 Introduction of Seismic Oceanography
395(3)
16.2 Thermohaline Structure Inversion
398(8)
16.2.1 Inversion method for temperature and salinity
398(1)
16.2.2 Inversion experiment of synthetic seismic data
399(3)
16.2.3 Inversion experiment of GO data (Huang et al., 2011)
402(4)
16.3 Discussion and Conclusion
406(2)
References
408(3)
17 Image Resolution Beyond the Classical Limit
411(28)
L.J. Gelius
17.1 Introduction
411(1)
17.2 Aperture and Resolution Functions
412(5)
17.3 Deconvolution Approach to Improved Resolution
417(7)
17.4 MUSIC Pseudo Spectrum Approach to Improved Resolution
424(10)
17.5 Concluding Remarks
434(2)
References
436(3)
18 Seismic Migration and Inversion
439(36)
Y.F. Wang
Z.H. Li
C.C. Yang
18.1 Introduction
439(1)
18.2 Migration Methods: A Brief Review
440(12)
18.2.1 Kirchhoff migration
440(1)
18.2.2 Wave field extrapolation
441(1)
18.2.3 Finite difference migration in ω - X domain
442(1)
18.2.4 Phase shift migration
443(1)
18.2.5 Stolt migration
443(3)
18.2.6 Reverse time migration
446(1)
18.2.7 Gaussian beam migration
447(1)
18.2.8 Interferometric migration
447(2)
18.2.9 Ray tracing
449(3)
18.3 Seismic Migration and Inversion
452(13)
18.3.1 The forward model
454(2)
18.3.2 Migration deconvolution
456(1)
18.3.3 Regularization model
457(1)
18.3.4 Solving methods based on optimization
458(4)
18.3.5 Preconditioning
462(2)
18.3.6 Preconditioners
464(1)
18.4 Illustrative Examples
465(3)
18.4.1 Regularized migration inversion for point diffraction scatterers
465(3)
18.4.2 Comparison with the interferometric migration
468(1)
18.5 Conclusion
468(3)
References
471(4)
19 Seismic Wave Fields Interpolation Based on Sparse Regularization and Compressive Sensing
475(34)
Y.F. Wang
J.J. Cao
T. Sun
C.C. Yang
19.1 Introduction
475(2)
19.2 Sparse Transforms
477(4)
19.2.1 Fourier, wavelet, Radon and ridgelet transforms
477(3)
19.2.2 The curvelet transform
480(1)
19.3 Sparse Regularizing Modeling
481(1)
19.3.1 Minimization in l0 space
481(1)
19.3.2 Minimization in l1 space
481(1)
19.3.3 Minimization in lp-lq space
482(1)
19.4 Brief Review of Previous Methods in Mathematics
482(3)
19.5 Sparse Optimization Methods
485(11)
19.5.1 l0 quasi norm approximation method
485(2)
19.5.2 l1norm approximation method
487(2)
19.5.3 Linear programming method
489(2)
19.5.4 Alternating direction method
491(2)
19.5.5 l1 norm constrained trust region method
493(3)
19.6 Sampling
496(1)
19.7 Numerical Experiments
497(6)
19.7.1 Reconstruction of shot gathers
497(1)
19.7.2 Field data
498(5)
19.8 Conclusion
503(1)
References
503(6)
20 Some Researches on Quantitative Remote Sensing Inversion
509(20)
H. Yang
20.1 Introduction
509(2)
20.2 Models
511(3)
20.3 A Priori Knowledge
514(2)
20.4 Optimization Algorithms
516(4)
20.5 Multi stage Inversion Strategy
520(4)
20.6 Conclusion
524(1)
References
525(4)
Index 529
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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