Ill-Posed Problems: Theory and Applications
Recent years have been characterized by the increasing amountofpublications in the field ofso-called ill-posed problems. This is easilyunderstandable because we observe the rapid progress of a relatively young branch ofmathematics, ofwhich the first results date back to about 30 years ago. By now, i...
| Main Authors: | , |
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| Format: | eBook |
| Language: | English |
| Published: |
Dordrecht
Springer Netherlands
1994, 1994
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| Edition: | 1st ed. 1994 |
| Series: | Mathematics and Its Applications
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 3.3 Application of Tikhonov’s scheme to Fredholm integral equations of the first kind
- 3.4 Tikonov’s scheme for nonlinear operator equations
- 3.5 Numerical implementation of Tikhonov’s scheme for solving operator equation
- 4 General technique for constructing linear RA for linear problems in Hilbert space
- 4.1 General scheme for constructing RA for linear problems with completely continuous operator
- 4.2 General case of constructing the approximating families and RA
- 4.3 Error estimates for solutions of the ill-posed problems. The optimal algorithms
- 4.4 Regularization in case of perturbed operator
- 4.5 Construction of linear approximating families and RA in Banach space
- 4.6 Stochastic errors. Approximation and regularization of the solution of linear problems in case of stochastic errors
- 5 Iterative algorithms for solvingnon-linear ill-posed problems with monotonic operators. Principle of iterative regularization
- 7 Iterative methods for solving non-linear ill-posed operator equations with non-monotonic operators
- 7.1 Iteratively regularized Gauss - Newton technique for operator equations
- 7.2 The other ways of constructing iterative algorithms for general ill-posed operator equations
- 8 Application of regularizing algorithms to solving practical problems
- 8.1 Inverse problems of image processing
- 8.2 Reconstructive computerized tomography
- 8.3 Computerized tomography of layered objects
- 8.4 Tomographic examination of objects with focused radiation
- 8.5 Seismic tomography in engineering geophysics
- 8.6 Inverse problems of acoustic sounding in wave approximation
- 8.7 Inverse problems of gravimetry
- 8.8 Problems oflinear programming
- Preface
- 1 General problems of regularizability
- 1.1 Definition of regularizing algorithm (RA)
- 1.2 General theorems on regularizability and principles of constructing the regularizing algorithms
- 1.3 Estimates of approximation error in solving the ill-posed problems
- 1.4 Comparison of RA. The concept of optimal algorithm
- 2 Regularizing algorithms on compacta
- 2.1 The normal solvability of operator equations
- 2.2 Theorems on stability of the inverse mappings
- 2.3 Quasisolutions of the ill-posed problems
- 2.4 Properties of ?-quasisolutions on the sets with special structure
- 2.5 Numerical algorithms for approximate solving the ill-posed problem on the sets with special structure
- 3 Tikhonov’s scheme for constructing regularizing algorithms
- 3.1 RA in Tikhonov’s scheme with a priori choice of the regularization parameter
- 3.2 A choice of regularization parameter with the use of the generalized discrepancy
- 5.1 Variational inequalities as a way of formulating non-linear problems
- 5.2 Equivalent transforms of variational inequalities
- 5.3 Browder-Tikhonov approximation for the solutions of variational inequalities
- 5.4 Principle of iterative regularization
- 5.5 Iterative regularization based on the zero-order techniques
- 5.6 Iterative regularization based on the first-order technique (regularized Newton technique)
- 5.7 RA for solving variational inequalities
- 5.8 Estimates of convergence rate of the iterative regularizing algorithms
- 6 Applications of the principle of iterative regularization
- 6.1 Algorithms for minimizing convex functionals. Solving the non-linear equations with monotonic operators
- 6.2 Algorithms for minimizing quadratic functionals. Non-linear procedures for solving linear problems
- 6.3 Iterative algorithms for solving general problems of mathematical programming
- 6.4 Algorithms to find the saddle points and equilibrium points in games