Ill-Posed Problems: Theory and Applications

Inbunden, Engelska, 1994

AvA. Bakushinsky,A. Goncharsky,Anatoly Bakushinsky

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This volume presents a unified approach to the solution of ill-posed problems, based on the concept of a regularizing algorithm (RA). This idea is then explored in the discussion of topics such as common conditions for the existence of regularizing algorithms, necessary and sufficient conditions of the approximations for linear problems, and the principle of iterative regularization for nonlinear problems. The majority of these issues have not previously been discussed in a monograph on ill-posed problems. The efficiency of many of the suggested algorithms should prove useful in their application to a wide range of practical problems. This volume can be read by anyone with a basic knowledge of functional analysis. This book should be of interest to applied mathematicians, engineers, and specialists in inverse problems.

Produktinformation

Utgivningsdatum1994-09-30
Mått160 x 241 x 20 mm
Vikt576 g
FormatInbunden
SpråkEngelska
SerieMathematics and Its Applications
Antal sidor258
Upplaga1994
FörlagKluwer Academic Publishers
ISBN9780792330738

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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.- 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.- 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.- 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.