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, impressive results have been achieved both in the theory ofsolving ill-posed problems and in the applicationsofalgorithms using modem computers. To mention just one field, one can name the computer tomography which could not possibly have been developed without modem tools for solving ill-posed problems. When writing this book, the authors tried to define the place and role of ill posed problems in modem mathematics. In a few words, we define the theory of ill-posed problems as the theory of approximating functions with approximately given arguments in functional spaces. The difference between well-posed and ill posed problems is concerned with the fact that the latter are associated with discontinuous functions. This approach is followed by the authors throughout the whole book. We hope that the theoretical results will be of interest to researchers working in approximation theory and functional analysis. As for particular algorithms for solving ill-posed problems, the authors paid general attention to the principles ofconstructing such algorithms as the methods for approximating discontinuous functions with approximately specified arguments. In this way it proved possible to define the limits of applicability of regularization techniques.

Inhalt
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.