S.F. Gilyazov and N.L. Gol'dman
Regularization of Ill-Posed Problems by Iteration Methods
Dordrecht: Kluwer Academic Publishers, 2000

This monograph presents new results in regularization of ill-posed problems by iteration methods, which is one of the most important and rapidly developing topics of the theory of ill-posed problems. These results are connected with the proposed united approach to the proof of regularizing properties of the `classical' iteration methods (steepest descent, conjugate direction) complemented by the stopping rule depending on the level of errors in the input data. Much emphasis is given to the choice of the iteration index as the regularization parameter and to the rate convergence estimates of the approximate solutions.

Descriptive regularization algorithms on the basis of conjugate gradient projection method utilizing shape constraints imposed on the sought solution are proposed. They are investigated for stable numerical solution of a wide class of ill-posed problems (the Fredholm integral equation of the first kind, inverse problems on the determination of boundary functions and coefficients of linear and quasilinear parabolic equations, etc.). Such algorithms ensure substantial savings in computational costs and universal.

Results of calculations for important applications in non-linear thermophysics (a continuous casting, the treatment of materials and perfection of heat-protective systems using laser and composite technologies) are also presented.

This book will be a useful resourse for specialists in the theory of partial differential and integral equations, in numerical analysis, theory and methods of solving ill-posed problems.