Ill-posed problems by Hadamard and their approximated solution by A.N. Tikhonov's regularization method
© V.I. Starostenko, S.M. Hovhannisian
Since the 60-s of the last century different methods of the approximate solution of the ill-posed problems by Hadamard began to be intensively solved and successively used. This trend became one of the central ones both in the computer mathematics and in the theory and practices of the interpretation of geophysical, astrophysical, physical, medical and other observations. Among all elaborated orientations, the most common method of solving ill-posed problems is A.N. Tikhonov's regularization method. The purpose of this paper is to outline briefly, strictly but in a sufficiently simple form the principles of the regularization method and to give the working algorithms of the solution of problems with linear and non-linear operators. The association between the regularization method and other known methods and algorithms is shown. The review is first of all oriented at the beginning geophysicists that wish to enter into the problematics (widely published presently) in the shortest time. That's why the review gives some elements of the functional analysis that facilitate the data presentation. In the paper preparation both widely known sources and the results of the own researches made by the authors were used. This review has been prepared on the occasion of the 95th anniversary of Academician A.N. Tikhonov, a prominent mathematician and geophysicist of the XXth century (30.10.1906-7.10.1993). The review is dedicated to his memory.