Algorithmic foundations of methods for solving ill-posed problems

Some questions of efficient implementation of regularizing algorithms for solving practical unstable problems that can be reduced to systems of linear algebraic equations are considered. Special attention is given to the choice of regularizing parameters (both theoretically justified and heuristic) in the framework of the theory of regularization. Evolution of the concepts of “solution” and “approximate method” is discussed for linear systems. The work was supported by the Russian Foundation for Basic Research (01-01-0398).