Development and Application of Ill-Posed Problems in the USSR

Abstract

This article describes the earlier stage of development of ill-posed problems in the Soviet Union where this area of mathematics originated. There are several types of the problems, such as Fredholm and Volterra integral equations of the first kind, algebraic systems with ill-conditioned matrix, optimal regulations, approximate Fourier summation, inverse heat conduction, etc, where ill-posed nature is a serious barrier to construct a stable solution. Different methods, in increasing order of generality, showing how to interpret and solve these types of problems are reviewed. The main point is to demonstrate that all approaches in solving the ill-posed problems can be based on common sense and intuition, though formalization is needed to explore the methodology in different fields of applied mathematics and engineering.