On a new notion of the solution to an ill-posed problem

A new understanding of the notion of the stable solution to ill-posed problems is proposed. The new notion is more realistic than the old one and better fits the practical computational needs. A method for constructing stable solutions in the new sense is proposed and justified. The basic point is: in the traditional definition of the stable solution to an ill-posed problem Au=fAu=f, where AA is a linear or nonlinear operator in a Hilbert space HH, it is assumed that the noisy data {fδ,δ}{fδ,δ} are given, ‖f−fδ‖≤δ‖f−fδ‖≤δ, and a stable solution uδ:=Rδfδuδ:=Rδfδ is defined by the relation limδ→0‖Rδfδ−y‖=0limδ→0‖Rδfδ−y‖=0, where yy solves the equation Au=fAu=f, i.e., Ay=fAy=f. In this definition yy and ff are unknown. Any f∈B(fδ,δ)f∈B(fδ,δ) can be the exact data, where B(fδ,δ):={f:‖f−fδ‖≤δ}B(fδ,δ):={f:‖f−fδ‖≤δ}.The new notion of the stable solution excludes the unknown yy and ff from the definition of the solution. The solution is defined only in terms of the noisy data, noise level, and an a priori information about a compactum to which the solution belongs.