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

摘要

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=f, where A is a linear or nonlinear operator in a Hilbert space H, it is assumed that the noisy data {fδ,δ} are given, ‖f−fδ‖≤δ, and a stable solution uδ:=Rδfδ is defined by the relation limδ→0‖Rδfδ−y‖=0, where y solves the equation Au=f, i.e., Ay=f. In this definition y and f are unknown. Any f∈B(fδ,δ) can be the exact data, where B(fδ,δ):={f:‖f−fδ‖≤δ}.The new notion of the stable solution excludes the unknown y and f 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.