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 A u = f , where A is a linear or nonlinear operator in a Hilbert space H , it is assumed that the noisy data { f δ , δ } are given, ‖ f − f δ ‖ ≤ δ , and a stable solution u δ : = R δ f δ is defined by the relation lim δ → 0 ‖ R δ f δ − y ‖ = 0 , where y solves the equation A u = f , i.e., A y = f . In this definition y and f are unknown. Any f ∈ B ( f δ , δ ) can be the exact data, where B ( f δ , δ ) : = { f : ‖ f − f δ ‖ ≤ δ } . w 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.