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

TL;DR

This paper introduces a more practical and realistic definition of stable solutions for ill-posed problems, removing the need to know the exact data and solution, and proposes a method for constructing such solutions.

Contribution

It proposes a new, more realistic notion of stable solutions for ill-posed problems and develops a method for constructing solutions under this new framework.

Findings

The new notion better fits practical computational needs.

A justified method for constructing stable solutions is proposed.

The traditional definition involves unknowns, which the new approach avoids.

Abstract

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 $\{f_\delta, \delta\}$ are given, $||f-f_\delta||\leq \delta$, and a stable solution $u_\d:=R_\d f_\d$ is defined by the relation $\lim_{\d\to 0}||R_\d f_\d-y||=0$, where $y$ solves the equation $Au=f$, i.e., $Ay=f$. In this definition $y$ and $f$ are unknown. Any $f\in B(f_\d,\d)$ can be the exact data, where $B(f_\d,\d):=\{f: ||f-f_\delta||\leq \delta\}$.The new notion of the stable solution excludes the unknown $y$ and $f$ from the definition of the solution.