Linear ill-posed problems and dynamical systems

TL;DR

This paper introduces a novel method for solving linear ill-posed problems in Hilbert spaces by using dynamical systems, proving convergence to the true solution even with data errors.

Contribution

It proposes a new approach involving a dynamical system to solve ill-posed linear equations, establishing existence, uniqueness, and convergence of solutions.

Findings

The method guarantees convergence of the solution u(t) to the true solution y.

The approach handles data errors in the right-hand side f.

Existence and uniqueness of the global solution u(t) are proven.

Abstract

A linear equation Au=f (1) with a bounded, injective, but not boundedly invertible linear operator in a Hilbert space H is studied. A new approach to solving linear ill-posed problems is proposed. The approach consists of solving a Cauchy problem for a linear equation in H, which is a dynamical system, proving the existence and uniqueness of its global solution u(t), and establishing that u(t) tends to a limit y, as t tends to infinity, and this limit y solves equation (1). The case when f in (1) is given with some error is also studied.