Convergence of the gradient method for ill-posed problems

TL;DR

This paper proves the convergence of the gradient descent method for solving ill-posed problems in Hilbert spaces, extending classical conditions to more general nonlinearity scenarios.

Contribution

It introduces new nonlinearity conditions that generalize classical tangential cone conditions, ensuring convergence of the gradient method for ill-posed problems.

Findings

Proves weak and strong convergence under new conditions

Extends classical Landweber iteration analysis

Provides theoretical foundation for broader problem classes

Abstract

We study the convergence of the gradient descent method for solving ill-posed problems where the solution is characterized as a global minimum of a differentiable functional in a Hilbert space. The classical least-squares functional for nonlinear operator equations is a special instance of this framework and the gradient method then reduces to Landweber iteration. The main result of this article is a proof of weak and strong convergence under new nonlinearity conditions that generalize the classical tangential cone conditions.