On a family of gradient type projection methods for nonlinear ill-posed problems

TL;DR

This paper introduces a unified family of gradient projection methods for nonlinear ill-posed problems satisfying the Tangential Cone Condition, extending existing methods and demonstrating improved convergence properties through theoretical analysis and numerical experiments.

Contribution

It develops a new family of projection methods that unify and extend gradient-based approaches, allowing convergence under a broader TCC range.

Findings

The new methods converge for a TCC twice as large as traditional methods.

Numerical experiments confirm the efficiency of the proposed methods.

The framework unifies Landweber, minimal error, and steepest descent methods.

Abstract

We propose and analyze a family of successive projection methods whose step direction is the same as Landweber method for solving nonlinear ill-posed problems that satisfy the Tangential Cone Condition (TCC). This family enconpasses Landweber method, the minimal error method, and the steepest descent method; thush providing an unified framework for the analysis of these methods. Moreover, we define in this family new methods which are convergent for the constant of the TCC in a range twice as large as the one required for the Landweber and other gradient type methods. The TCC is widely used in the analysis of iterative methods for solving nonlinear ill-posed problems. The key idea in this work is to use the TCC in order to construct special convex sets possessing a separation property, and to succesively project onto these sets. Numerical experiments are presented for a nonlinear 2D elliptic parameter identification problem, validating the efficiency of our method.