Some convergence results of a modified asymptotical regularization gradient method for nonlinear ill-posed operator equation

Abstract

A modified asymptotical regularization gradient method is presented for nonlinear ill-posed operator equations. Employing a priori and a posteriori rules respectively, we obtain some convergence results under a modified source condition. The method is stable with respect to perturbations of the source condition. If the modified source condition holds, the method is of optimal order under a priori rule. The method can be taken as the generalization of the existing asymptotical gradient method. From the method, many other convergent iterative methods can be constructed. Our numerical results show that the algorithm is valid.

Keywords:

• ill-posed
• nonlinear operator equation
• asymptotical regularization
• gradient method
• convergence

2000 AMS Subject Classifications :

• 47J06
• 47J25
• 65J20

Acknowledgements

The authors are grateful to the anonymous referees for their very helpful comments. This project is supported by Scientific Research Funds of Hunan Provincial Education Department (08C667,08C025) and Huaihua University (HHUQ2008-01).