A modified steepest descent method for solving non-smooth inverse problems

Abstract

The classical steepest descent method is applicable only on the inverse problems for which the forward operator is Gâteaux differentiable. In this paper, we propose an extension of the classical steepest descent method so that it can be used for solving non-smooth nonlinear ill-posed problems. Basically, we incorporate the Bouligand subderivative of the forward mapping in order to propose the extension. We study the convergence analysis of the proposed method by assuming the boundedness of the Bouligand subderivative as well as a modified tangential cone condition. In addition to this, we discuss an important feature, i.e., strongly convergent regularizing nature of the proposed method. Finally, we provide the numerical simulations to show the practicality of the proposed method and compare our results with that of existing methods in the literature.