A fast two-point gradient method for solving non-smooth nonlinear ill-posed problems

Abstract

We propose and analyze a fast two-point gradient (TPG) method for solving non-smooth ill-posed inverse problems where the forward operator is merely directionally but not Gâteaux differentiable. This method is seen as a combination of a derivative-free Landweber iteration and a general case of Nesterov’s acceleration scheme. Since the forward mapping is not Gâteaux differentiable in our case, the standard analysis is not applicable to the convergence analysis. Under certain assumptions on the combination parameters, we therefore provide a new convergence analysis of the proposed method also with the help of the concept of asymptotic stability and a generalized tangential cone condition. The design of the TPG method involves the choices of the combination parameters which are carefully discussed. Moreover, the TPG method is applied to an inverse source problem for a non-smooth semilinear elliptic PDE where a Bouligand subdifferential can be used in place of the non-existing Gâteaux derivative, and the corresponding Bouligand two-point gradient iteration is shown to be a convergent regularization scheme. Numerical simulations are presented to illustrate the advantages over the Bouligand–Landweber iteration.