A fast generalized two-point homotopy perturbation iteration with a learned initial value for nonlinear ill-posed problems

Abstract

In this paper, a new fast generalized iteration is proposed for solving nonlinear ill-posed problems in which forward operators may not be Gâteaux differentiable. We confirm that the generalized iteration constructed by the homotopy perturbation method and the two-point gradient method is an iterative regularization method. In addition, the physics-informed neural network is used to generate the initial value required for the iteration to converge faster and avoid falling into local minima. There is a key idea to use the modified discrete backtracking search algorithm to determine the combination parameters in each iteration. Since the forward operators may not be derivable in the process of theoretical analysis, we approximate it by the Bouligand sub-differential, which is proposed in Clason and Nhu (2019). The concept of asymptotic stability is introduced, which together with a generalized tangential cone condition proves the convergence and regularity of this method. Finally, several smooth and non-smooth numerical examples are carried out to demonstrate the efficiency and superior performance of the proposed method.