Accelerated homotopy perturbation iteration method for a non-smooth nonlinear ill-posed problem

Abstract

In this paper, a fast iterative algorithm based on J-order homotopy perturbation method is proposed for the nonlinear ill-posed problem whose forward operator is not Gâteaux differentiable. The Bouligand subderivative of the forward operator is utilized to replace the Fréchet derivative in iteration system. The sequential subspace optimization technique is introduced to accelerate the convergence speed, which regards the correction term of homotopy perturbation as multiple search directions to update the new iterate. To this end, the current iteration is sequentially projected to the stripes whose width is determined by search directions, the nonlinearity of the forward operator and noise level. We present the convergence analysis based on the asymptotic stability estimates and a generalized tangential cone condition. Numerical experiments are performed to illustrate the effectiveness of the proposed method.