Further study on homotopy perturbation iteration for nonlinear inverse problems

In this paper, we propose two versions of the homotopy perturbation iteration with inexact inner solvers (HP-IIS) for nonlinear inverse problems in both Hilbert and Banach spaces. These approaches involve the integration of an inexact inner solver to estimate the solution of the minimization problem, since solving this problem at each iteration step exactly in general is impossible in practical applications. Based on the ϵ-subdifferential calculus, we establish the convergence analysis of the proposed methods in both spaces. To validate the theoretical results, we implement the proposed HP-IIS methods through parameter identification inverse problems, auto-convolution inverse problems and Robin coefficient reconstruction problems. Numerical results demonstrate that while the exact solution to minimization problem remains intractable, the HP-IIS methods yield reconstructions that are quantitatively favorable. Furthermore, the proposed HP-IIS methods are still effective in scenarios where the data is corrupted by non-Gaussian noise.