An iteration regularizaion method with general convex penalty for nonlinear inverse problems in Banach spaces

TL;DR

This paper introduces a new iterative regularization method with convex penalties for nonlinear inverse problems in Banach spaces, leveraging homotopy perturbation to improve solution feature detection and computational efficiency.

Contribution

It presents a novel iterative scheme based on homotopy perturbation and convex analysis, with proven convergence and stability for nonlinear inverse problems in Banach spaces.

Findings

Reduces computational time compared to Landweber iteration.

Effectively detects features like sparsity and piecewise constancy.

Validated through numerical simulations in parameter identification.

Abstract

In this paper, we discuss the construction, analysis and implementation of a novel iterative regularization scheme with general convex penalty term for nonlinear inverse problems in Banach spaces based on the homotopy perturbation technique, in an attempt to detect the special features of the sought solutions such as sparsity or piecewise constant. By using tools from convex analysis in Banach spaces, we provide a detailed convergence and stability results for the presented algorithm. Numerical simulations for one-dimensional and two-dimensional parameter identification problems are performed to validate that our approach is competitive in terms of reducing the overall computational time in comparison with the existing Landweber iteration with general convex penalty.