Convergence analysis of a regularized iterative scheme for solving nonlinear problems

Abstract

Nonlinear inverse and ill-posed problems occur in many practical applications and the regularization techniques are employed to get a stable approximate solution for the same. Although many schemes are available in literature, iterative regularization techniques are the most commonly used approaches. One such important method is the Levenberg-Marquardt scheme. However, the scheme involves computation of the Fréchet derivative at every iterate which makes it tedious and the restrictive assumptions on it often difficult to verify for practical scenarios. In this paper, we propose a simplified Levenberg-Marquardt scheme that has two benefits. Firstly, computation of the Fréchet derivative is required only once at the initial point and secondly, the convergence and optimal convergence rate of the method is established with weaker assumptions as compared to the standard method. We also provide numerical examples to illustrate the theory and, results clearly illustrate the advantages of the proposed scheme over the standard method.