A modified Levenberg–Marquardt scheme for solving a class of parameter identification problems

Abstract

Parameter identification problems in PDEs are special class of nonlinear inverse problems which has many applications in science and technology. One such application is the Electrical Impedance Tomography (EIT) problem. Although many methods are available in literature to tackle nonlinear problems, the computation of Fréchet derivative is often a bottle neck for deriving the solution. Moreover, many assumptions are required to establish the convergence of such methods. In this paper, we propose a modified form of Levenberg–Marquardt scheme which does not require the knowledge of exact Fréchet derivative, instead, uses an approximate form of it and at the same time, no additional assumptions are required to establish the convergence of the scheme. We illustrate the theoretical result through numerical examples. In order to ensure that the proposed scheme can be applied to practical problems, we have applied the scheme to EIT problem and the reconstruction process clearly demonstrates that the method can be successfully applied to practical problems.

Keywords:

• Parameter identification problems
• electrical impedance tomography
• nonlinear ill-posed problems
• regularization
• iterative method

AMS Classifications:

• 65J10
• 65J20
• 65J22
• 47L10

Acknowledgments

We profoundly thank the unknown referee(s) for their careful reading of the manuscript and valuable suggestions that significantly improved the presentation of the paper as well.

Disclosure statement

No potential conflict of interest was reported by the author(s).