The Levenberg-Marquardt Iteration for Numerical Inversion of the Power Density Operator

TL;DR

This paper analyzes the convergence of the Levenberg-Marquardt algorithm for reconstructing conductivity in a hybrid imaging problem, providing theoretical insights and validating with simulated data.

Contribution

It offers a convergence analysis in infinite dimensions and demonstrates the algorithm's effectiveness through numerical experiments.

Findings

Convergence established under injectivity and noise-free data assumptions.

Algorithm successfully reconstructs conductivity from simulated measurements.

Provides theoretical foundation for iterative methods in hybrid conductivity imaging.

Abstract

In this paper we develop a convergence analysis in an infinite dimensional setting of the Levenberg-Marquardt iteration for the solution of a hybrid conductivity imaging problem. The problem consists in determining the spatially varying conductivity $\sigma$ from a series of measurements of power densities for various voltage inductions. Although this problem has been very well studied in the literature, convergence and regularizing properties of iterative algorithms in an infinite dimensional setting are still rudimentary. We provide a partial result under the assumptions that the derivative of the operator, mapping conductivities to power densities, is injective and the data is noise-free. Moreover, we implemented the Levenberg-Marquardt algorithm and tested it on simulated data.