The inverse conductivity problem via the calculus of functions of bounded variation

TL;DR

This paper introduces a new method for solving the inverse conductivity problem using functions of bounded variation, proposing a BV-based framework and a specialized minimization scheme validated through numerical experiments.

Contribution

The work develops a BV space-based approach for the inverse conductivity problem and introduces a novel minimization scheme tailored to this functional space.

Findings

Effective reconstruction of conductivity profiles demonstrated.

Numerical experiments validate the theoretical approach.

BV framework improves stability and accuracy.

Abstract

In this work, a novel approach for the solution of the inverse conductivity problem from one and multiple boundary measurements has been developed on the basis of the implication of the framework of BV - functions. The space of the functions of bounded variation is here recommended as the most appropriate functional space hosting the conductivity profile under reconstruction. For the numerical solution, we propose and implement a suitable minimization scheme of an enriched - constructed herein - functional, by exploiting the inner structure of BV - space. Finally, we validate and illustrate our theoretical results with numerical experiments.