Numerical approach on inverse problem of 3D isotropic conductivity by resistivity network

Abstract

MREIT is a problem of finding the conductivity of internal body by internal current density with Maxwell's equation. The well known method is solving the following equation

$\nabla \cdot (\sigma \nabla u) = 0 in \Omega$ $\sigma \nabla u = g on \partial \Omega$

where u is electrical potential, $\sigma$ is conductivity. Here by Ohm's law the relation of current density J and u, $\sigma$ is $J = -\sigma \nabla u$, and with this relation the MREIT gives a way to construct the conductivity.

In the meantime, another approach was made, using another equation to construct the conductivity. Using the irrotationality of the electrical field we derive $\nabla \times (rJ)$ where r is an inverse of conductivity and is called resistivity. With such relation Mingi Lee [6] find a non-iterative method for construction in 2 dimension case with resistivity network modeling. In this thesis we approach numerically the 3 dimension isotropic resistivity construction with similar idea.