Network approach to conductivity recovery

Abstract

Inverse Problem on MREIT is a problem that finding electrical impedance of internal body by internal electrical relation governed by maxwell`s equation. The simplified equations are following.

\begin{displaymath} \left \{ \begin{array}{ccccc} -\nabla \cdot (\sigma \nabla u) & = & 0 & \textrm{in} & \Omega \\ -\sigma \nabla u & = & g & \textrm{on} & \partial \Omega \end{array}, \right. \end{displaymath}

where $u(\mathbf{x})$ is a electrical potential, $\sigma(\mathbf{x})$ is a electrical conductivity, and $-\sigma \nabla u(:=\mathbf{J})$ is a current density(current passing through unit volume). The objective function is $\sigma(\mathbf{x})$, provided $\mathbf{J}$ or magnetic field density $\mathbf{B}$, particularly only component $B_z$. This kind of equation actually is very abundant in various modeling, it will not be just a electrical progress to resolve that inverse problem but many equillibrium model or diffusive model. In view of this, It was not a brand new approach to approximate it as a discrete network model[21], there are many instances in area mechanical force balancing, or even in electrical model. So we introduce electrical network approach here in connection with finite difference or integral form of equations, and how it will solve them. This will provide simplified framework in the relation between $\mathbf{J}$, $B_z$, $\sigma$. In the first chapter, we introduce about MREIT, in 2nd chapter, we give a brief history on MREIT problem since 1992. In 3rd chapter, we connect our network approach to the others` research and try to give linearized explanation. In 4th chapter, we report our result of numerical simulation using network approach. We are going to mention that it is very stable to noise and solved very nice and fast way.