Discretization and Convergence of the EIT Optimal Control Problem in 2D and 3D Domains

TL;DR

This paper studies the discretization and convergence of an optimal control approach to Electrical Impedance Tomography (EIT) in 2D and 3D, proving that finite difference discretizations converge to the continuous problem.

Contribution

It introduces a new Sobolev-Hilbert space framework and proves convergence of finite difference discretizations for EIT optimal control in 2D and 3D.

Findings

Finite difference discretization converges to the continuous EIT control problem.

A new Sobolev-Hilbert space is introduced for analysis.

Convergence is established in both 2D and 3D domains.

Abstract

We consider Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity and potential in the body based on the measurement of the boundary voltages on the $m$ electrodes for a given electrode current. The variational formulation is pursued in the optimal control framework, where electrical conductivity and boundary voltages are control parameters, and the cost functional is the norm declinations of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. EIT optimal control problem is fully discretized using the method of finite differences. New Sobolev-Hilbert space is introduced, and the convergence of the sequence of finite-dimensional optimal control problems to EIT coefficient optimal control problem is proved both with respect to functional and control in 2- and 3-dimensional domains.