# Adaptive Reconstruction for Electrical Impedance Tomography with a Piecewise Constant Conductivity

## TL;DR

This paper introduces an adaptive numerical method for electrical impedance tomography that reconstructs piecewise constant conductivities using Tikhonov regularization, a posteriori error estimators, and mesh refinement, with proven convergence.

## Contribution

It develops a novel adaptive algorithm combining residual-based error estimation with a Modica-Mortola penalty for improved conductivity reconstruction.

## Key findings

- The adaptive method converges to a solution of the continuous optimality system.
- Numerical examples demonstrate the effectiveness and convergence of the proposed algorithm.
- The approach effectively reconstructs piecewise constant conductivities from boundary measurements.

## Abstract

In this work we propose and analyze a numerical method for electrical impedance tomography of recovering a piecewise constant conductivity from boundary voltage measurements. It is based on standard Tikhonov regularization with a Modica-Mortola penalty functional and adaptive mesh refinement using suitable a posteriori error estimators of residual type that involve the state, adjoint and variational inequality in the necessary optimality condition and a separate marking strategy. We prove the convergence of the adaptive algorithm in the following sense: the sequence of discrete solutions contains a subsequence convergent to a solution of the continuous necessary optimality system. Several numerical examples are presented to illustrate the convergence behavior of the algorithm.