Electrical Impedance Tomography with Restricted Dirichlet-to-Neumann Map Data

TL;DR

This paper introduces a globally convergent numerical method for reconstructing isotropic electrical conductivity in EIT using restricted boundary data, employing Carleman estimates and convexification techniques.

Contribution

The paper develops a novel convexification approach with Carleman estimates for stable EIT reconstruction from limited boundary data.

Findings

The method achieves global convergence to the true solution.

Numerical examples show high accuracy and stability.

The approach effectively handles restricted boundary data.

Abstract

We propose a new numerical method to reconstruct the isotropic electrical conductivity from measured restricted Dirichlet-to-Neumann map data in electrical impedance tomography (EIT) model. "Restricted Dirichlet-to-Neumann (DtN) map data" means that the Dirichlet and Neumann boundary data for EIT are generated by a point source running either along an interval of a straight line or along a curve located outside of the domain of interest. We "convexify" the problem via constructing a globally strictly convex Tikhonov-like functional using a Carleman Weight Function. In particular, two new Carleman estimates are established. Global convergenceto the correct solution of the gradient projection method for this functional is proven. Numerical examples demonstrate a good performance of this numerical procedure.