A high order discontinuous Galerkin method for the recovery of the conductivity in Electrical Impedance Tomography

TL;DR

This paper introduces a high order discontinuous Galerkin method for solving the nonlinear inverse problem of Electrical Impedance Tomography, demonstrating high accuracy and efficiency in reconstructing interior conductivities from boundary measurements.

Contribution

The paper develops a novel third order DG method for EIT, including a new optimization formulation and theoretical proof of minimizer existence, with extensive numerical validation.

Findings

High accuracy in conductivity reconstruction

Efficient solution for complex 2D problems

Effective handling of discontinuous conductivities

Abstract

In this work, we develop an efficient high order discontinuous Galerkin (DG) method for solving the Electrical Impedance Tomography (EIT). EIT is a highly nonlinear ill-posed inverse problem where the interior conductivity of an object is recovered from the surface measurements of voltage and current flux. We first propose a new optimization problem based on the recovery of the conductivity from the Dirichlet-to-Neumann map to minimize the mismatch between the predicted current and the measured current on the boundary. And we further prove the existence of the minimizer. Numerically the optimization problem is solved by a third order DG method with quadratic polynomials. Numerical results for several two-dimensional problems with both single and multiple inclusions are demonstrated to show the high {accuracy and efficiency} of the proposed high order DG method. Analysis and computation for discontinuous conductivities are also studied in this work.