Convexification With the Viscocity Term for Electrical Impedance Tomography

TL;DR

This work develops a globally convergent convexification method for a 2D coefficient inverse problem in Electrical Impedance Tomography by introducing a viscosity term that yields a tractable two-elliptic-PDE system. A Carleman-weighted functional $J_{\lambda,\alpha}$ is minimized for each angular parameter to recover an auxiliary field from which the unknown conductivity $\sigma$ is reconstructed via a stabilized Poisson-type problem solved with the Quasi-Reversibility Method. The authors prove a Carleman estimate, establish global strong convexity and gradient-descent convergence for large $\lambda$, and demonstrate robust, accurate reconstructions of complex inclusions (including CT-like abdominal shapes) under modest noise. The approach provides a principled route to global convergence in EIT reconstructions and shows practical promise for challenging media with high-contrast inclusions. Overall, the method combines Carleman-based convexification, a viscosity regularization, and a robust numerical pipeline to achieve reliable conductivity imaging from boundary measurements.

Abstract

A version of the globally convergent convexification numerical method is constructed for the problem of Electrical Impedance Tomography in the 2D case. An important element of this version is the presence of the viscosity term. Global convergence analysis is carried out. Results of numerical experiments are presented.

Figures (10)

• Figure 1: A schematic diagram of our measurement setup. The small black square indicates the domain $\Omega$ defined in (\ref{['2.20']}). The large blue circle is the domain $G$ where the forward problem (\ref{['2.8']}), (\ref{['2.9']}) is solved to generate the data for the inverse problem. The red circle is the circle $C_{A}$ in (\ref{['2.1']}) and small red discs on it indicate positions of the point source.
• Figure 2: The results of the solution of the forward problem (\ref{['2.8']}), (\ref{['2.9']}) for the case when the inclusion has the shape of the letter '$A$' with $\sigma _{a}=2$ inside of this letter, see (\ref{['8.01']}). Top $\mathbf{x}\in G$, bottom $\mathbf{x}\in \Omega .$ The positions of the source are: $\mathbf{x}_{0} = (3.5, 1.5)$ (1st column), $\mathbf{x}_{0} = (1.5, 3.5)$ (2nd column), $\mathbf{x}_{0} = (-0.5, 1.5)$ (3rt column), $\mathbf{x}_{0} = (1.5, -0.5)$ (4th column). The red star is the source position.
• Figure 3: Test 1. The reconstructed coefficient $\sigma( \mathbf{x} )$, where the function $\sigma( \mathbf{x} )$ is given in 8.01 with $\sigma_{a}=2$ inside of the letter 'A'. We test different values of $\lambda$.
• Figure 4: Test 1. The convergence behavior of $\left\vert \nabla J_{\lambda ,\alpha }\left( r,s\right) \left( \varphi \right) \right\vert$ with the iterations of fmincon for $\varphi =\pi$ and the optimal triple of parameters $\left( \alpha ,\varepsilon ,\lambda \right)$ as in (\ref{['8.6']}). The function $\sigma \left( \mathbf{x}\right)$ is given in (\ref{['8.01']}) with $\sigma _{a}=2$ inside of the letter '$A$'. The $y-$axis corresponds to the value of $\left\vert \nabla J_{\lambda ,\alpha }\left( r,s\right) \left( \varphi \right) \right\vert$.
• Figure 5: Test 2. Exact (left) and reconstructed (right) coefficient $\sigma (\mathbf{x})$ with $\sigma_{a}=4$ (first row) and $\sigma_{a}=8$ (second row) inside of the letter 'A' as in (\ref{['8.01']}). The inclusion/background contrasts in (\ref{['8.02']}) are respectively $4:1$ and $8:1$. The reconstructions of both shapes of inclusions and the inclusion/background contrasts (\ref{['8.02']}) are accurate, although the image in the second row is more blurred.