A Carleman Semi-Discrete Convexification Method Combined With Deep Learning for Electrical Impedance Tomography

TL;DR

This paper introduces a semi-discrete Carleman convexification method combined with deep learning to improve the solution process for Electrical Impedance Tomography, ensuring global convergence and providing a reliable starting point for training.

Contribution

It develops a new semi-discrete convexification approach with h-strong convexity, enabling accurate initial guesses for deep learning in nonlinear inverse problems.

Findings

Numerical experiments show successful reconstruction of complex media structures.

The method guarantees convergence without requiring a good initial guess.

The approach is computationally feasible for practical EIT problems.

Abstract

In this paper, a new semi-discrete version of the Carleman estimate-based convexification globally convergent numerical method is developed. It is used for the delivery of the starting point for the training procedure of deep learning. An important feature of the continuous version of the convexification method is that its convergence to the true solution is independent on the availability of a good first guess about this solution. A new concept of the h-strong convexity is introduced, where h is the grid step size in the semi-discrete version of the convexification method. The h -strong convexity allows to obtain an a priori accuracy estimate of the starting point for the training step of the deep learning procedure. This approach is demonstrated for a highly nonlinear problem of Electrical Impedance Tomography. Results of numerical experiments for complicated media structures demonstrate the computational feasibility of this procedure.