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

Abstract

In this talk, we develop a new semi-discrete version of the Carleman estimate-based convexification method, a globally convergent numerical approach for nonlinear inverse problems. The semi-discrete convexification method is used to generate an accurate starting point for the subsequent deep learning training procedure. An important feature of the continuous convexification method is that its convergence to the true solution does not require the availability of a sufficiently accurate initial guess.

We introduce a new concept of (h)-strong convexity, where (h) denotes the grid step size in the semi-discrete formulation. This property enables us to derive an a priori accuracy estimate for the starting point supplied to the deep learning stage. The proposed approach is applied to the highly nonlinear inverse problem of Electrical Impedance Tomography. Numerical experiments involving complicated conductivity structures demonstrate the accuracy, robustness, and computational feasibility of the combined semi-discrete convexification and deep learning procedure.