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

Michael V. Klibanov, Kirill V. Golubnichiy, Benjamin Jiang

TL;DR

A new semi-discrete version of the Carleman estimate-based convexification globally convergent numerical method is developed and 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.

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.

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

TL;DR

A new semi-discrete version of the Carleman estimate-based convexification globally convergent numerical method is developed and 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.

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.
Paper Structure (30 sections, 208 equations, 5 figures, 1 table)

This paper contains 30 sections, 208 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Statements of Forward and Inverse Problems
  3. The Carleman Estimate-Based Convexification Method for Coefficient Inverse Problem (\ref{['2-2']})-(\ref{['2.6']})
  4. Transformation procedure
  5. The Carleman estimate
  6. The globally strongly convex CWF-weighted Tikhonov-like functional
  7. The accuracy of the minimizer
  8. The Semi-Discrete Analogue of the Functional $J_{\kappa ,\alpha }$
  9. Discretization
  10. Some function spaces
  11. Semi-discretization of the functional $J_{\kappa ,\alpha }$
  12. Theorems About the Semi-Discrete Functional $J_{h,\kappa ,\alpha }$
  13. The $h-$strong convexity of $J_{h,\kappa ,\alpha }$
  14. Accuracy estimates
  15. Minimizing sequence
  16. ...and 15 more sections

Figures (5)

  • Figure 1: A schematic diagram of our measurements. The large disk is the domain $P_{D}\left( a,b\right)$ where the forward problem is solved. The smaller circle is the circle $E_{B}\left( a,b\right)$ where point sources are located$.$ The square is our domain of interest $\Omega ,$ where the CIP is solved.
  • Figure 1: Architecture of the proposed DnCNN + ResUNet model with SEBlock. The grayscale input image is denoised by DnCNN and then passed to a SE-enhanced ResUNet for reconstruction.
  • Figure 1: Left column: image obtained by the semi-discrete version of the convexification method on a too coarse grid with the grid step size $h_{1}=0.1$ as in (\ref{['7.4']}). Right column: the true image. Middle column: the reconstructed image after deep learning. These results are visibly worse than those obtained with the a finer coarse grid step size $h_{2}=0.05$ as in (\ref{['7.5']}) in the next Figure \ref{['fig:qualitative_results']}. Apparently, the coarse grid step size $h_{1}=0.1$ is too large.
  • Figure 2: Left column: the image obtained by the semi-discrete version of the convexification method on a coarse mesh with a finer coarse grid with step size $h_{2}=0.05$ as in (\ref{['7.5']}). The a priori accuracy estimate of the starting point is given in (\ref{['7.2']}). Right column: the true image. The same characters are used in this column as in \ref{['fig:poor_mesh_results']}. Middle column: the reconstructed image after deep learning. One can observe an excellent reconstruction accuracy in the middle column. Our algorithm recovers quite well intricate structures and sharp contours.
  • Figure 3: This is the second series of images with the grid step size of a finer coarse grid $h_{2}=0.05$ as in (\ref{['7.5']}). Left column: image obtained by the semi-discrete version of the convexification method with $h_{2}=0.05.$ Right column: the true image. Middle column: the reconstructed image after deep learning. The same high reconstruction accuracy as the one of Figure \ref{['fig:qualitative_results']} is observed.