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A monotonicity-based globalization of the level-set method for inclusion detection

Bastian Harrach, Houcine Meftahi

TL;DR

The paper tackles the problem of recovering inclusions in conductivity from boundary measurements in electrical impedance tomography by merging a globally convergent monotonicity method with a flexible level-set approach. It leverages the Neumann-to-Dirichlet map $\Lambda(\sigma)$ and its Fréchet derivative to generate robust initial guesses (via Loewner monotonicity and monotonicity-based regularization) for a level-set evolution driven by a Kohn–Vogelius functional. The authors derive and employ monotonicity tests, formulate a regularized reconstruction, and implement a level-set refinement that can also recover the conductivity contrast $\sigma_1$; partial boundary data suffice in the Kohn–Vogelius framework. Numerical experiments show that the combined approach yields faster, more stable reconstructions and robust shape/parameter identification under noise and data limitations, outperforming either method alone.

Abstract

We focus on a geometrical inverse problem that involves recovering discontinuities in electrical conductivity based on boundary measurements. This problem serves as a model to introduce a shape recovery technique that merges the monotonicity method with the level-set method. The level-set method, commonly used in shape optimization, often relies heavily on the accuracy of the initial guess. To overcome this challenge, we utilize the monotonicity method to generate a more precise initial guess, which is then used to initialize the level-set method. We provide numerical results to illustrate the effectiveness of this combined approach.

A monotonicity-based globalization of the level-set method for inclusion detection

TL;DR

The paper tackles the problem of recovering inclusions in conductivity from boundary measurements in electrical impedance tomography by merging a globally convergent monotonicity method with a flexible level-set approach. It leverages the Neumann-to-Dirichlet map and its Fréchet derivative to generate robust initial guesses (via Loewner monotonicity and monotonicity-based regularization) for a level-set evolution driven by a Kohn–Vogelius functional. The authors derive and employ monotonicity tests, formulate a regularized reconstruction, and implement a level-set refinement that can also recover the conductivity contrast ; partial boundary data suffice in the Kohn–Vogelius framework. Numerical experiments show that the combined approach yields faster, more stable reconstructions and robust shape/parameter identification under noise and data limitations, outperforming either method alone.

Abstract

We focus on a geometrical inverse problem that involves recovering discontinuities in electrical conductivity based on boundary measurements. This problem serves as a model to introduce a shape recovery technique that merges the monotonicity method with the level-set method. The level-set method, commonly used in shape optimization, often relies heavily on the accuracy of the initial guess. To overcome this challenge, we utilize the monotonicity method to generate a more precise initial guess, which is then used to initialize the level-set method. We provide numerical results to illustrate the effectiveness of this combined approach.
Paper Structure (12 sections, 7 theorems, 51 equations, 9 figures)

This paper contains 12 sections, 7 theorems, 51 equations, 9 figures.

Table of Contents

  1. Introduction
  2. A model problem formulation
  3. Shape recovery via the monotonicity method
  4. Loewner monotonicity
  5. Standard and linearized monotonicity method
  6. The monotonicity-based regularization method
  7. Numerical results with the monotonicity method
  8. Shape recovery via the Kohn-Vogelius functional
  9. Algorithm and numerical results
  10. Combined monotonicty and level-set method for shape reconstruction
  11. Numerical results for shape and parameter reconstruction
  12. Conclusion

Key Result

Lemma 1

Let $\sigma, \tau\in L^\infty_{+}(\Omega)$ be two conductivities and let $g\in L^2_{\diamond}(\partial\Omega)$ be an applied boundary current. Let $u_1= u^g_{\sigma}, u_2=u^g_{\tau}\in H^1_{\diamond}(\Omega)$. Then and

Figures (9)

  • Figure 1: Reconstruction with the linearized monotonicity method in the case of noise-free data
  • Figure 2: Reconstruction with the linearized monotonicity method in the case of noisy data, $\delta=0.1\%$
  • Figure 3: Reconstruction with the monotonicity-based regularization method in the case of noise-free data.
  • Figure 4: Reconstruction with the monotonicity-based regularization method in the case of noisy data, $\delta=10\%$.
  • Figure 5: Reconstruction with the levelset method. First row: initialization (solid contours) and true inclusion (dashed contour). Second row: reconstruction with free noise (solid contours) and true inclusion (dashed contours). Convergence occurs in 70, 79, 87 and 60 iterations (from left to right). Third row: reconstruction with noise level $\eta =0.01$ (solid contours) and true inclusion (dashed contours). Convergence occurs in 78, 82, 125 and 110 iterations (from left to right). Fourth row: reconstruction with noise level $\eta =0.03$ (solid contours) and true inclusion (dashed contours). Convergence occurs in 105, 90, 175 and 120 iterations (from left to right).
  • ...and 4 more figures

Theorems & Definitions (14)

  • Lemma 1
  • proof
  • Corollary 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • ...and 4 more