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The Monotonicity Principle for Magnetic Induction Tomography

Antonello Tamburrino, Gianpaolo Piscitelli, Zhengfang Zhou

Abstract

The inverse problem treated in this article consists in reconstructing the electrical conductivity from the free response of the system in the magneto-quasi-stationary (MQS) limit. The MQS limit corresponds to a diffusion PDE. In this framework, a key role is played by the Monotonicity Principle, that is a monotone relation connecting the unknown material property to the (measured) free-response. MP is relevant as basis of noniterative and real-time imaging methods. Monotonicity Principles have been found in many different physical problems governed by PDEs of different nature. Despite its rather general nature, each different physical/mathematical context requires to discover the proper operator showing MP. For doing this, it is necessary to develop ad-hoc mathematical approaches tailored on the specific framework. In this article, we prove a monotonic relationship between the electrical resistivity and the time constants characterizing the free-response for MQS systems. The key result is the representation of the induced current density through a modal representation. The main result is based on the analysis of an elliptic eigenvalue problem, obtained from separation of variables.

The Monotonicity Principle for Magnetic Induction Tomography

Abstract

The inverse problem treated in this article consists in reconstructing the electrical conductivity from the free response of the system in the magneto-quasi-stationary (MQS) limit. The MQS limit corresponds to a diffusion PDE. In this framework, a key role is played by the Monotonicity Principle, that is a monotone relation connecting the unknown material property to the (measured) free-response. MP is relevant as basis of noniterative and real-time imaging methods. Monotonicity Principles have been found in many different physical problems governed by PDEs of different nature. Despite its rather general nature, each different physical/mathematical context requires to discover the proper operator showing MP. For doing this, it is necessary to develop ad-hoc mathematical approaches tailored on the specific framework. In this article, we prove a monotonic relationship between the electrical resistivity and the time constants characterizing the free-response for MQS systems. The key result is the representation of the induced current density through a modal representation. The main result is based on the analysis of an elliptic eigenvalue problem, obtained from separation of variables.

Paper Structure

This paper contains 17 sections, 10 theorems, 68 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Statement of the Problem
  3. Mathematical Model for the Eddy Current Problem
  4. The Eigenvalue Problem
  5. The First Eigenvalue
  6. The Higher Eigenvalues
  7. Monotonicity of eigenvalues
  8. Variational Characterization
  9. The Monotonicity Principle for the Eigenvalues
  10. Interpretation of the results
  11. The Completeness of the basis
  12. Orthogonality and decoupling
  13. Circuit interpretation
  14. Ill-posedness of the problem
  15. Shape Reconstruction, converse of Monotonicity and Bounds
  16. ...and 2 more sections

Key Result

Lemma 3.1

Let $\mathbf{v}_{j}\in H_L(\Omega)$. If $\mathbf{v}_{j}\rightharpoonup\mathbf{v}$ in ${L}^{2}\left( \Omega;\mathbb{R}^3\right)$, then $\mathbf{v}\in H_L(\Omega)$.

Figures (4)

  • Figure 1: Representation of a typical Magnetic induction tomography.
  • Figure 2: The domain $\Omega$ (dashed region) and two typical elements $\bf w$ and $\bf u$ of $H_L(\Omega)$. Both vector fields present closed streamlines. The streamlines for $\bf w$ are homotopic to a point, whereas the streamlines for $\bf u$ circulate around the cavity.
  • Figure 3: Interpretation of equation (\ref{['ode_f2']}) in terms of electrical circuit.
  • Figure 4: The decoupled systems. $i_c$ is the inducted current produced by the vector potential ${\bf A}_s$ produced by the source.

Theorems & Definitions (20)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 10 more