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Monotonicity Principle in Tomography of Nonlinear Conducting Materials

Antonio Corbo Esposito, Luisa Faella, Gianpaolo Piscitelli, Ravi Prakash, Antonello Tamburrino

Abstract

We treat an inverse electrical conductivity problem which deals with the reconstruction of nonlinear electrical conductivity starting from boundary measurements in steady currents operations. In this framework, a key role is played by the Monotonicity Principle, which establishes a monotonic relation connecting the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). Monotonicity Principles are the foundation for a class of non-iterative and real-time imaging methods and algorithms. In this article, we prove that the Monotonicity Principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions. Then, we show that apart from linear and $p$-Laplacian cases, it is impossible to transfer this Monotonicity result from the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce a new boundary operator, identified as an Average DtN operator.

Monotonicity Principle in Tomography of Nonlinear Conducting Materials

Abstract

We treat an inverse electrical conductivity problem which deals with the reconstruction of nonlinear electrical conductivity starting from boundary measurements in steady currents operations. In this framework, a key role is played by the Monotonicity Principle, which establishes a monotonic relation connecting the unknown material property to the (measured) Dirichlet-to-Neumann operator (DtN). Monotonicity Principles are the foundation for a class of non-iterative and real-time imaging methods and algorithms. In this article, we prove that the Monotonicity Principle for the Dirichlet Energy in nonlinear problems holds under mild assumptions. Then, we show that apart from linear and -Laplacian cases, it is impossible to transfer this Monotonicity result from the Dirichlet Energy to the DtN operator. To overcome this issue, we introduce a new boundary operator, identified as an Average DtN operator.

Paper Structure

This paper contains 18 sections, 7 theorems, 96 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Foundations of the problem
  3. The physical problem
  4. The Electrical Conductivity
  5. Scalar potential and Dirichlet Energy functional
  6. Well-posedness of the Forward Problem
  7. Connection among $\sigma$, $J_\sigma$ and $Q_\sigma$
  8. The DtN operator.
  9. The Gâteaux derivative of the Dirichlet Energy
  10. A convergence result
  11. The perturbation of the Dirichlet Energy with respect to the minimizer
  12. The perturbation of $\mathbb F_\sigma$ with respect to the boundary values
  13. Monotonicity Principle
  14. Monotonicity Principle for the Dirichlet Energy
  15. Connection between Dirichlet Energy and DtN operator
  16. ...and 3 more sections

Key Result

Lemma 3.1

Let $\Omega$ be an open bounded domain with Lipschitz boundary and $f,\varphi\in X_\diamond$. Then where $u^{f+\varepsilon \varphi}\in W^{1,p}(\Omega)$ is the minimizer of gminimum corresponding to the boundary data $f+\varepsilon \varphi$.

Figures (3)

  • Figure 1: Impact of (H4) on the constitutive relationship in terms of electrical conductivity (a) and current density (b). Solid lines corresponds to the upper and lower constraints to either $\sigma$ or ${\bf J}$.
  • Figure 2: The nonlinear case. For any given spatial point in the region $\Omega$, (a) the electrical conductivity $\sigma(\cdot,E)$ is the secant line to the graph of the function $J_\sigma(\cdot,E)$; (b) $Q_\sigma (\cdot, E)$ is the area of the sub-graph of $J_\sigma(\cdot, E)$.
  • Figure 3: The linear case. For any given spatial point of the region $\Omega$, (a) the electrical conductivity is both the secant and the tangent line to the graph of the function $J_\sigma(\cdot, E)$; (b) the area of the subgraph, that is $Q_\sigma (\cdot, E)$, and the area of the super-graph are both equal to a half of the ohmic power density $J\! E$ absorbed by the system.

Theorems & Definitions (13)

  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • Proposition 3.4
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • ...and 3 more