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Imaging of nonlinear materials via the Monotonicity Principle

Vincenzo Mottola, Antonio Corbo Esposito, Gianpaolo Piscitelli, Antonello Tamburrino

Abstract

Inverse problems, which are related to Maxwell's equations, in the presence of nonlinear materials is a quite new topic in the literature. The lack of contributions in this area can be ascribed to the significant challenges that such problems pose. Retrieving the spatial behaviour of some unknown physical property, from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. Furthermore, this complexity grows exponentially in the presence of nonlinear materials. In the tomography of linear materials, the Monotonicity Principle (MP) is the foundation of a class of non-iterative algorithms able to guarantee excellent performances and compatibility with real-time applications. Recently, the MP has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background for this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The proposed method is intendend for all problems governed by the quasilinear Laplace equation, i.e. static problems involving nonlinear materials. In this paper, we provide some preliminary results which give the foundation of our method and some extended numerical examples.

Imaging of nonlinear materials via the Monotonicity Principle

Abstract

Inverse problems, which are related to Maxwell's equations, in the presence of nonlinear materials is a quite new topic in the literature. The lack of contributions in this area can be ascribed to the significant challenges that such problems pose. Retrieving the spatial behaviour of some unknown physical property, from boundary measurements, is a nonlinear and highly ill-posed problem even in the presence of linear materials. Furthermore, this complexity grows exponentially in the presence of nonlinear materials. In the tomography of linear materials, the Monotonicity Principle (MP) is the foundation of a class of non-iterative algorithms able to guarantee excellent performances and compatibility with real-time applications. Recently, the MP has been extended to nonlinear materials under very general assumptions. Starting from the theoretical background for this extension, we develop a first real-time inversion method for the inverse obstacle problem in the presence of nonlinear materials. The proposed method is intendend for all problems governed by the quasilinear Laplace equation, i.e. static problems involving nonlinear materials. In this paper, we provide some preliminary results which give the foundation of our method and some extended numerical examples.
Paper Structure (29 sections, 5 theorems, 64 equations, 15 figures, 2 tables)

This paper contains 29 sections, 5 theorems, 64 equations, 15 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Mathematical Model
  3. Connection to physical problems
  4. Magnetostatic case
  5. Electrostatic case
  6. Steady currents case
  7. Imaging Method
  8. The MP for the inverse obstacle problem
  9. Imaging Method
  10. Design of the Test Anomalies
  11. Well separated material properties
  12. Not well separated material properties
  13. Testing condition $\overline{\Lambda}_{T}\nleqslant\overline{\Lambda}_{A}$
  14. Basic idea and main result
  15. Upper bound to $\langle \overline{\Lambda}_A(f),f\rangle$
  16. ...and 14 more sections

Key Result

Theorem 3.4

Let $\gamma_1$ and $\gamma_2$ be two material properties satisfying (H1)-(H4), then

Figures (15)

  • Figure 1: Geometry of the sample under test for the magnetostatic case. In white the domain $\Omega$, in green the anomaly $A$ and in gray the material surrounding the domain $\Omega$ characterized by an infinite magnetic permeability. A prescribed surface current density $\mathbf{J}_s$ (in yellow) is imposed on $\partial \Omega$.
  • Figure 2: A: well separated ($c_{nl}^l>c_{bg}^u$) material properties. B: well-separated ($c_{nl}^u<c_{bg}^l$) material properties. C: not well separated material properties.
  • Figure 3: Left: the actual configuration. Center: the real anomaly is included in a known anomaly $F$. Right: the nonlinear material in $F$ is replaced by a linear one.
  • Figure 4: Left: Test anomaly not included in $A$. Right: Material properties involved.
  • Figure 5: Not well separated material properties. The non linear material property $\gamma$ considered here is the relative magnetic permeability of M$330-50$ Electrical Steel. The horizontal axis corresponds to the magnetic field magnitude $H$, i.e. $s=H/(1 \text{A/m})$.
  • ...and 10 more figures

Theorems & Definitions (17)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Theorem 3.4: Monotonicity Principle, art:Co21MPMETHODS
  • Remark 3.5
  • Proposition 4.1
  • Remark 4.2
  • Remark 4.3
  • Lemma 4.4
  • proof
  • ...and 7 more