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Tomography of nonlinear materials via the Monotonicity Principle

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

TL;DR

This work tackles inverse imaging of nonlinear materials by aiming to recover a nonlinear anomaly $A$ embedded in a linear background from boundary magnetostatic measurements. It introduces a non-iterative approach grounded in the Monotonicity Principle (MP) for nonlinear elliptic PDEs, leveraging the average Dirichlet-to-Neumann operator $\overline{\Lambda}$ and a test-anomaly framework. To overcome the nonlinearity of $\overline{\Lambda}$, the method precomputes boundary potentials from linear problems and uses eigenfunctions of $\overline{\Lambda}_F^u - \overline{\Lambda}_T^l$ to select probing potentials $f_0$, enabling a monotonicity test that yields a reconstructed region $\tilde{A}$ as a union of test anomalies $T_k$. Numerical experiments in a circular domain with a nonlinear ferromagnetic anomaly demonstrate real-time feasibility and robustness to noise, while addressing multi-component geometries as an avenue for future work.

Abstract

In this paper we present a first non-iterative imaging method for nonlinear materials, based on Monotonicity Principle. Specifically, we deal with the inverse obstacle problem, where the aim is to retrieve a nonlinear anomaly embedded in linear known background. The Monotonicity Principle (MP) is a general property for various class of PDEs, that has recently generalized to nonlinear elliptic PDEs. Basically, it states a monotone relation between the point-wise value of the unknown material property and the boundary measurements. It is at the foundation of a class of non-iterative imaging methods, characterized by a very low execution time that makes them ideal candidates for real-time applications. In this work, we develop an inversion method that overcomes some of the peculiar difficulties in practical application of MP to imaging of nonlinear materials, preserving the feasibility for real-time applications. For the sake of clarity, we focus on a specific application, i.e. the Magnetostatic Permeability Tomography where the goal is retrieving the unknown (nonlinear) permeability by boundary measurements in DC operations. This choice is motivated by applications in the inspection of boxes and containers for security. Reconstructions from simulated data prove the effectiveness of the presented method.

Tomography of nonlinear materials via the Monotonicity Principle

TL;DR

This work tackles inverse imaging of nonlinear materials by aiming to recover a nonlinear anomaly embedded in a linear background from boundary magnetostatic measurements. It introduces a non-iterative approach grounded in the Monotonicity Principle (MP) for nonlinear elliptic PDEs, leveraging the average Dirichlet-to-Neumann operator and a test-anomaly framework. To overcome the nonlinearity of , the method precomputes boundary potentials from linear problems and uses eigenfunctions of to select probing potentials , enabling a monotonicity test that yields a reconstructed region as a union of test anomalies . Numerical experiments in a circular domain with a nonlinear ferromagnetic anomaly demonstrate real-time feasibility and robustness to noise, while addressing multi-component geometries as an avenue for future work.

Abstract

In this paper we present a first non-iterative imaging method for nonlinear materials, based on Monotonicity Principle. Specifically, we deal with the inverse obstacle problem, where the aim is to retrieve a nonlinear anomaly embedded in linear known background. The Monotonicity Principle (MP) is a general property for various class of PDEs, that has recently generalized to nonlinear elliptic PDEs. Basically, it states a monotone relation between the point-wise value of the unknown material property and the boundary measurements. It is at the foundation of a class of non-iterative imaging methods, characterized by a very low execution time that makes them ideal candidates for real-time applications. In this work, we develop an inversion method that overcomes some of the peculiar difficulties in practical application of MP to imaging of nonlinear materials, preserving the feasibility for real-time applications. For the sake of clarity, we focus on a specific application, i.e. the Magnetostatic Permeability Tomography where the goal is retrieving the unknown (nonlinear) permeability by boundary measurements in DC operations. This choice is motivated by applications in the inspection of boxes and containers for security. Reconstructions from simulated data prove the effectiveness of the presented method.
Paper Structure (4 sections, 1 theorem, 13 equations, 2 figures)

This paper contains 4 sections, 1 theorem, 13 equations, 2 figures.

Table of Contents

  1. Introduction and statement of the problem
  2. Main idea and Proposed Method
  3. Numerical Examples
  4. Conclusions

Key Result

Proposition 1

Let $A\subset\Omega$ the anomalous region filled by the nonlinear magnetic permeability $\mu_{NL}$. Let $T\subset\Omega$ the region occupied by the test anomaly and let $F\subset\Omega$ such that $A\subseteq F$. If $\overline{\Lambda}_F^u-\overline{\Lambda}_T^l$ has negative eigenvalues, then any ei

Figures (2)

  • Figure 1: Left: Magnetic permeabilities involved. Right: Test anomaly $T$, Actual anomaly $A$ included in the region $F$.
  • Figure 2: Reconstructions carried out by the proposed method. In white the estimated anomaly $\tilde{A}$ while the boundary of the anomaly $A$ is in red. The noise level is equal to $\eta=10^{-3}$.

Theorems & Definitions (1)

  • Proposition 1