arXiv subjects

Vincenzo Mottola

Publications and source records attributed to Vincenzo Mottola.

10 recordsLinked to original sources

arXiv2026

Monotonicity Principle and "p-Laplace Signature" for Tomography in Nonlinear Elliptic Inverse Problems

This paper proposes a framework for treating the inverse obstacle problem for nonlinear elliptic equations with nonlinear materials. The problem is challenging because nonlinear materials exhibit a rich diversity of scenarios to consider, since nonlinearity can take different forms. In this article, after categorizing the nonlinearities into a few fundamental classes, a dedicated imaging method is proposed for each class, derived by combining two powerful concepts: the Monotonicity Principle (MP) and the $p-$Laplace Signature (pLS). The Monotonicity Principle (MP), recently extended to nonlinear materials, provides a monotonic relationship between the material property and the measured quantity (the Average Dirichlet-to-Neumann map) that can be \lq\lq inverted\rq\rq \ to find the shape of anomalies. The $p-$Laplace Signature (pLS) allows for modelling the solution of an elliptic PDE with nonlinear materials, for large or small boundary data, in terms of a proper $p-$Laplace equation that captures the essence (the signature) of the problem. For example, pLS with $p=2$ allows the reduction of a nonlinear elliptic PDE to a linear one, providing a powerful bridge for applying imaging methods and algorithms developed for linear materials. In this contribution, the two pillars of MP and pLS are combined in new imaging methods to enlarge the class of nonlinearity that can be treated within the inverse obstacle problem. Moreover, the theoretical limits of the methods are provided in the ideal case of noise-free measurements: outer-support reconstruction when $p=2$, and convex-hull reconstruction when $p\neq2$.

eess.SP↗

arXiv2026

The Kernel Method for Electrical Resistance Tomography

This paper treats the inverse problem of retrieving the electrical conductivity of a material starting from boundary measurements in the framework of Electrical Resistance Tomography (ERT). In particular, the focus is on non-iterative reconstruction methods suitable for real-time applications. In this work, the Kernel Method, a new non-iterative reconstruction method for Electrical Resistance Tomography, is presented. The imaging algorithm addresses the problem of retrieving one or more anomalies of arbitrary shape, topology, and size embedded in a known background (the inverse obstacle problem). The foundation of the Kernel Method is based on the idea that if a proper current density applied at the boundary (Neumann data) of the domain exists such that it is able to produce the same measurements with and without the anomaly, then this boundary source produces a power density that vanishes in the region occupied by the anomaly, when applied to the problem involving the background material only. This new tomographic method has a simple numerical implementation that requires a very low computational cost. In this paper, the theoretical foundation of the Kernel Method is provided, and an extensive numerical campaign proves the effectiveness of this new imaging method.

math.NA↗

arXiv2026

Adjoint-Based Gradient Evaluation for Metasurface Inverse Design via Affine Geometric Transformations

The sharp increasing in fabrication capabilities of nanomaterials, and complex structures such as meta-surfaces and metalens, has opened to the possibility of employing them for accurately control the electromagnetic field, beyond the possibility ensured by traditional devices. The demand for large scale structures and more complex functionalities from meta-surfaces lead to the research for advanced techniques of inverse design, able to conjugate the ability to produce effective designs and limited computational cost. Among the various approaches for inverse design of large meta-surfaces, the ones based on the adjoint variable method are appealing since able to ensure a minimal computational cost for the gradient computation of the cost function. In this work, a systematic methodology for the application of the adjoint variable method for large meta-surface design is presented. The method is based on: (i) a parametrization of the relevant geometric parameters of the meta-atoms, (ii) the fast computation of the gradient with respect such parameters, allowing for the implementation of general affine transformations during the optimization process. The main findings are first theoretically justified and a numerical validation is provided to show the effectiveness of the proposed approach.

math.NA↗

arXiv2026

QR-Recursive Compression of Volume Integral Equations for Electromagnetic Scattering by Large Metasurfaces

In this paper, a novel QR decomposition-based compression scheme is combined with a volume integral equations method for the fast and efficient numerical computation of the scattering of electromagnetic fields from large scale metasurfaces, via an iterative approach. The underlying problem is of a multiscale nature. Indeed, these metasurfaces are made of a large collection of interacting sub-wavelength scatterers, thus making the numerical computation of the solution very challenging. More specifically, the paper proposes a tailored version of a QR decomposition-based compression for a volume integral equation, together with a proper preconditioner that exploits the geometrical structure of the array, in order to achieve a fast and accurate iterative solver, in view of realistic applications. Numerical examples prove the effectiveness of the method in efficiently modeling metasurfaces made by thousands of particles.

math.NA↗

arXiv2025

Invariants in Eddy Current Testing via Dimensional Analysis

The Buckingham's $π$, theorem has been recently introduced in the context of Non destructive Testing \& Evaluation (NdT\&E) , giving a theoretical basis for developing simple but effective methods for multi-parameter estimation via dimensional analysis. Dimensional groups, or $π-$groups, allow for the reduction of the number of parameters affecting the dimensionless measured quantities. In many real-world applications, the main interest is in estimating only a subset of the variables affecting the measurements. An example is estimating the thickness and electrical conductivity of a plate from Eddy Current Testing data, regardless of the lift-off of the probe, which may be either uncertain and/or variable. Alternatively, one may seek to estimate thickness and lift-off while neglecting the influence of the electrical conductivity, or to estimate the electrical conductivity and the lift-off, neglecting the thickness. This is where the concept of invariants becomes crucial. An invariant transformation is a mathematical mapping that makes the measured signal independent of one or more of these uncertain parameters. Invariant transformations provide a way to isolate useful signals from uncertain ones, improving the accuracy and reliability of the NdT results. The main contribution of this paper is a systematic method to derive \emph{invariant} transformations for frequency domain Eddy Current Testing data, via dimensional analysis. The proposed method is compatible with real-time and in-line operations. After its theoretical foundation is introduced, the method is validated by means of experimental data, with reference to configurations consisting of plates with different thicknesses, electrical conductivity, and lift-off. The experimental validation proves the effectiveness of the method in achieving excellent accuracy on a wide range of parameters of interest.

eess.SP↗

arXiv2025

The inverse obstacle problem for nonlinear inclusions

The Monotonocity Principle (MP), stating a monotonic relationship between a material property and a proper corresponding boundary operator, is attracting great interest in the field of inverse problems, because of its fundamental role in developing real time imaging methods. Moreover, under quite general assumptions, a MP for elliptic PDEs with nonlinear coefficients has been established. This MP provided the basis for introducing a new imaging method to deal with the inverse obstacle problem, in the presence of nonlinear anomalies. This constitutes a relevant novelty because there is a general lack of quantitative and physic based imaging method, when nonlinearities are present. The introduction of a MP based imaging method poses a set of fundamental questions regarding the performance of the method in the presence of noise. The main contribution of this work is focused on theoretical aspects and consists in proving that (i) the imaging method is stable and robust with respect to the noise, (ii) the reconstruction approaches monotonically to a well-defined limit, as the noise level approaches to zero, and that (iii) the limit contains the unknown set and is contained in the outer boundary of the unknown set. Results (i) and (ii) come directly from the Monotonicity Principle, while results (iii) requires to prove the so-called Converse of the Monotonicity Principle, a theoretical sults of fundamental relevance to evaluate the ideal (noise-free) performances of the imaging method. The results are provided in a quite general setting for Calderòn problem, and proved for three wide classes where the nonlinearity of the anomaly can be either bounded from infinity and zero, or bounded from zero only, or bounded by infinity only. These classes of constitutive relationships cover the wide majority of cases encountered in applications.

math.AP↗

arXiv2024

Modal Decomposition in Numerical Computation of Eddy Current Transients

A methodology to reduce the computational cost of time domain computations of eddy currents problems is proposed and implemented in a parallel computing environment. It is based on the modal decomposition of the current density and it is applicable even in presence of injected currents into the electrodes of a conducting domain. Using a theta-method integration algorithm, the performances of the the proposed approach are compared against those of a classical method based on the Cholesky factorization, for a case of interest from eddy current nondestructive testing. For this large eddy current problem (number of unknowns greater than 100k, number of time steps of interest equal to 100k) the proposed solution method is shown to be much faster than those based on standard time integration schemes.

cs.CE↗

arXiv2024

Piecewise nonlinear materials and Monotonicity Principle

This paper is focused on the Monotonicity Principle (MP) for nonlinear materials with piecewise growth exponent. This results are relevant because enables the use of a fast imaging method based on MP, to the wide class of problems with two or more materials, where at least one is nonlinear. The treatment is very general and allows to model a wide variety of practical configurations such as, for instance, Superconducting (SC) or Perfect Electrical Conducting (PEC) or Perfect Electrical Insulating (PEI) materials. A key role is played by the average Dirichlet-to-Neumann operator, introduced in [Corbo Esposito et. al, Inverse Problems 2021], where the MP for a single type of nonlinearity was treated. Realistic numerical examples confirm the theoretical findings.

math.AP↗

arXiv2024

Tomography of nonlinear materials via the Monotonicity Principle

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.

math.NA↗

arXiv2023

Imaging of nonlinear materials via the Monotonicity Principle

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.

math.NA↗