arXiv subjects

Luisa Faella

Publications and source records attributed to Luisa Faella.

7 recordsLinked to original sources

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↗

arXiv2024

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 Calder\`on 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↗

arXiv2023

Junction of ferroelectric thin cylinders

In this paper, starting from a non-convex and nonlocal $3D-$variational model for the electric polarization in a ferroelectric material, and using an asymptotic process based on dimensional reduction, we analyze junction phenomena for two vertical joined ferroelectric thin cilynders with differents small thicknesses. We obtain three variational models for thin wire joined to a thin film, depending on the limit of the ratio between these two thicknesses.

math.AP↗

arXiv2023

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.

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arXiv2020

Monotonicity Principle in Tomography of Nonlinear Conducting Materials

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.

math.AP↗

arXiv2019

A homogenization bending shell theory for multiscale materials from $3D$ nonlinear elasticity

We derive homogenized bending shell theories starting from three dimensional nonlinear elasticity. The original three dimensional model contains three small parameters: the two homogenization scales $\varepsilon$ and $\varepsilon^2$ of the material properties and the thickness $h$ of the shell. Depending on the asymptotic ratio of these three parameters, we obtain different asymptotic theories.

math.AP↗

arXiv2013

Uniform resolvent convergence for strip with fast oscillating boundary

In a planar infinite strip with a fast oscillating boundary we consider an elliptic operator assuming that both the period and the amplitude of the oscillations are small. On the oscillating boundary we impose Dirichlet, Neumann or Robin boundary condition. In all cases we describe the homogenized operator, establish the uniform resolvent convergence of the perturbed resolvent to the homogenized one, and prove the estimates for the rate of convergence. These results are obtained as the order of the amplitude of the oscillations is less, equal or greater than that of the period. It is shown that under the homogenization the type of the boundary condition can change.

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