arXiv subjects

Camille Carvalho

Publications and source records attributed to Camille Carvalho.

10 recordsLinked to original sources

arXiv2026

Solving forward and inverse wave scattering via boundary integral equations and deep learning. Applications to cloaking design

We propose a deep learning framework based on an encoder-decoder architecture for the design and evaluation of cloaking devices, demonstrated in this work for two-dimensional wave propagation governed by the Helmholtz equation. The cloaks under consideration are concentric layered media surrounding the object, whose geometry and material parameters determine the scattering response. We consider circular and object-fitted layer configurations and parameterize all designs by the layer thicknesses, enabling a unified representation for direct comparison of different cloaks for the same object. Training data are generated using a boundary element formulation suitable for geometries where analytic solutions are not available, and neural networks are trained with standard hyperparameters on geometry-specific datasets. The proposed approach is applied to circular, star-shaped, and kite-shaped objects. Results show that object-fitted configurations consistently outperform simpler circular-layer designs in scattering reduction, highlighting the importance of geometry in cloaking performance. Overall, we present a flexible, data-driven approach for systematic comparison of cloaking strategies, with potential extension to more complex geometries and broadband settings.

physics.comp-ph↗

arXiv2026

Discretization in Multilayered Media with High Contrasts: Is It All About the Boundaries?

Wave propagation in multilayered media with high material contrasts poses significant numerical challenges, as large variations in wavenumbers lead to strong reflections and complex transmission of the incoming wave field. To address these difficulties, we employ a boundary integral formulation thereby avoiding volumetric discretization. In this framework, the accuracy of the numerical solution depends strongly on how the material interfaces are discretized. In this work, we demonstrate that standard meshing strategies based on resolving the maximum wavenumber across the domain become computationally inefficient in multilayered configurations, where high wavenumbers are confined to localized subdomains. Through a systematic study of multilayer transmission problems, we show that no simple discretization rule based on the maximum wavenumber or material contrasts emerges. Instead, the wavenumber of the background (exterior) medium plays a dominant role in determining the optimal boundary resolution. Building on these insights, we propose an adaptive approach that achieves uniform accuracy and efficient computation across multiple layers. Numerical experiments for a range of multilayer configurations demonstrate the scalability and robustness of the proposed approach.

math.NA↗

arXiv2022

Scattering resonances in unbounded transmission problems with sign-changing coefficient

It is well-known that classical optical cavities can exhibit localized phenomena associated to scattering resonances, leading to numerical instabilities in approximating the solution. This result can be established via the ``quasimodes to resonances'' argument from the black-box scattering framework. Those localized phenomena concentrate at the inner boundary of the cavity and are called whispering gallery modes. In this paper we investigate scattering resonances for unbounded transmission problems with sign-changing coefficient (corresponding to optical cavities with negative optical properties, for example made of metamaterials). Due to the change of sign of optical properties, previous results cannot be applied directly, and interface phenomena at the metamaterial-dielectric interface (such as the so-called surface plasmons) emerge. We establish the existence of scattering resonances for arbitrary two-dimensional smooth metamaterial cavities. The proof relies on an asymptotic characterization of the resonances, and showing that problems with sign-changing coefficient naturally fit the black box scattering framework. Our asymptotic analysis reveals that, depending on the metamaterial's properties, scattering resonances situated closed to the real axis are associated to surface plasmons. Examples for several metamaterial cavities are provided.

math.AP↗

arXiv2021

Modified representations for the close evaluation problem

When using boundary integral equation methods, we represent solutions of a linear partial differential equation as layer potentials. It is well-known that the approximation of layer potentials using quadrature rules suffer from poor resolution when evaluated closed to (but not on) the boundary. To address this challenge, we provide modified representations of the problem's solution. Similar to Gauss's law used to modify Laplace's double-layer potential, we use modified representations of Laplace's single-layer potential and Helmholtz layer potentials that avoid the close evaluation problem. Some techniques have been developed in the context of the representation formula or using interpolation techniques. We provide alternative modified representations of the layer potentials directly (or when only one density is at stake). Several numerical examples illustrate the efficiency of the technique in two and three dimensions.

math.NA↗

arXiv2021

Quadrature by Parity Asymptotic eXpansions (QPAX) for scattering by high aspect ratio particles

We study scattering by a high aspect ratio particle using boundary integral equation methods. This problem has important applications in nanophotonics problems, including sensing and plasmonic imaging. To illustrate the effect of parity and the need for adapted methods in presence of high aspect ratio particles, we consider the scattering in two dimensions by a sound-hard, high aspect ratio ellipse. This fundamental problem highlights the main challenge and provide valuable insights to tackle plasmonic problems and general high aspect ratio particles. For this problem, we find that the boundary integral operator is nearly singular due to the collapsing geometry from an ellipse to a line segment. We show that this nearly singular behavior leads to qualitatively different asymptotic behaviors for solutions with different parities. Without explicitly taking this nearly singular behavior and this parity into account, computed solutions incur a large error. To address these challenges, we introduce a new method called Quadrature by Parity Asymptotic eXpansions (QPAX) that effectively and efficiently addresses these issues. We first develop QPAX to solve the Dirichlet problem for Laplace's equation in a high aspect ratio ellipse. Then, we extend QPAX for scattering by a sound-hard, high aspect ratio ellipse. We demonstrate the effectiveness of QPAX through several numerical examples.

math.NA↗

arXiv2020

Close evaluation of layer potentials in three dimensions

We present a simple and effective method for evaluating double-and single-layer potentials for Laplace's equation in three dimensions close to the boundary. The close evaluation of these layer potentials is challenging because they are nearly singular integrals. The method we propose is based on writing these layer potentials in spherical coordinates where the point at which their kernels are peaked maps to the north pole. An N-point Gauss-Legendre quadrature rule is used for integration with respect to the the polar angle rather than the cosine of the polar angle. A 2N-point periodic trapezoid rule is used to compute the integral with respect to the azimuthal angle which acts as a natural and effective averaging operation in this coordinate system. The numerical method resulting from combining these two quadrature rules in this rotated coordinate system yields results that are consistent with asymptotic behaviors of the double- and single-layer potentials at close evaluation distances. In particular, we show that the error in computing the double-layer potential, after applying a subtraction method, is quadratic with respect to the evaluation distance from the boundary, and the error is linear for the single-layer potential. We improve upon the single-layer potential by introducing an alternate approximation based on a perturbation expansion and obtain an error that is quadratic with respect to the evaluation distance from the boundary.

math.NA↗

arXiv2019

A Fully Fourth Order Accurate Energy Stable Finite Difference Method for Maxwell's Equations in Metamaterials

We present a novel fully fourth order in time and space {finite difference method for the time domain Maxwell's equations} in metamaterials. We consider a Drude metamaterial model for the material response to incident electromagnetic fields. We consider the second order formulation of the system of partial differential equations that govern the evolution in time of electric and magnetic fields along with the evolution of the polarization and magnetization current densities. Our discretization employs fourth order staggering in space of different field components and the modified equation approach to obtain fourth order accuracy in time. Using the energy method, we derive energy relations for the continuous models, and design numerical schemes that preserve a discrete analogue of the energy relation. Numerical simulations are provided in one and two dimensional settings to illustrate fourth order convergence as well as compare with second order schemes.

math.NA↗

arXiv2018

Asymptotic approximations for the close evaluation of double-layer potentials

When using the boundary integral equation method to solve a boundary value problem, the evaluation of the solution near the boundary is challenging to compute because the layer potentials that represent the solution are nearly-singular integrals. To address this close evaluation problem, we apply an asymptotic analysis of these nearly singular integrals and obtain an asymptotic approximation. We derive the asymptotic approximation for the case of the double-layer potential in two and three dimensions, representing the solution of the interior Dirichlet problem for Laplace's equation. By doing so, we obtain an asymptotic approximation given by the Dirichlet data at the boundary point nearest to the interior evaluation point plus a nonlocal correction. We present numerical methods to compute this asymptotic approximation, and we demonstrate the efficiency and accuracy of the asymptotic approximation through several examples. These examples show that the asymptotic approximation is useful as it accurately approximates the close evaluation of the double-layer potential while requiring only modest computational resources.

math.NA↗

arXiv2017

Asymptotic analysis for close evaluation of layer potentials

We study the evaluation of layer potentials close to the domain boundary. Accurate evaluation of layer potentials near boundaries is needed in many applications, including fluid-structure interactions and near-field scattering in nano-optics. When numerically evaluating layer potentials, it is natural to use the same quadrature rule as the one used in the Nyström method to solve the underlying boundary integral equation. However, this method is problematic for evaluation points close to boundaries. For a fixed number of quadrature points, $N$, this method incurs $O(1)$ errors in a boundary layer of thickness $O(1/N)$. Using an asymptotic expansion for the kernel of the layer potential, we remove this $O(1)$ error. We demonstrate the effectiveness of this method for interior and exterior problems for Laplace's equation in two dimensions.

math.NA↗

arXiv2016

On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients

We investigate in a $2$D setting the scattering of time-harmonic electromagnetic waves by a plasmonic device, represented as a non dissipative bounded and penetrable obstacle with a negative permittivity. Using the $\textrm{T}$-coercivity approach, we first prove that the problem is well-posed in the classical framework $H^1_{\text{loc}} $ if the negative permittivity does not lie in some critical interval whose definition depends on the shape of the device. When the latter has corners, for values inside the critical interval, unusual strong singularities for the electromagnetic field can appear. In that case, well-posedness is obtained by imposing a radiation condition at the corners to select the outgoing black-hole plasmonic wave, that is the one which carries energy towards the corners. A simple and systematic criterion is given to define what is the outgoing solution. Finally, we propose an original numerical method based on the use of Perfectly Matched Layers at the corners. We emphasize that it is necessary to design an $\textit{ad hoc}$ technique because the field is too singular to be captured with standard finite element methods.

math.NA↗