arXiv subjects

Lyes Rahmouni

Publications and source records attributed to Lyes Rahmouni.

15 recordsLinked to original sources

arXiv2023

On some Operator Filtering Strategies Based on Suitably Modified Green's Functions

Recent contributions showed the benefits of operator filtering for both preconditioning and fast solution strategies. While previous contributions leveraged laplacian-based filters, in this work we introduce and study a different approach leveraging the truncation of appropriately chosen spectral representations of operators' kernels. In this contribution, the technique is applied to the operators of the 2D TE- and TM-electric field integral equations (EFIE). We explore two different spectral representations for the 2D Green's function that lead to two distinct types of filtering of the EFIE operators. Numerical results corroborate the effectiveness of the newly proposed approaches, also in the Calderón preconditioned EFIE

math.NA↗

arXiv2022

Laplacian Filtered Loop-Star Decompositions and Quasi-Helmholtz Laplacian Filters: Definitions, Analysis, and Efficient Algorithms

Quasi-Helmholtz decompositions are fundamental tools in integral equation modeling of electromagnetic problems because of their ability of rescaling solenoidal and non-solenoidal components of solutions, operator matrices, and radiated fields. These tools are however incapable, per se, of modifying the refinement-dependent spectral behavior of the different operators and often need to be combined with other preconditioning strategies. This paper introduces the new concept of filtered quasi-Helmholtz decompositions proposing them in two incarnations: the filtered Loop-Star functions and the quasi-Helmholtz Laplacian filters. Because they are capable of manipulating large parts of the operators' spectra, new families of preconditioners and fast solvers can be derived from these new tools. A first application to the case of the frequency and h-refinement preconditioning of the electric field integral equation is presented together with numerical results showing the practical effectiveness of the newly proposed decompositions.

math.NA↗

arXiv2022

Fast Direct Solvers for Integral Equations at Low-Frequency Based on Operator Filtering

This paper focuses on fast direct solvers for integral equations in the low-to-moderate-frequency regime obtained by leveraging preconditioned first kind or second kind operators regularized with Laplacian filters. The spectral errors arising from boundary element discretizations are properly handled by filtering that, in addition, allows for the use of low-rank representations for the compact perturbations of all operators involved. Numerical results show the effectiveness of the approaches and their effectiveness in the direct solution of integral equations.

math.NA↗

arXiv2022

On the Fast Direct Solution of a Preconditioned Electromagnetic Integral Equation

This work presents a fast direct solver strategy for electromagnetic integral equations in the high-frequency regime. The new scheme relies on a suitably preconditioned combined field formulation and results in a single skeleton form plus identity equation. This is obtained after a regularization of the elliptic spectrum through the extraction of a suitably chosen equivalent circulant problem. The inverse of the system matrix is then obtained by leveraging the Woodbury matrix identity, the low-rank representation of the extracted part of the operator, and fast circulant algebra yielding a scheme with a favorable complexity and suitable for the solution of multiple right-hand sides. Theoretical considerations are accompanied by numerical results both of which are confirming and showing the practical relevance of the newly developed scheme.

math.NA↗

arXiv2022

Laplacian Filters for Integral Equations: Further Developments and Fast Algorithms

This paper extends the concept of Laplacian filtered quasi-Helmholtz decompositions we have recently introduced, to the basis-free projector-based setting. This extension allows the discrete analyses of electromagnetic integral operators spectra without passing via an explicit Loop-Star decomposition as previously done. We also present a fast scheme for the evaluation of the filters in quasi linear complexity in the total number of unknowns. Together with the fact that only a logarithmic number of these filters are required for solving the h-refinement breakdown of electric field integral equation, this results in an effective preconditioner that rivals Calderón strategies in performance without relying on barycentric refinements. Numerical results confirm the theoretically predicted behavior and the effectiveness of the approach.

eess.IV↗

arXiv2021

On a Low-Frequency and Contrast Stabilized Full-Wave Volume Integral Equation Solver for Lossy Media

In this paper we present a new regularized electric flux volume integral equation (D-VIE) for modeling high-contrast conductive dielectric objects in a broad frequency range. This new formulation is particularly suitable for modeling biological tissues at low frequencies, as it is required by brain epileptogenic area imaging, but also at higher ones, as it is required by several applications including, but not limited to, transcranial magnetic and deep brain stimulation (TMS and DBS, respectively). When modeling inhomogeneous objects with high complex permittivities at low frequencies, the traditional D-VIE is ill-conditioned and suffers from numerical instabilities that result in slower convergence and in less accurate solutions. In this work we address these shortcomings by leveraging a new set of volume quasi-Helmholtz projectors. Their scaling by the material permittivity matrix allows for the re-balancing of the equation when applied to inhomogeneous scatterers and thereby makes the proposed method accurate and stable even for high complex permittivity objects until arbitrarily low frequencies. Numerical results, canonical and realistic, corroborate the theory and confirm the stability and the accuracy of this new method both in the quasi-static regime and at higher frequencies.

eess.IV↗

arXiv2020

On the modeling of brain fibers in the EEG forward problem via a new family of wire integral equations

Source localization based on electroencephalography (EEG) has become a widely used neuroimagining technique. However its precision has been shown to be very dependent on how accurately the brain, head and scalp can be electrically modeled within the so-called forward problem. The construction of this model is traditionally performed by leveraging Finite Element or Boundary Element Methods (FEM or BEM). Even though the latter is more computationally efficient thanks to the smaller interaction matrices it yields and near-linear solvers, it has traditionally been used on simpler models than the former. Indeed, while FEM models taking into account the different media anisotropies are widely available, BEM models have been limited to isotropic, piecewise homogeneous models. In this work we introduce a new BEM scheme taking into account the anisotropies of the white matter. The boundary nature of the formulation allows for an efficient discretization and modelling of the fibrous nature of the white matter as one-dimensional basis functions, limiting the computational impact of their modelling. We compare our scheme against widely used formulations and establish its correctness in both canonical and realistic cases.

physics.med-ph↗

arXiv2020

A New Preconditioner for the EFIE Based on Primal and Dual Graph Laplacian Spectral Filters

The Electric Field Integral Equation (EFIE) is notorious for its ill-conditioning both in frequency and h-refinement. Several techniques exist for fixing the equation conditioning problems based on hierarchical strategies, Calderon techniques, and related technologies. This work leverages on a new approach, based on the construction of tailored spectral filters for the EFIE components which allow the block renormalization of the EFIE spectrum resulting in a provably constant condition number for the equation. This is achieved without the need for a barycentric refinement and with low computational overhead compared with other schemes. In particular, only sparse matrices are required in addition to the EFIE original matrix. Numerical results will show the robustness of our scheme and its application to the solution of realistic problems.

math.NA↗

arXiv2020

A Quasi-Helmholtz Projector Stabilized Full Wave Solver Encompassing the Eddy Current Regime

Despite its several qualities, the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation for simulating scattering by dielectric media suffers from numerical instabilities and severe ill-conditioning at low frequencies. While this drawback has been the object of numerous solution attempts in the standard low-frequency breakdown regime for scattering problems, the eddy-current regime requires a specific treatment. In this contribution, we present an extension of the recently introduced quasi-Helmholtz projectors based preconditioning of the PMCHWT to obtain an equation stable at low frequencies and specifically in three regimes relevant for eddy currents analyses: (i) when the frequency decreases with a constant conductivity, (ii) when the conductivity increases at fixed frequency and (iii) when the frequency decreases while keeping the product of the frequency and the conductivity constant. Being based on quasi-Helmholtz projectors our new strategy does not further degrade the conditioning of the original equation and is compatible with existing fast solvers. The resulting full-wave formulation is capable of smoothly transitioning from simulations at high frequencies to the different low-frequency regimes which encompass, in particular, eddy currents applications. Numerical results demonstrate the validity of our approach in all the regimes with a special emphasis given to eddy currents.

physics.comp-ph↗

arXiv2020

Diffusion MRI Consistent Wire Models for Efficient Solutions of the Anisotropic Forward Problem in Electroencephalography

The surface Boundary Element Method (BEM) is one of the most commonly employed formulations to solve the forward problem in electroencephalography, but the applicability of its classical incarnations is lamentably limited to piece-wise homogeneous media. Several head tissues, however, are strongly anisotropic due to their complex underlying micro-structure. This implies that standard boundary integral formulations oversimplify the electrical properties of the head and produce unrealistic solutions, something that drastically limits the suitability and impact of BEM technologies to brain imaging. This contribution addresses this issue by observing that the brain anisotropy in the white matter is due to the presence of neuronal wire-like structures. We then extend the well known wire integral equations used for high frequency problems to the imperfectly conducting quasi-static case and we propose a new hybrid wire/surface/volume integral equation. When applied on multimodal magnetic resonance images combined with tractography, this new approach can flexibly and realistically handle the conductivity anisotropy in any head compartment providing high level of accuracy and efficiency. The beneficial properties of the new formulation together with its impact on brain imaging is demonstrated via numerical results on both canonical and realistic case scenarios.

eess.IV↗

arXiv2020

A Hybrid Volume-Surface-Wire Integral Equation for the Anisotropic Forward Problem in Electroencephalography

Solving the electroencephalography (EEG) forward problem is a fundamental step in a wide range of applications including biomedical imaging techniques based on inverse source localization. State-of-the-art electromagnetic solvers resort to a computationally expensive volumetric discretization of the full head to account for its complex and heterogeneous electric profile. The more efficient, popular in biomedical imaging circles, but unfortunately oversimplifying Boundary Element Method (BEM) relies instead on a piecewise-uniform approximation that severely curbs its application in high resolution EEGs. This contribution lifts the standard BEM contraints by treating the local anisotropies with adequate wire and thin volume integral equations that are tailored to specific structures of the fibrous white matter and the inhomogeneous skull. The proposed hybrid integral equation formulation thereby avoids the full volumetric discretization of the head medium and allows for a realistic and efficient BEM-like solution of the anisotropic EEG forward problem. The accuracy and flexibility of the proposed formulation is demonstrated through numerical experiments involving both canonical and realistic MRI-based head models.

physics.comp-ph↗

arXiv2019

Symmetric Galerkin boundary element method for computing the quantum states of the electron in a piecewise-uniform mesoscopic system

The quantum behavior of charge carriers in semiconductor structures is often described in terms of the effective mass Schrödinger equation, neglecting the rapid fluctuations of the wave function on the scale of the atomic lattice. For systems with piecewise-constant mass and potential energy, this amounts to solving a set of Helmholtz equations with wavenumbers dictated by the physical parameters of each homogeneous subregion. Making use of the Green function method, the system of differential equations can be expressed in boundary integral form to enable efficient numerical solution. In the present study, this strategy is applied in combination with a Galerkin technique to compute the energy spectrum and the wave functions of the electron in a mesoscopic structure composed of two regions. The proposed formulation differs from those presented before for the same scenario in that it implements a symmetric discretization of the four Helmholtz boundary integral operators, which leads to compact expressions and very accurate results.

physics.comp-ph↗

arXiv2019

Handling Anisotropic Conductivities in the EEG Forward Problem with a Symmetric Formulation

The electroencephalography (EEG) forward problem, the computation of the electric potential generated by a known electric current source configuration in the brain, is a key step of EEG source analysis. In this problem, it is often desired to model the anisotropic conductivity profiles of the skull and of the white matter. These profiles, however, cannot be handled by standard surface integral formulations and the use of volume finite elements is required. Leveraging on the representation theorem using an anisotropic fundamental solution, this paper proposes a modified symmetric formulation for solving the EEG forward problem by a surface integral equation which can take into account anisotropic conductivity profiles. A set of numerical results is presented to corroborate theoretical treatments and to show the impact of the proposed approach on both canonical and real case scenarios.

physics.med-ph↗

arXiv2019

A Calderon Regularized Symmetric Formulation for the Electroencephalography Forward Problem

The symmetric formulation of the electroencephalography (EEG) forward problem is a well-known and widespread equation thanks to the high level of accuracy that it delivers. However, this equation is first kind in nature and gives rise to ill-conditioned problems when the discretization density or the brain conductivity contrast increases, resulting in numerical instabilities and increasingly slow solutions. This work addresses and solves this problem by proposing a new regularized symmetric formulation. The new scheme is obtained by leveraging on Calderon identities which allow to introduce a dual symmetric equation that, combined with the standard one, results in a second kind operator which is both stable and well-conditioned under all the above mentioned conditions. The new formulation presented here can be easily integrated into existing EEG imaging packages since it can be obtained with the same computational technology required by the standard symmetric formulation. The performance of the new scheme is substantiated by both theoretical developments and numerical results which corroborate the theory and show the practical impact of the new technique.

physics.med-ph↗

arXiv2016

Conforming Discretizations of Boundary Element Solutions of the Electroencephalography Forward Problem

In this paper we present a new discretization strategy for the boundary element formulation of the Electroencephalography (EEG) forward problem. Boundary integral formulations, classically solved with the Boundary Element Method (BEM), are widely used in high resolution EEG imaging because of their recognized advantages in several real case scenarios. Unfortunately however, it is widely reported that the accuracy of standard BEM schemes is limited, especially when the current source density is dipolar and its location approaches one of the brain boundary surfaces. This is a particularly limiting problem given that during an high-resolution EEG imaging procedure, several EEG forward problem solutions are required for which the source currents are near or on top of a boundary surface. This work will first present an analysis of standardly discretized EEG forward problems, reporting on a theoretical issue of some of the formulations that have been used so far in the community. We report on the fact that several standardly used discretizations requires the expansion term to be a square integrable function. Instead, those techniques are not consistent when a more appropriate mapping is considered. Such a mapping allows the expansion function term to be a less regular function, thus sensibly reducing the need for mesh refinements and low-precisions handling strategies that are currently required. These mappings, however, require a different and conforming discretization which must be suitably adapted to them. To do this we adopt a mixed discretization based on dual boundary elements residing on a suitably defined dual mesh. Finally we show how the resulting EEG problems has favorable properties with respect to previously proposed schemes and we show their applicability to real case modeling scenarios obtained from MRI data.

physics.med-ph↗