arXiv subjects

Francesco P. Andriulli

Publications and source records attributed to Francesco P. Andriulli.

44 records · Page 3Linked to original sources

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↗

arXiv2019

Magnetic and Combined Field Integral Equations Based on the Quasi-Helmholtz Projectors

Boundary integral equation methods for analyzing electromagnetic scattering phenomena typically suffer from several of the following problems: (i) ill-conditioning when the frequency is low; (ii) ill-conditioning when the discretization density is high; (iii) ill-conditioning when the structure contains global loops (which are computationally expensive to detect); (iv) incorrect solution at low frequencies due to current cancellations; (v) presence of spurious resonances. In this paper, quasi-Helmholtz projectors are leveraged to obtain a magnetic field integral equation (MFIE) formulation that is immune to drawbacks (i)-(iv). Moreover, when this new MFIE is combined with a regularized electric field integral equation, a new quasi-Helmholtz projector combined field integral equation is obtained that also is immune to (v). Numerical results corroborate the theory and show the practical impact of the newly proposed formulations.

physics.comp-ph↗

arXiv2019

Vortex Waves and Channel Capacity: Hopes and Reality

Several recent contributions have envisioned the possibility of increasing currently exploitable maximum channel capacity of a free space link, both at optical and radio frequencies, by using vortex waves, i.e. carrying Orbital Angular Momentum (OAM). Our objective is to disprove these claims by showing that they are in contradiction with very fundamental properties of Maxwellian fields. We demonstrate that the Degrees of Freedom (DoF) of the field cannot be increased by the helical phase structure of electromagnetic vortex waves beyond what can be done without invoking this property. We also show that the often-advocated over-quadratic power decay of OAM beams with distance does not play any fundamental role in the determination of the channel DoF.

physics.optics↗

arXiv2018

On a Refinement-Free Calderón Multiplicative Preconditioner for the Electric Field Integral Equation

We present a Calderón preconditioner for the electric field integral equation (EFIE), which does not require a barycentric refinement of the mesh and which yields a Hermitian, positive definite (HPD) system matrix allowing for the usage of the conjugate gradient (CG) solver. The resulting discrete equation system is immune to the low-frequency and the dense-discretization breakdown and, in contrast to existing Calderón preconditioners, no second discretization of the EFIE operator with Buffa-Christiansen (BC) functions is necessary. This preconditioner is obtained by leveraging on spectral equivalences between (scalar) integral operators, namely the single layer and the hypersingular operator known from electrostatics, on the one hand, and the Laplace-Beltrami operator on the other hand. Since our approach incorporates Helmholtz projectors, there is no search for global loops necessary and thus our method remains stable on multiply connected geometries. The numerical results demonstrate the effectiveness of this approach for both canonical and realistic (multi-scale) problems.

math.NA↗

arXiv2016

A Low-Frequency and Refinement Stable Impedance Boundary Condition EFIE

In this contribution, a discretisation of the IBC EFIE is introduced that (i) yields the correct solution at arbitrarily small frequencies, (ii) requires for its solution a number of matrix vector products bounded as the frequency tends to zero and as the mesh density increases. The low frequency stabilisation is based on a projector-based discrete Helmholtz splitting, rescaling, and recombination that depends on the low frequency behaviour of both the EFIE operator and the surface impedance condition. The dense mesh stabilisation is a modifcation of the Perfect Electric Conductor operator preconditioning approach taking into account the effect on the singular value spectrum of the IBC term.

physics.comp-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↗

arXiv2015

A Regularized Boundary Element Formulation for Contactless SAR Evaluations within Homogeneous and Inhomogeneous Head Phantoms

This work presents a Boundary Element Method (BEM) formulation for contactless electromagnetic field assessments. The new scheme is based on a regularized BEM approach that requires the use of electric measurements only. The regularization is obtained by leveraging on an extension of Calderon techniques to rectangular systems leading to well-conditioned problems independent of the discretization density. This enables the use of highly discretized Huygens surfaces that can be consequently placed very near to the radiating source. In addition, the new regularized scheme is hybridized with both surfacic homogeneous and volumetric inhomogeneous forward BEM solvers accelerated with fast matrix-vector multiplication schemes. This allows for rapid and effective dosimetric assessments and permits the use of inhomogeneous and realistic head phantoms. Numerical results corroborate the theory and confirms the practical effectiveness of all newly proposed formulations.

physics.med-ph↗

arXiv2015

On the Hierarchical Preconditioning of the Combined Field Integral Equation

This paper analyzes how hierarchical bases preconditioners constructed for the Electric Field Integral Equation (EFIE) can be effectively applied to the Combined Field Integral Equation (CFIE). For the case where no hierarchical solenoidal basis is available (e.g., on unstructured meshes), a new scheme is proposed: the CFIE is implicitly preconditioned on the solenoidal Helmholtz subspace by using a Helmholtz projector, while a hierarchical non-solenoidal basis is used for the non-solenoidal Helmholtz subspace. This results in a well-conditioned system. Numerical results corroborate the presented theory.

math.NA↗