Rajendra Mitharwal

On the Hierarchical Preconditioning of the PMCHWT Integral Equation on Simply and Multiply Connected Geometries

We present a hierarchical basis preconditioning strategy for the Poggio–Miller–Chang–Harrington–Wu–Tsai (PMCHWT) integral equation considering both simply and multiply connected geometries. To this end, we first consider the direct application of hierarchical basis preconditioners, developed for the electric field integral equation (EFIE), to the PMCHWT. It is notably found that, whereas for the EFIE a diagonal preconditioner can be used for obtaining the hierarchical basis scaling factors, this strategy is catastrophic in the case of the PMCHWT since it leads to a severely ill-conditioned PMCHWT system in the case of multiply connected geometries. We then proceed to a theoretical analysis of the effect of hierarchical bases on the PMCHWT operator for which we obtain the correct scaling factors and a provably effective preconditioner for both low frequencies and mesh refinements. Numerical results will corroborate the theory and show the effectiveness of our approach.

A mixed discretized surface-volume integral equation for solving EEG forward problems with inhomogeneous and anisotropic head models

This work presents a new integral formulation to solve the EEG forward problem for potentially inhomogeneous and anisotropic head conductivity profiles. The formulation has been obtained from a surface/volume variational expression derived from Greens third identity and then solved in terms of both surface and volume unknowns. These unknowns are expanded with suitably chosen basis functions which systematically enforce transmission conditions. Finally, by leveraging on a mixed discretization, the equation is tested within the framework of a Petrov-Galerkins scheme. Numerical results show the high level of accuracy of the proposed method, which compares very favourably with those obtained with existing, finite element, schemes.

Handling the low-frequency breakdown of the PMCHWT integral equation with the quasi-Helmholtz projectors

This contribution presents a quasi-Helmholtz projectors based regularization of the low frequency breakdown of the PMCHWT integral equation. The PMCHWT equation in the low-frequency regime shows an ill-conditioned behavior inherited from the Electric Field Integral Operators it contains. The stabilization via quasi-Helmholtz projectors, differently from the use of standard Loop-Star/Tree decompositions, does not introduce an additional mesh-size-related ill-conditioning and it applies smoothly to both simply and non-simply connected geometries. The presentation of the main formulation will be complemented by numerical results demonstrating the effectiveness and accuracy of the proposed scheme.

On the Multiplicative Regularization of Graph Laplacians on Closed and Open Structures With Applications to Spectral Partitioning

A new regularization technique for graph Laplacians arising from triangular meshes of closed and open structures is presented. The new technique is based on the analysis of graph Laplacian spectrally equivalent operators in terms of Sobolev norms and on the appropriate selection of operators of opposite differential strength to achieve a multiplicative regularization. In addition, a new 3-D/2-D nested regularization strategy is presented to deal with open geometries. Numerical results show the advantages of the proposed regularization as well as its effectiveness when used in spectral partitioning applications.

A novel volume integral equation for solving the Electroencephalography forward problem

In this paper, a novel volume integral equation for solving the Electroencephalography forward problem is presented. Differently from other integral equation methods standardly used for the same purpose, the new formulation can handle inhomogeneous and fully anisotropic realistic head models. The new equation is obtained by a suitable use of Greens identities together with an appropriate handling of all boundary conditions for the EEG problem. The new equation is discretized with a consistent choice of volume and boundary elements. Numerical results shows validity and convergence of the approach, together with its applicability to real case models obtained from MRI data.

Two volume integral equations for the inhomogeneous and anisotropic forward problem in electroencephalography

Abstract This work presents two new volume integral equations for the Electroencephalography (EEG) forward problem which, differently from the standard integral approaches in the domain, can handle heterogeneities and anisotropies of the head/brain conductivity profiles. The new formulations translate to the quasi-static regime some volume integral equation strategies that have been successfully applied to high frequency electromagnetic scattering problems. This has been obtained by extending, to the volume case, the two classical surface integral formulations used in EEG imaging and by introducing an extra surface equation, in addition to the volume ones, to properly handle boundary conditions. Numerical results corroborate theoretical treatments, showing the competitiveness of our new schemes over existing techniques and qualifying them as a valid alternative to differential equation based methods.

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.

Regularized formulations for spectral graph partitioning

In this paper, we introduce a novel regularization technique for the spectral partitioning of a mesh that relies on a efficient preconditioning of the associated graph Laplacian. The regularization is obtained by leveraging on fractional order Sobolev norms obtained with integral operators and by linking the Laplacian and the operators with suitably chosen Gram matrices that connect the underlying discretization spaces. The numerical results support the developed theory when applied to some of the realistic examples arising in Computational Electro-magnetics applications.