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

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 Calder\'on preconditioned EFIEComment: 3 pages, 3 figures, to be published in ICEAA 202

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

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

International audienceThe 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

Mixed discretization formulations for the direct EEG problem

International audienceThis work presents a new discretization scheme for the integral equation based Electroencephalography direct problem. The scheme is based on mixed discretizations and presents level of accuracy that are higher than those obtained with currently available formulations. The discretization scheme is conforming with respect to the Sobolev space mappings of all operators involved. Numerical results show the effectiveness of the new approac

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

International audienceThis 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