A New Refinement-Free Preconditioner for the Symmetric Formulation in Electroencephalography

TL;DR

This paper introduces a novel, refinement-free preconditioner for the symmetric formulation in EEG forward problems, resulting in well-conditioned systems that improve computational efficiency without mesh refinement.

Contribution

The work proposes a spectral analysis-based preconditioning strategy that avoids barycentric mesh refinement, producing a symmetric, positive-definite system for EEG modeling.

Findings

Achieves well-conditioned, symmetric system matrices

Validates effectiveness on realistic head models

Enables efficient iterative solutions

Abstract

Widely employed for the accurate solution of the electroencephalography forward problem, the symmetric formulation gives rise to a first kind, ill-conditioned operator ill-suited for complex modelling scenarios. This work presents a novel preconditioning strategy based on an accurate spectral analysis of the operators involved which, differently from other Calder\'on-based approaches, does not necessitate the barycentric refinement of the primal mesh (i.e., no dual matrix is required). The discretization of the new formulation gives rise to a well-conditioned, symmetric, positive-definite system matrix, which can be efficiently solved via fast iterative techniques. Numerical results for both canonical and realistic head models validate the effectiveness of the proposed formulation.