We present a Calder\'on preconditioning scheme for the symmetric formulation of the forward electroencephalographic (EEG) problem that cures both the dense discretization and the high-contrast breakdown. Unlike existing Calder\'on schemes presented for the EEG problem, it is refinement-free, that is, the electrostatic integral operators are not discretized with basis functions defined on the barycentrically-refined dual mesh. In fact, in the preconditioner, we reuse the original system matrix thus reducing computational burden. Moreover, the proposed formulation gives rise to a symmetric, positive-definite system of linear equations, which allows the application of the conjugate gradient method, an iterative method that exhibits a smaller computational cost compared to other Krylov subspace methods applicable to non-symmetric problems. Numerical results corroborate the theoretical analysis and attest of the efficacy of the proposed preconditioning technique on both canonical and realistic scenarios.
@article{arxiv.2211.01967,
title = {On a Calder\'on preconditioner for the symmetric formulation of the electroencephalography forward problem without barycentric refinements},
author = {Viviana Giunzioni and John E. Ortiz G. and Adrien Merlini and Simon B. Adrian and Francesco P. Andriulli},
journal= {arXiv preprint arXiv:2211.01967},
year = {2023}
}