On a Calder\'on preconditioner for the symmetric formulation of the electroencephalography forward problem without barycentric refinements

TL;DR

This paper introduces a refinement-free Calderón preconditioning scheme for the symmetric EEG forward problem that improves numerical stability and computational efficiency without barycentric mesh refinements.

Contribution

It proposes a novel, refinement-free Calderón preconditioner for the symmetric EEG forward problem that reduces computational costs and enhances numerical stability.

Findings

Preconditioner effectively handles dense discretization and high-contrast issues.

System becomes symmetric and positive-definite, enabling efficient conjugate gradient solutions.

Numerical tests confirm improved performance on realistic scenarios.

Abstract

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.