  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.
