On the preconditioning of the symmetric formulation for the EEG forward problem by leveraging on calderon formulas

Among Boundary Element Methods (BEMs) for solving the Electroencephalography (EEG) forward problem, the symmetric formulation is one of the most accurate and widely adopted ones. However, differently from less precise standard BEM formulations, such as the double layer potential and its adjoint, the symmetric formulation is first kind in nature. This results in practice in a condition number of the associated linear system that grows when the average mesh size decreases (as it happens in the case of highly detailed models). This work solves this drawback by proposing a novel preconditioning strategy that leverages on Calderon formulas. The preconditioned equation can be easily integrated into an existing EEG symmetric implementations, it shows conditioning properties that are (theoretically provably) independent of the discretization, and in practice orders of magnitude better than the unpreconditioned symmetric formulation.