Comparing iterative solvers for linear systems associated with the finite difference discretisation of the forward problem in electro-encephalographic source analysis

Model-based reconstruction of electrical brain activity from electro-encephalographic measurements is of growing importance in neurology and neurosurgery. Algorithms for this task involve the solution of a 3D Poisson problem on a realistic head geometry obtained from medical imaging. In the model, several compartments with different conductivities have to be distinguished, leading to a problem with jumping coefficients. Furthermore, the Poisson problem needs to be solved repeatedly for different source contributions. Thus efficient solvers for this subtask are required. Experience with different iterative solvers is reported, i.e. successive over-relaxation, (preconditioned) conjugate gradients and algebraic multigrid, for a discretisation based on cell-centred finite differences. It was found that: first, the multigrid-based solver performed the task 1.8-3.5 times faster, depending on the platform, than the second-best contender; secondly, there was no need to introduce a reference potential that forced a unique solution; and, thirdly, neither the grid- nor matrix-based implementation of the solvers consistently gave a smaller run time.