  This paper introduces a novel numerical method for the inverse problem of
electroencephalography(EEG). We pose the inverse EEG problem as an optimal
control (OC) problem for Poisson's equation. The optimality conditions lead to
a variational system of differential equations. It is discretized directly in
finite-element spaces leading to a system of linear equations with a sparse
Karush-Kuhn-Tucker matrix. The method uses finite-element discretization and
thus can handle MRI-based meshes of almost arbitrary complexity. It extends the
well-known mixed quasi-reversibility method (mQRM) in that pointwise noisy data
explicitly appear in the formulation making unnecessary tedious interpolation
of the noisy data from the electrodes to the scalp surface. The resulting
algebraic problem differs considerably from that obtained in the mixed
quasi-reversibility, but only slightly larger. An interesting feature of the
algorithm is that it does not require the formation of the lead-field matrix.
Our tests, both with spherical and MRI-based meshes, demonstrates that the
method accurately reconstructs cortical activity.
