Transforming electric and magnetic field operators of arbitrary frequency into always well-conditioned and Hermitian positive definite operators.
Theoretical and numerical investigations, algorithmic aspects, and high frequency treatments:
We have obtained the first existing integral equation which, while improving the conditioning of the original equation, results in a Hermitian positive definite operator independent of the sign of the original problem.
Low frequency treatments - handling low-frequency breakdowns, topological multiple connectivity:
We had to face a substantial challenge that has been solved here for the first time: the magnetic kind operators, which are fundamental for both the low and the high frequency regime, cannot be stabilized at low frequency for non-simply connected geometries without using increasingly high numbers of integration points for decreasingly low frequencies. This had, up to today, a substantially negative impact on the possibility of obtaining always stable and fast invertible equations for arbitrary frequencies. In completing this task, we have solved this problem by obtaining the first scheme that disentangles the integration complexity cost from the stabilization.
Numerical discretizations and serial implementations:
Several standard and novel techniques have been applied to obtain the required discretizations. Among the new ones, we had to build a dual discretization to prepare the work on the scalar layers. We have validated it on head models in view of Workpackage 4 and, as a byproduct, we have also exploited the duality offered by the discretization to obtain increased precision while solving the associated inverse problems.
