Electric scalar potential estimations for non-invasive brain activity detection through multinode Shepard method

Electric scalar potential estimation is a key step for non-invasive investigations of brain activity with high time resolutions. The neural sources can be reconstructed by solving a typical inverse problem based on a forward problem formulated as a set of boundary value problems coupled by interface conditions. In this paper, we propose a Shepard multinode method to numerically estimate electric scalar potentials via collocation. The method is based on a special kind of inverse distance weighting partition of unity method to increase polynomial precision, approximation order, and accuracy of the classical Shepard approximation. The barycentric form, through the use of cardinal basis function, is adopted to generate Shepard interpolants with high polynomial reproduction order. Comparisons with the analytic solution and the standard Shepard method are provided to assess the proposed approach.