Under quasi-static assumptions, Maxwell’s equations governing the spatial behaviour of the electromagnetic fields lead to partial differential equations (PDE) of elliptic type in domains of R³.

We will mainly consider the case of an electrical potential solution to such a PDE, which depends on the electrical conductivity and of the current density of the medium. In electroencephalography (EEG), an important application in medical engineering, a spherical model of the head is classically taken, with piecewise constant conductivity in spherical layers representing the scalp, the skull and the brain, and pointwise dipolar current sources located within the brain. It appears that the potential is then solution to Laplace equation in the outermost layers and to Laplace-Poisson equation in the innermost one (where the sources are situated). The inverse EEG source problem is the following, assuming the conductivity val- ues to be known: being given a set of pointwise values of the electrical potential (measured by electrodes) on the scalp, and the current flux, find the quantity, the locations, and the moments of the pointwise dipolar sources in the brain. We will see how to solve this issue by a first data transmission step from the scalp to the cortex (cortical mapping), followed with a singularities localisation step, performed using best approximation techniques on planar slices [1, 2]. Numerical examples will be presented, from the software FindSources3D [3]. Maxwell’s and Newton’s equation are to the effect that under suitable assumptions, many other physical quantities, like magnetic or gravitational potentials, possess such a PDE model. The particular examples of magnetic plasma confinment in tokamaks and of some geodesy problems will be briefly discussed.

References:

[1] L. Baratchart, J. Leblond, J.-P. Marmorat, Inverse source problem in a 3D ball from best meromorphic approximation on 2D slices, Elec. Trans. Numerical Anal- ysis (ETNA), 25, 41–53, 2006.

[2] M. Clerc, J. Leblond, J.-P. Marmorat, T. Papadopoulo, Source localization in EEG using rational approximation on plane sections, Inverse Problems, 28, 055018, 2012.

[3] FindSources3D, http://www-sop.inria.fr/apics/FindSources3D/