The inverse source problem for Maxwell's equations

Abstract

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Inverse source problems for eddy current equations

“…On the other hand, He and Romanov [1] show that the location and the polarization of a current dipole in a conducting object can be uniquely determined by measuring at a fixed frequency the magnetic field and its normal derivative on the whole surface. The same result has been obtained by Ammari et al [2] from the knowledge of the tangential component of either the electric or the magnetic field on Γ. Albanese and Monk [3] have characterized which part of a volume source confined in Ω C can be uniquely identified from measurements of the tangential component of the electric field on Γ. Moreover, they also prove uniqueness of the inverse source problem if the source is supported on the surface of a-priori known subdomain contained in Ω C or if it is the sum a finite number of dipole sources.…”

“…In this section we investigate the uniqueness of the inverse source problem assuming that the unknown source J e is a function in (L 2 (Ω C )) 3 . First we will prove that without additional information, the source cannot be reconstructed from the knowledge of the tangential component of the electric field on Γ.…”

Electro–magneto-encephalography for a three-shell model: distributed current in arbitrary, spherical and ellipsoidal geometries

“…the case of a collection of dipoles, is analysed in Dassios & Fokas (preprint a,b) for spherical and ellipsoidal geometries, respectively. For other related important works, see El Badia & Ha-Duong (2000), Jerbi et al (2002), Nara & Ando (2003), Nolte & Dassios (2005), Albanese & Monk (2006), Peng et al (2006), Nara et al (2007) and Leblond et al (preprint). This paper is organized as follows: the equations needed for EEG and MEG in a three-shell model are derived in §2; this is done for the sake of completeness so that this paper is self-contained.…”

Mathematical Foundation of Electroencephalography

“…Albanese and Monk [28] illustrated that it is not possible to recreate a three-dimensional current based on EEG measurements. This result has been practically demonstrated in Ref.…”