Inverse source problems for eddy current equations

Abstract

Abstract. We study the inverse source problem for the eddy current approximation of Maxwell equations. As for the full system of Maxwell equations, we show that a volume current source cannot be uniquely identified by the knowledge of the tangential components of the electromagnetic fields on the boundary, and we characterize the space of non-radiating sources. On the other hand, we prove that the inverse source problem has a unique solution if the source is supported on the boundary of a subdomain or if it is the sum of a finite number of dipoles. We address the applicability of this result for the localization of brain activity from electroencephalography and magnetoencephalography measurements.

Optimal control of an eddy current problem with a dipole source

“…Despite the adopted simplifications, several mathematical difficulties are here present: the most important, as mentioned, is that our state equation is an eddy current system with a Dirac distribution as source. We propose an approach that seems new in this context; the resolution of the problem is split into three steps, the first one being the determination of a fundamental solution to deal with the singularity at x 0 (this idea has been already used to tackling some inverse problems; see for instance Wolters et al [24] and Alonso Rodríguez et al [17]). After that, the specific structure of the eddy current problem leads to a state variable that is composed by two terms, a vector one and (the gradient of) a scalar one.…”

Existence and Regularity for Eddy Current System with Nonsmooth Conductivity

“…For a very general survey on eddy currents with discontinuous conductivity and related numerics, with applications to advanced medical diagnostics, the interested reader is referred instead to the nice treatise [3], where inverse problems are also considered. We refer to [4] for issues related to the source identification from boundary EM measurement.…”

Some remarks on a class of inverse problems related to the parabolic approximation to the Maxwell equations: a controllability approach

“…The inverse source problem for the time‐harmonic Maxwell equations is considered in , where it is shown that the problem of finding a volume current density from surface measurements does not have a unique solution (whereas with additional a priori information, uniqueness can be established). The inverse source problem for the eddy current approximation of the time‐harmonic Maxwell equations is studied in : non‐uniqueness and uniqueness results are furnished, and the results are applied for the localisation of brain activity from electroencephalography and magnetoencephalography measurements. In , it is established that the eddy current model approximates the full Maxwell equations up to the second order with respect to frequency if and only if an additional condition is imposed on the current source; otherwise, it is a first‐order approximation.…”