Direct algorithm for multipolar sources reconstruction

This paper proposes an identification algorithm for identifying multipolar sources F   in the elliptic equation Δu+μu=FΔu+μu=F from boundary measurements. The reconstruction question of this class of sources appears naturally in Helmholtz equation (μ>0)(μ>0) and in biomedical phenomena particularly in EEG/MEG problems (μ=0)(μ=0) and bioluminescence tomography (BLT) applications (μ<0)(μ<0). Previous works have treated the inverse multipolar source problems, only for equations with μ=0μ=0, using algebraic approaches depending on the complex calculation of determinants. Knowing that the novelty in our method concerns several points, the principal one is its simplicity where its proof is not founded on the determinants calculation and its ease in implementation. Moreover, this work involves the general form of equations considering μ∈Rμ∈R and at the same time considers a more general type of sources than former related works including sources having small compact support within a finite number of subdomains. Finally, some numerical results are shown to prove the robustness of our identification algorithm.