Direct algorithm for multipolar sources reconstruction

Abstract

This paper proposes an identification algorithm for identifying multipolar sources F in the elliptic equation Δu + μu = F from boundary measurements. The reconstruction question of this class of sources appears naturally in Helmholtz equation (μ > 0) and in biomedical phenomena particularly in EEG/MEG problems (μ = 0) and bioluminescence tomography (BLT) applications (μ < 0). Previous works have treated the inverse multipolar source problems, only for equations with μ = 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 ∈ 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.

Some remarks on the small electromagnetic inhomogeneities reconstruction problem

“…Let Ω ⊂ R 3 be an open bounded domain with a Lipschitz boundary Γ. Suppose that Ω contains m small inhomogeneities D j of the form (1)…”

“…The main novelty presented in this paper lies in the proposed identification method and the established stability result, using only a single Cauchy data at a fixed frequency. This method is employed priorly in the problem of sources reconstruction [1] and is based on the use of a reciprocity gap functional and specific test functions that lead to a number of algebraic relationships. Specifically, in here, the related developed relationships are utilized in order to recover the parameters of the domain inhomogeneities.…”

“…with S j , and B j respectively are the center, the diameter and the support of the inhomogeneities D j defined in (1) and ψ is a function that is evaluated using a single layer potential operator and another function denoted ϕ . Actually, for all X ∈ Γ and each multi-index , the couple (ϕ , ψ ) is the unique solution to…”

“…The idea behind our identification method, as developed in [1], consists in choosing particular test functions v in H κ0 , as v a n (X) = (x + iy) n e iκ0z , X = (x, y, z) ∈ Ω.…”

“…Thus, the number, the 2−dimensional projections and other characteristics of the homogeneities are reconstructed, modulo 6 , based on the resolution of the algebraic relationships (13), whose identification algorithm is obtained in [1] and is recalled briefly in Section 4 for the convenience of the reader. 4.…”