A boundary element implementation of the reciprocity gap method for the numerical reconstruction of multiple inclusions

We present a numerical implementation of the reciprocity gap functional method for the reconstruction of multiple disjoint inclusions with different constant conductivities. The approach is formulated through a boundary element discretization of the associated transmission problem, leading to a block integral system defined on the union of the interior boundaries. For each sampling point, the method requires the solution of a regularized linear system whose behavior is analyzed with respect to discretization and regularization parameters. We extend the theoretical sampling characterization to the case of multiple connected components and discuss the structural modifications induced by the increased dimension of the interior nullspace. The proposed algorithm is validated through several numerical experiments, including configurations with smooth and non-smooth geometries, and tests under additive Gaussian noise. A quantitative comparison with the Linear Sampling Method is provided, highlighting similarities and differences in conditioning and stability. The results confirm that the reciprocity gap approach can be effectively implemented within a boundary element framework and provides reliable localization of multiple inclusions in both noise-free and perturbed settings.