In this work, a new pointwise source reconstruction method is proposed. From a single pair of boundary measurements, we want to completely characterize the unknown set of pointwise sources, namely, the number of sources and their locations and intensities. The idea is to rewrite the inverse source problem as an optimization problem, where a Kohn-Vogelius type functional is minimized with respect to a set of admissible pointwise sources. The resulting second-order reconstruction algorithm is non-iterative and thus very robust with respect to noisy data. Finally, in order to show the effectiveness of the devised reconstruction algorithm, some numerical experiments into two spatial dimensions are presented. Copyright Therefore, solving the inverse potential problem (3) in C ı . / means to find m ,˛ i and x i , which denote the number, intensities and locations of the unknown pointwise sources, respectively. In particular, we are going to address two distinct situations with respect to the domain D, which are Case 1: We suppose that D is a bounded domain in R 2 and that m coincides with @D, namely m D @D. Therefore, total boundary measurement .u , q / on m is available. See sketch in Figure 1(a). Case 2: We assume that D is the half-plane in R 2 and that m is a subset of @D, namely m¨@ D. Therefore, only partial boundary measurement .u , q / on m is available. See sketch in Figure 1(b).Because by assumption, the domain contains the support of the unknown source b , the inverse problem associated with each one of the aforementioned cases can be defined in instead of D. Let us denote the boundary of by @ . In the Case 1, we have D D and m D @ . While in the Case 2, m¨@ and its complement is denoted by D @ n m , where is a fictitious boundary used in the reformulation of the inverse problem in the next section. See sketch in Figure 1.
