A projective method for an inverse source problem of the Poisson equation

This paper proposes a method for reconstructing the positions, strengths, and number of point sources in a three-dimensional (3D) Poisson field from boundary measurements. Algebraic relations are obtained, based on multipole moments determined by the sources and data on the boundary of a domain. To solve for the source parameters with efficient use of data, we select the necessary number of equations from them in the following two ways: (1) the use of those starting from lower-degree multipole moments; and (2) the use of combined ones involving infinitely higher-degree multipole moments. We show that both methods are based on the projection of 3D sources onto a two-dimensional space: the xy-plane for the first one and the Riemann sphere which is set to contain the domain for the second one. We also show that they share the same fundamental equations which can be solved by a procedure proposed by El-Badia and Ha-Duong (2000 Inverse Problems 16 651-63). Numerical simulations show that projection onto the xy-plane is more appropriate for sources scattered in the middle of the domain, whereas projection onto the Riemann sphere is more appropriate for sources concentrated close to the boundary of the domain. We also give an appropriate method of measurement for the Riemann sphere projection.