This paper proposes a method for reconstructing the positions, strengths, and number of point sources in a three-dimensional (3-D) 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 manners : 1) the use of ones starting from lower degree multipole moments, or 2) the use of combined ones involving infinitely higher degree multipole moments. We show that both methods are based on the projection of 3-D sources onto a 2-D space : the xy-plane for the first one and the Riemann sphere for the second one. We also show that they are formulated as a unified moment problem in which the unknown parameters can be explicitly reconstructed. 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.

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An eigenvalue of an operator takes an important role to understand an nonlinear phenomenon in science and engineering. Especially, it often becomes a key value when we consider a behavior of dynamical systems. This article concerns with such an eigenvalue problem for matrices and differential operators. We first deal with a theoretical framework for it by introducing a concept of spectrum, and consider a matrix eigenvalue problem. Then we concern with an infinite dimensional eigenvalue problem, that is to obtain eigenvalues and eigenvectors for an operator in infinite dimensional domain of definition. We explain it from classical procedure to our recent results making use of the numerical verification methods. We also deal with some applications of the verified eigenvalues and eigenvectors to algebla, another numerical verification methods for nonlinear problems and stability analysis of bifurcation phenomenon in hydrodinamics.

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