リーマン球面への射影によるポアソン方程式のソース項同定法

概要

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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.
日本応用数理学会の論文
2003-09-25