Determination of locations of point-like masses in an inverse source problem of the Poisson equation
Many inverse problems are formulated by inverse problems of partial differential equations. In this article, an inverse source problem of the Poisson equation is discussed with a boundary element approach and discrete Fourier transform. The case that several point-like masses are placed in a disk domain is studied. The author considers the problem to determine the mass positions from the potential and flux on the boundary. In Section 2 an expression of the logarithmic potential using boundary integrals is derived. A Fourier expansion of the logarithmic potentials of point-like masses is discussed in Section 3, and an algebraic relation between Fourier coefficients and mass positions is also given. Section 4 shows error estimations for the discrete Fourier transform of the logarithmic potential. In Section 5 several procedures to solve the problem are proposed, and to estimate errors of numerical solutions. Section 6 shows some numerical experiments and illustrates the effectiveness of the algorithms.
scientific article; zbMATH DE number 739556
scientific article