A Genetic Algorithm for Pointwise Source Reconstruction by the Method of Fundamental Solutions

Abstract

Inverse source reconstruction problems offer great potential for\napplications of interest to engineering, such as the identification of\npolluting sources, and to medicine, such as electroencephalography, to\ncite at least two relevant examples. From a mathematical point of view,\nthe identification of a concentrated source (intensity and location)\ncorresponds to the identification of the centroid (location) and size\n(intensity) of a distributed source. On the other hand, from a numerical\npoint of view, it is observed that the use of domain discretization\nmethods is intrinsically associated with the introduction of numerical\nnoise in reconstruction algorithms, which is strongly inadvisable since\ninverse problems are reckoned to be ill-posed. The objective of this\nwork is to explore, in the context of a Poisson problem and taking into\naccount a numerical point of view, a new reconstruction algorithm based\non the method of fundamental solutions, where a source point adequately\nrepresents the pointwise source within the domain. The inverse problem\nis reformulated as an optimization problem solved through a genetic\nalgorithm. Finally, numerical examples are performed to analyze the\naccuracy of the proposed algorithm for two and three dimensions.