Advances in the greedy optimization algorithm for nodes and collocation points using the method of fundamental solutions

Abstract The Method of Fundamental Solutions and Boundary Element Method commonly involve development of approximation functions featuring linear combinations of basis functions dened over the problem domain and boundary. These basis functions are selected with respect to the governing partial differential equation (PDE) being approximated, and are customarily fundamental solutions of the PDE operator. Points where singularities occur are known as source points and are traditionally uniformly distributed outside the problem domain. In this paper, basis functions with singularities at source points are examined with interest in optimizing the placement of the corresponding nodes to reduce computational error while also reducing the number of nodes involved. A motivation for such an optimization is the reduction in matrix solver requirements in solving large dense matrix systems. An algorithm is explored that has been shown to provide such an optimization capability with a 3D case study in steady state heat transport. It is shown that by including the nodal position coordinates as additional variables to be optimized, the resulting approximation function is improved in computational accuracy by increasing the number of degrees of freedom to be optimized.