Application of a simulated annealing algorithm in the optimal placement of the source points in the method of the fundamental solutions

The placement of source points constitutes a key issue for the method of the fundamental solutions. In particular, for problems with singularities of any kind the determination of the optimal placement of source points becomes relevant, as no linear combination of arbitrarily located source points can guarantee a reasonable approximation to the solution. This paper investigates the use of a "Simulated Annealing" algorithm in the optimal placement of source points in singular problems. The algorithm is essentially an iterative random search with adaptive moves along the coordinate directions. It permits uphill moves under the control of a probabilistic criterion, thus tending to avoid the first local minima encountered. The proposed methodology is employed with a variety of test problems. Results are compared to those of an analytical optimisation routine and their relatively merits and disadvantages discussed. Simulated annealing is shown to be an attractive option for the optimisation of singular problems, with a high rate of success, and able to solve problems for which analytical optimisation routines fail.