@phdthesis{Schwartze2020,
  type      = {Master Thesis},
  author    = {Nicolai Schwartze},
  title     = {Computational intelligence methods for solving partial differential equations},
  journal    = {Computational Intelligence Methoden zum L{\"o}sen partieller Differentialgleichungen},
  doi       = {10.25924/opus-3693},
  pages     = {XI, 120},
  year      = {2020},
  abstract  = {This master thesis investigates a Computational Intelligence-based method for solving PDEs. The proposed strategy formulates the residual of a PDE as a fitness function. The solution is approximated by a finite sum of Gauss kernels. An appropriate optimisation technique, in this case JADE, is deployed that searches for the best fitting parameters for these kernels. This field is fairly young, a comprehensive literature research reveals several past papers that investigate similar techniques. To evaluate the performance of the solver, a comprehensive testbed is defined. It consists of 11 different Poisson equations. The solving time, the memory consumption and the approximation quality are compared to the state of the art open-source Finite Element solver NGSolve. The first experiment tests a serial JADE. The results are not as good as comparable work in the literature. Further, a strange behaviour is observed, where the fitness and the quality do not match. The second experiment implements a parallel JADE, which allows to make use of parallel hardware. This significantly speeds up the solving time. The third experiment implements a parallel JADE with adaptive kernels. It starts with one kernel and introduce more kernels along the solving process. A significant improvement is observed on one PDE, that is purposely built to be solvable. On all other testbed PDEs the quality-difference is not conclusive. The last experiment investigates the discrepancy between the fitness and the quality. Therefore, a new kernel is defined. This kernel inherits all features of the Gauss kernel and extends it with a sine function. As a result, the observed inconsistency between fitness and quality is mitigated. The thesis closes with a proposal for further investigations. The concepts here should be reconsidered by using better performing optimisation algorithms from the literature, like CMA-ES. Beyond that, an adaptive scheme for the collocation points could be tested. Finally, the fitness function should be further examined.},
  language  = {en}
}