Masterthesis

Computational Intelligence Methods for Solving Partial Differential Equations

For earning a Master of Science in Mechatronics from Vorarlberg University of Applied Science

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.

NGSolve GUI Prerequisites

for a more detailed description look at
https://ngsolve.org/docu/latest/install/usejupyter.html

python version 3.7 with virtual environment

  > conda create -n ngsolve anaconda python=3.7

change virtual envorinment to ngsolve:
```
  > conda activate ngsolve
```
ipykernel version 4.10 in newly created virtual environment
```
  > conda install ipykernel==4.10.0
```

install ngsolve as recommended by https://ngsolve.org/downloads

  > conda config --add channels conda-forge 
  > conda config --add channels ngsolve 
  > conda install ngsolve

jupyter notebook extensions for ngsolve

  > jupyter nbextension install --user --py ngsolve

run ngsolve gui-examples under ./code/examples/ngsolve_gui.ipynb