A Level-Set Framework for Shape Optimisation

For my PhD in applied mathematics, I considered the application of the level-set method to shape optimisation. This is not an entirely new connection, and a lot of literature exists that uses level sets in applications. There exists also a lot of theory around level-set equations and time evolution of level-set geometries, but mostly in a quite abstract setting.

My main contribution, therefore, is the formulation of a level-set framework that is particularly tailored towards shape-sensitivity analysis and other questions necessary for shape optimisation. This framework is based on a Hopf-Lax formula for the time evolution of level-set shapes, similar in spirit to the classical Fast Marching Method. (But note that my focus is not mainly on numerical algorithms but also on theoretical conclusions from this formulation.)

A lot of the program code developed during my research has been published as free software in the level-set package for GNU Octave.

Thesis: The actual thesis with some error corrections applied. The official version is available at the university library.
Poster about the Thesis: Presentation of my thesis and PhD research on a poster. This is, of course, very incomplete and compact, but may give a brief overview.
The Level-Set Package for GNU Octave: Brief overview of my level-set package as given during OctConf 2015.
Presentation Slides: Some slides I prepared for my presentation of the thesis during the defense.
Image-Segmentation Movie: A particular optimisation descent run for the image-segmentation problem discussed in Chapter 6 of the thesis. It shows the evolution of the current shape throughout multiple iterations, with the current steepest-descent speed field indicated by the heatmap in the background.
Descent with Elliptic PDE: The descent for a PDE-constrained problem according to Chapter 8.