Research and Science

During my education and studies, I already worked more or less intensively on different topics, in particular mathematical image processing and related optimisation topics, also for my PhD about shape optimisation. Besides that, I'm also interested in economics and social sciences (see my Master's thesis) and several other topics:

Computer Science and algorithms
Cryptography and Steganography
Blockchain technology and P2P networks
Artificial Intelligence and neural networks
Particle Physics and related topics of modern physics

For some of the works and publications below, I wrote computer code and have additional data. I'm usually willing to release all of that under a free license, please contact me if you are interested.

Selected Papers

A Hopf-Lax Formula for the Time Evolution of the Level-Set Equation and a New Approach to Shape Sensitivity Analysis: This paper discusses how the viscosity solution of the classical level-set equation can be found by solving an Eikonal equation. This yields a theoretical justification for the application of the Fast-Marching Method. Furthermore, my representation of the solution also allows to draw further theoretical conclusions, including the formation of a new shape-sensitivity calculus. First published in Interfaces and Free Boundaries 18(3), published by the European Mathematical Society.
Self-Consistent Gradient Flow for Shape Optimisation: A new idea for shape optimisation on the example of image segmentation, which bypasses the inefficient convergence of gradient descent and is at the same time not dependent on second-order derivatives. It is based on a construction of the gradient flow that can be efficiently evaluated in practice. Open access publication thanks to FWF.
Geometric Constraints in Descent Methods for Shape Optimisation: In this paper, I discuss different projection methods for shape optimisation with geometric constraints. The original publication is available at www.esaim-m2an.org, copyright by EDP Sciences and SMAI.
Measure-Theoretic Properties of Level Sets of Distance Functions: Geometric analysis of the surface measure of evolving level sets for the case of a constant normal speed. The final publication is available at link.springer.com.
Game Channels for Trustless Off-Chain Interactions in Decentralized Virtual Worlds: This paper describes a new protocol that can be applied to blockchain-based game worlds (such as Huntercoin) to scale them in theory to infinite size and enable near real-time interactions. Since the original publication, I've been able to improve the design further.
Difficulty Control for Blockchain-Based Consensus Systems: A paper with statistical analysis of Bitcoin's difficulty control and possible (academic) improvements to it. The final publication is available at link.springer.com.

Theses

A Level-Set Framework for Shape Optimisation: My PhD thesis in applied mathematics that treats the use of level-set methods in the context of shape optimisation. While such a use is not new in applications, my main contribution is a new theoretical framework for shape-sensitivity analysis and other important questions.
Political Power and Socio-Economic Inequality: My master's thesis in mathematics, where a model for uneven power distribution and social inequality shows a first-order phase transition between equal and inequal societies. See the page for more information.
Stochastic Variational Approaches to Non-Hermitian Quantum-Mechanical Problems: My master's thesis in theoretical physics. I consider, how quantum-mechanical resonances can be treated using the complex scaling method and other related techniques. For solving the resulting non-Hermitian eigenvalue problems, I propose a generalisation of the stochastic variational method and Rayleigh-Ritz principle to complex eigenvalues. The thesis' content was presented by my supervisor Willibald Plessas at IC-MSquare 2016, with a short overview paper published in JPCS.
A Measure Theoretic Approach to Image Segmentation Framed in Terms of Intensities: This is my bachelor's thesis in mathematics. I developed a method for image segmentation (on gray-level images) that is based on a measure-theoretic representation of the image and results in the K means algorithm.
Jules Verne's Journey Through Interplanetary Space: My bachelor's thesis in physics. I considered several aspects of Jules Verne's Off on a Comet from the point of view of (today's) science, modelled and simulated them and interpeted how much Verne was scientifically correct. The book contains a lot of numerical figures. Note that during the Modellierungswoche mit Mathematik 2012 I lead a group where we worked on a similar topic, investigating partially the same and partially different aspects of Jules Verne's books.
Allgemeine Algebra: My Fachbereichsarbeit in mathematics, dealing with defining numbers and operations on them. In German.

Various Other Publications

Tip Optimization: This is the report about a project done during a course in mathematical modelling together with my co-student Doris Koinegg about optimal strategy for assigning customers of a restaurant to tables. We did theoretic and numerical analysis based on stochastics, and got some nice results.
Calling the GPU from GNU Octave: This is a seminar report, where I use a basic image denoising problem as example to demonstrate (and test) how to do calculations on a GPU (specifically based on Nvidia's CUDA system) from within GNU Octave via compiling the CUDA code into an Oct-File (special kind of shared library which allows definition of Octave functions in native C++ code).
Some Mathematical Aspects of Fairness: A seminar paper dealing with some approaches to fairness. In particular, I treat inequity aversion (Fehr & Schmidt 1999) in game theory, matching problems and the Gale & Shapley algorithm, and voting systems and in particular Arrow's impossibility theorem as well as some ideas to escape it.
Automatic Differentiation with ADOL-C and Complex Numbers: My seminar report describing how automatic differentiation with ADOL-C can be used also for complex numbers, in the particular case for my work on a stochastic variational method for problems of non-Hermitian quantum mechanics. (Which became my master's thesis and is listed above.)