Teaching - Applied Mathematics - Numerics and Control

Master thesis projects

Our research group is looking to supervise master student projects! Below are two ideas for possible research directions that are suitable as master thesis projects. If you are interested, do not hesitate to contact us. Do not be put off if you do not understand the details of the project descriptions below. Your willingness to learn new concepts and your passion are the most important factors.

Project 1: Optimisation algorithms

Many computational tasks, such as training neural networks (machine learning), require finding optimal solutions to high dimensional problems. The need for highly efficient and robust optimisation algorithms is ubiquitous in modern applied mathematics. It is, therefore, of interest to take a closer look at optimisation algorithms, acceleration techniques and tricks to make algorithms more robust. For this, it is necessary to understand why some of the existing optimisation algorithms work better than others. The use of classical mathematical notions for the analysis of state-of-the-art optimisation algorithms is just heading off!

We are looking for an interested student who is keen to implement and compare different optimisation algorithms in their master thesis and to discover whether and how underlying mathematical structure helps to improve existing algorithms. The student should have some programming skills (any language such as MATLAB, Python, C++, ...) and be interested in learning about optimisation algorithms and mathematical structures such as Hamiltonian systems and symplecticity.

Project 2: Backward error analysis

Ordinary differential equations are ubiquitous in mathematical models that describe, for instance, the motions of planets, molecules, or stock prices. As the equations are typically complicated, exact solutions are not immediately available but need to be approximated using numerical methods. Efficient and reliable numerical methods are, thus, of high importance. They provide approximate solutions at certain time-steps. There are two conflicting ideas to obtain good numerical methods:

The method adapts the time-steps during the computation. The step-size is chosen based on local error estimates or on the position of the motion in phase space. This can lead to highly accurate and efficient methods.
The method is designed to preserve mathematical structure of the original equation. This means that if exact solutions preserve certain integrals of motion, exhibit certain symmetries, lie on periodic orbits, so do the numerically computed solutions. This gives rise to robust numerical methods which show excellent long-term behaviour and produce solutions which share important properties of exact solutions and look qualitatively correct.

These two approaches are hard to combine: unfortunately, structure preserving numerical integrators lose their favourable properties when combined with adaptive time-stepping strategies.

Mathematical tools, such as backward error analysis, can not only be used to understand why some numerical methods work better than others, they can also be used to improve existing methods. We are looking for a student who is interested in trying out recently developed mathematical tools to improve step-size adaptive numerical methods while preserving underlying mathemtical structure. The student should have some programming skills (any language such as MATLAB, Python, C++, ...) and be interested in learning about backward error analysis and state-of-the-art numerical methods.