Master and Bachelor theses

This page presents a selection of possible topics for Bachelor's and Master's theses. Most of them deal with current research topics. If you don't like any of the suggested topics (or perhaps already have one of your own) and still want to write a thesis with me, just get in touch. Contact me

The list is roughly divided into Bachelor's and Master's theses, but this division is not mandatory.

Many topics require programming in one form or another. Python or C++ is usually used for this. DUNE is also frequently used.

I recommend reading the Handbook of Writing for the Mathematical Sciences. The Faculty of Mathematics provides LaTeX templates for theses.

Bachelor theses

Integration over spherical triangles

In the numerics lecture you usually only learn the numerical approximation of integrals on an interval, but of course other areas of integration are also covered. For example, how could you construct quadrature rules for functions defined on spherical triangles?

Calculating the matrix exponential function

The article Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later contains tons of exciting math. Who would have thought it would be so difficult to calculate the matrix exponential function? Here you can choose a few of the 19 dubious methods and examine and compare them in more detail.

Diffeological spaces

Diffeological spaces are a generalization of the manifolds known from differential geometry. They can be used to elegantly describe various mathematical objects that are "not smooth enough" for normal differential geometry. The aim of this work is to understand the concept of diffeological space and to apply it to various function spaces.

Literature: Iglesias-Zemmour: "Diffeology", American Mathematical Society

The topic can easily be extended to a Master's thesis.

And there is more...

  • Automatic differentiation (literature: Griewank, Walther, "Evaluating Derivatives")
  • Barycentric rational interpolation (literature e.g: Floater, Hormann,"Barycentric rational interpolation with no poles and high rates of approximation", Numerical Mathematics, 2007)
  • Subdivision surfaces
  • Biharmonic equation / C^1 elements
  • Lie group integrators
  • Hierarchical matrices
  • Matrix functions
  • Convex analysis for spectral functions
  • Direct assembly of the inverse mass matrix
  • Optimization on matrix manifolds
  • ...

Master theses

Functional a posteriori error estimates

(Joint supervision with Prof. Dirk Pauly)

In the paper Solution Theory, Variational Formulations, and Functional a Posteriori Error Estimates for General First-Order Systems with Applications to Electro-Magneto-Statics and More, Dirk Pauly proves the well-posedness of certain first-order PDEs using Hilbert complex techniques. (In his own words, the paper is "zu lang, aber einfach -- nicht abschrecken lassen!") He also constructs an a posteriori error estimator, i.e., an algorithm to compute upper and lower bounds for the error of numerical PDE solutions. However, while this algorithm has been proven to be correct, it has never actually been tried out in practice. Therefore, the goal of this master thesis is to understand the error estimator, implement it using Dune, and investigate how well it does in practice.

Boundary element methods (BEM)

Classical solution theory of PDEs allows to replace certain elliptic PDEs by integral equations on the domain boundary. This can be helpful, e.g., if the domain is partly unbounded, or has a very complicated boundary. The Boundary Element Method (BEM) tries to solve the integral equation numerically. The approach has several important advantages: A boundary grid is typically much smaller than a grid for the full domain, and it is usually much easier to construct. The price to pay is that the stiffness matrix is dense all of a sudden, and assembly and solving can become costly if not done right. The goal of this thesis is to understand the method, and to implement a prototype using Dune.

Literature: Sauter, Schwab: Boundary Element Methods

The Butcher group

(joint supervision with Prof. Uli Krähmer).

The classical proof for the convergence order of the Runge-Kutta time-stepping method for ODEs involves computing Taylor expansions of the ODE and of the Runge-Kutta scheme (see [1] for a description). To cope with the resulting mess of different mixed nested derivatives, Cayley and Butcher had the idea to label the derivatives by rooted trees. Surprisingly, there is a Lie group structure behind these trees. Even more surprisingly, the same group was found independently later when studying Hopf algebras and quantum field theory [2]. The aim of this project is to understand both ways to arrive at the Butcher group, and to describe them in a single coherent document.

Kitaev models

(joint supervision with Prof. Uli Krähmer)

Kitaev models are theoretical models for a quantum computer. They are believed to be particularly robust against the influence of the environment (such as cosmic rays) because of a feature called "topological protection". In its simple-most form, a Kitaev model consists of a graph inscribed into a high-genus compact surface. Associated to each edge of the graph is a vector with d complex entries (a "qubit"). These vectors interact and this interaction is modelled by the action of certain linear operators, essentially discrete stationary variations of Schrödinger's equation. For a robust quantum computer one needs information about the spectrum of the linear operators, i.e., their set of eigenvalues. The goal of this project is to understand the model, numerically compute the spectra, and to determine spaces that are invariant under all the linear operators.

Finite elements with Bernstein polynomials

Finite elements are a technique for solving partial differential equations. The unknown function is approximated by piecewise polynomials. The important Lagrange finite elements use piecewise Lagrange polynomials as a basis. However, Bernstein polynomials can also be used instead. These have the advantage that they are non-negative on every element. This has advantages if constraints have to be taken into account, e.g. if the solution must not exceed an upper bound at certain points.

In this project, this approach is to be tried out. Candidates with a penchant for analysis can try to estimate the discretization error. More practically oriented candidates can try to build a multigrid method for the obstacle problem.

GENERIC

The GENERIC formalism is an extension of classical Hamiltonian mechanics (which describes energy-conserving processes) to processes that also have dissipative components. Dissipative processes are processes that release energy, such as friction or plastic deformation. A central question is how structure-preserving time integration methods can look like for such systems.

Literature: H.C. Öttinger, Beyond Equilibrium Thermodynamics. Wiley, 2004

Prerequisites: Interest in time discretizations, mechanics, and perhaps a little differential geometry

Multigrid for primal plasticity or fracture in Python

The article [1] describes a multigrid method that can be used to quickly simulate the plastic deformation of simple materials. A good C++ implementation of this algorithm exists, but both the mathematical description and the C++ implementation are somewhat unwieldy. The aim of this work is to understand the multigrid algorithm, to create a simple Python implementation and to describe it well. If you are not interested in plastic deformation, you can try the same project for the cracking problem in [2].

Literature:

Prerequisites: Interest in programming in Python

Crack propagation in a plastic material (ductile fracture)

In [1] it is described how quasi-static crack propagation in a brittle material (e.g. glass) can be modeled and simulated. However, many materials behave elastoplastically: if the loads are not too small, they initially deform permanently, and cracks only occur if the loads become even greater.

Various models for such crack formation have already been proposed in the engineering literature (see e.g. [3]). In principle, these models combine the crack formation model from [1] with the elastoplastic material model from [2]. However, mathematically thorough investigations are still pending, which will require similar techniques as in [1] and [2], e.g. variational inequalities, convex analysis and some non-smooth analysis.

In principle, this topic can be turned into two master's theses: One focusing on mathematical analysis: there one would prove the convexity, smoothness and existence statements from [1] for the more general model. On the other hand, one can also write a master's thesis on the algorithms described in [1,2]. These should generalize fairly directly to the case of cracking in elastoplastic materials. A C++/Dune implementation for cracking in elastic materials already exists.

Literature:

Non-associative plasticity

Non-associative plasticity refers to models for elastoplastic materials that, for certain reasons, do not fit into the nice convex-analytical framework in [1]. In the quite recent paper [2] it is now claimed that this is possible. The aim of this work is to understand [2] and to investigate the consequences of the newly discovered structures for the construction of solution algorithms.

Literature:

Dynamic crack propagation

In [1] it is described how quasi-static crack propagation in a brittle material (e.g. glass) can be modeled and simulated. However, the models there neglect the fact that undulations can occur in such materials, which may lead to further cracks. In order to take such effects into account, the kinetic energy must be incorporated into the model. This changes the character of the equation and new time-stepping methods are required (in particular, the new equation should be describable by the GENERIC formalism described above). The aim of this work is to understand such an extended model, to apply a time discretization to it, and to implement it in C++/Dune (an implementation for quasi-static cracking already exists).

Literature: [1] C. Gräser, D. Kienle, and O. Sander: Truncated Nonsmooth Newton Multigrid for phase-field brittle-fracture problems, arXiv: 2007.12290, 2020

Isogeometric analysis

The buzzword "isogeometric analysis" refers to the finite element method. However, spline functions are used for interpolation instead of the normal Lagrange polynomials. The advantage is that in many cases the time-consuming grid generation can be avoided. In many technical applications, the areas are also described by splines, and isogeometric analysis can be used to calculate directly on them.

The aim of this project is to understand isogeometric analysis and to incorporate the spline basis functions into DUNE.

And there is more...

  • Discretization and implementation of a model for Cosserat plasticity
  • de Casteljau-FE
  • The Thompson metric
  • The Landau-Lifshitz-Gilbert equation for micromagnetism
  • Particle methods for solving partial differential equations (e.g. the maximum entropy method)
  • Whitney forms of higher order
  • Optimization on manifolds
  • ...

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Claudia Hess
Last modified: Jul 11, 2025