Studies

Many physical processes can be modeled by partial differential equations (PDEs), which typically need to be solved numerically. Our research focuses on the development and analysis of modern finite element methods for PDEs.

If you are interested in a thesis topic or a student project, please take a look here. A selection of completed thesis projects can be found here.

The Research Group in Variational Analysis and Continuous Optimization works on optimization as a central area of applied mathematics. Its applications range from the social and natural sciences to engineering and finance.

Continuous optimization is often closely connected with branches of analysis. Its central theoretical contributions concern the analysis of nonsmooth and set-valued objects. The tools of classical analysis, including derivatives, integrals and resolvents, are contained in the more modern language of variational analysis, which forms the theoretical foundation of mathematical optimization.

If you are considering writing a Bachelor's or Master's thesis in optimization, you are expected to have attended a basic course in optimization or operations research and at least one of the seminars regularly offered by Prof. Luke. Master's students are expected to have attended at least one semester of Prof. Luke's special lectures on variational analysis or numerical optimization.

Topics are determined individually, depending on the student's background and the current research activities of the working group.

Required lectures: Numerics I and II, as well as either Optimization or Scientific Computing.

Topics for Bachelor's and Master's theses may come from the following areas:

• Numerical Linear Algebra, e.g. fast algorithms, special matrices and eigenvalues
• Numerical Analysis, especially approximation theory, quadrature formulas and stable algorithms
• Numerical Fourier Analysis, e.g. fast Fourier transforms, trigonometric transforms and wavelets
• Applications in signal and image processing, e.g. denoising and compression
• Inverse problems in signal and image processing
• Mathematical foundations of deep learning
• Applications in cooperation with industrial partners or interdisciplinary projects

Topics are assigned individually, depending on the mathematical background and interests of the candidate. Examples of previous Bachelor's and Master's thesis topics can be found here:

Examples of previous theses

Prerequisites for a Bachelor's thesis:

• Numerical Mathematics I
• Numerical Mathematics II or Functional Analysis or one lecture from the Image and Geometry Processing lecture series, e.g. Fourier Analysis

Prerequisites for a Master's thesis:

• At least two of the following lectures: Numerical Mathematics II, Functional Analysis, lectures from the “Image and Geometry Processing” series, or Approximation Theory

Further information on the lectures can be found here:

Teaching of the Mathematical Signal and Image Processing group

The research focus of the group lies on problems from the applied sciences, especially physics. Typical goals include the evaluation of indirect measurement data and modeling aspects, which often require the numerical solution of inverse problems.

Current applications:

• Identification of material parameters and forces in active matter
• Traction force microscopy and inverse problems in elasticity
• Dynamic computed tomography, especially on the nanoscale
• Terahertz tomography for nondestructive testing

Methodological focus areas:

• fast regularization techniques
• parameter identification problems for partial differential equations
• combinations of machine learning methods and regularization schemes

Topics for Bachelor's and Master's theses are usually selected based on the student's background and interests. Successful completion of courses in Numerical Mathematics is required for both Bachelor's and Master's theses. Knowledge of inverse problems, optimization, functional analysis, numerical methods for partial differential equations, scientific computing or deep learning is helpful, but not required.