Numerical analysis, machine learning and scientific computing

In this research field we are developing advanced computational methods centered around efficient solution strategies for partial differential equations.

In numerical analysis, we focus on developing and analysing advanced discretization schemes, including their stability and convergence properties, using, e.g., tools from functional analysis. Particular emphasis is on numerical multiscale methods, unfitted approximation schemes and model order reduction. Moreover, we contribute substantially to the advancement of numerical methods for high dimensional and stochastic PDEs as well as for high dimensional optimization and control problems.

Machine learning allows to detect low-dimensional hidden structures in data or in existing models, which gives rise to efficient surrogate modelling in close relationship to model order reduction. As many machine learning methods are hard to interpret and not well understood, we are interested in their mathematical foundation as well as in the combination of machine learning and classical numerical techniques such that rigorous error control becomes feasible.

Regarding scientific computing, we focus on the design, implementation, and application of efficient algorithms and software tools for solving large-scale mathematical models arising in real-world problems, e.g., in physics, engineering, biology, medicine or finance. This includes high-performance computing and the development of efficient solvers for the underlying linear and non-linear systems. Together with international collaborators, we offer these methods as part of open-source software frameworks which build the basis for many research projects.