Research

The working group deals with the development and analysis of numerical methods for solving partial differential equations that occur in materials science and geometry. Based on statements regarding the existence and uniqueness of solutions to nonlinear differential equations, time step methods and finite element methods are investigated with regard to stability and convergence. The approximation methods developed in this way are tested experimentally with the aid of powerful computers and allow the suitability of the underlying mathematical models for practical predictions to be assessed.

The working group focuses on the analysis and numerics of variational problems and the associated gradient flows. The topics range from problems in microstructure formation during the minimization of non-convex energies to the evolution of interfaces in media with random obstacles. Particular attention is paid to the mathematical derivation of effective macroscopic models from microscopic behavior and their numerical implementation.

The working group deals with the theoretical and numerical analysis of nonlinear partial differential equations. These are treated using techniques and ideas from a wide variety of fields, such as functional analysis, function space theory, and numerical error analysis. A priori estimates and limit processes play a central role. The problems addressed are mostly motivated by questions from fluid mechanics or geometry.

My areas of research include the following:

• Approximation properties of deep neural networks
• Machine Learning
• Numerical methods for stochastic and deterministic partial differential equations
• Numerical and stochastic analysis
• Computational Stochastics