The main field of research is the development and the analysis of efficient numerical algorithms for operator equations. Realistic problems, e.g., in 3D, usually give rise to systems with a huge number of degrees of freedom, so that adaptive strategies are necessary to increase efficiency. We are particularly interested in adaptive schemes that are based on quite recently developed new basis functions, the so-called wavelets. The strong analytical properties of wavelets can be used to design reliable a posteriori error estimators. These error estimators give rise to adaptive refinement strategies that are guaranteed to converge with optimal order for a large class of problems, including operators of negative order. We are currently concerned with the generalization of these approaches to nonlinear problems, inverse problems and stochastic evolution equations, respectively.
Besides this, we are working on Besov regularity theory of partial differential equations and on Computational Harmonic Analysis (shearlet theory).
Moreover, we started to work on the mathematical modelling and numerical simulation of microbiological systems. In particular, we are concerned with the development of mathematical models to describe cell polarity.
Within SYNMIKRO we are primarily concerned with the mathematical modeling and the numerical simulation of microbiological systems. The overall goal is to gain some deeper understanding of the underlying system by analyzing suitable models. In the long run, this additional knowledge will be used to make reliable predictions for the successful design of biological experiments. At the moment, we mainly focus on models for cell polarity, in particular related with the bacterium Myxococcus xanthus. The directions of movement of M. xanthus are closely related with protein oscillations inside the cell. Therefore, first of all we studied the following fundamental question: are these oscillations necessarily induced by an external trigger or can oscillating systems with few parameters function without an external force? To this end, we developed a first model with two proteins. The basic assumptions are: interactions between the proteins only take place at the cell poles, and the proteins are transported through the cell by diffusion. At the cell poles, the same laws for interactions hold for both proteins, and no pole is specified. Indeed, it turned out that within this setting oscillations without any external trigger exist! In the near future, this model will be generalized and refined. We will take stochastic influences into account and discuss other interaction scenarios. Moreover, systems with more proteins will be studied.
Another focus is the development of suitable algorithms for the analysis of microscopical images that naturally occur as part of the analysis of microbiological systems. The key words are i.a. segmentation and tracking. These research projects will probably also have some impact on other projects in SYNMIKRO.
Within SYNMIKRO, we are collaborating with the groups of Eckhardt (Physics), Lenz (Physics), Søgaard-Andersen (MPI) and Kostina (Mathematics).