Recent Research Topics
 Coordination of the priority program SPP1324
 From september 2008 till december 2014 the DFG priority programm SPP 1324 was coordinated by Prof. Dahlke.
 Adaptive wavelet methods for operator equations
 Adaptive methods for elliptic problems
The Aim of this project is the development of adaptive wavelet methods for operator equations, especially for elliptic partial differential equations. The equivalence of Sobolev norms and weighted sequence norms of wavelet coefficients allows the construction of reliable and efficient error estimator. They lead to adaptive schemes whose convergence can be proven.
The current focus of research is:
 Adaptive wavelet frame methods
 Adaptive wavelet domain decomposition methods
 Adaptive wavelet schemes for nonlinear elliptic problems
 Adaptive methods using tensor wavelets
 Development of adaptive wavelet method based on quarkonial decompositions
 Development of adaptive methods for the pPoisson equation (project)
We collaborate with Rob Stevenson, Massimo Fornasier, Lars Diening and Thorsten Raasch.

 Adaptive schemes for parabolic problems
The aim is the generalization of the existing adaptive strategies for
elliptic problems to parabolic equations. To this end, one
has to derive a suitable time step control strategy in time direction.
Then, in each time step, one can use the wellknown adaptive strategies for elliptic problems
to obtain an efficient space discretization.

 Adaptive methods for inverse problems
The development of adaptive methods for inverse problems is currently an important challenge. The known adaptive schemes work well as long as the underlying operator is boundedly invertible. Due to data and operator errors, this is not the case for inverse problems.
The following issues are currently under investigation:
 Development of adaptive wavelet methods for inverse parabolic problems (project)
 Development of acceleration strategies using decreasing thresholding for linear and nonlinear inverse problems
We collaborate with Peter Maaß, Rob Stevenson, Thorsten Raasch and Massimo Fornasier.

 Adaptive methods for stochastic partial differential equations
The adaptive treatment of stochastic partial differential becomes harder through singularities caused by stochastic termes.
We currently focus on the following topics:
 Development of adaptive schemes for elliptic and parabolic stochastic partial differential equations using Wavelet Frame methods (project)
 Construction of stochastic processes in Besov spaces
We cooperate with Klaus Ritter.


 Regularity theory of partial differential equation
The motivation for this project is the following fundamental question: For which cases is it possible to gain efficiency by using adaptive schemes when compared with nonadaptive (uniform) methods? The convergence rate of nonadaptive algorithms depends in general on the Sobolev regularity of the exact solution. A theoretical investigation shows that the convergence rate of adaptive schemes is determined by the regularity measured in nonclassical function spaces, the Besov spaces. Therefore, to ensure that adaptivity really pays, it is necessary to investigate the Besov regularity of the exact solution.
We currently focus on the following topcis:
 Regularity theory of nonlinear elliptic equation (project)
 Regularity theory for NavierStokes equation
 Regularity theory for integral operators on manifolds (project)
 Regularity theory for stochastic partial differential equation (project, project)
We cooperate with Felix Lindner, René L. Schilling, Winfried Sickel and Lars Diening.


 Shearlets
Shearlets are recently developed affine representation systems that are wellsuited for the detection of directional information. In contrast to other systems such as curvelets or contourlets, shearlets are associated with a square integrable representation of a specific group, the full shearlet group. This enables the application of powerful tools from harmonic analysis. (Further information).
The following topics are currently studied:
 Construction and analysis of shearlet coorbit spaces
 Analysis of the relations of the shearlet groups to other groups
 Shearlet transforms on manifolds
 Applications of shearlets in image processing
We collaborate with Gabriele Steidl, Gerd Teschke, Ernesto de Vito and Filippo de Mari.

 Coorbit Theory
The coorbit theory has been developed by H. Feichtinger and K.H. Gröchenig. Based on square integrable group representations, this theory allows the construction of new smoothness spaces where the regularity is measured by the decay of the associated voice transform. Moreover, coorbit theory provides a natural way to construct atomic decompositions and even Banach frames for these spaces. Two important special cases are the homogeneous Besov spaces and the modulation spaces. The Besov spaces can be interpreted as coorbits associated with the affine groups, whereas the modulation spaces are the coorbit spaces related with the WeylHeisenberg groups.
The following issues are in the focus of our researches:
 Relations to the shearlet theory, inhomogeneous shearlet coorbit spaces
 The construction of coorbit spaces for nonintegrable group representations
 Coorbit spaces and frames on manifolds
We collaborate with Gabriele Steidl, Gerd Teschke, Ernesto de Vito and Filippo de Mari.

 Mathematical Modelling of Microbial Systems
Within the LOEWE Centre for Synthetic Micro Biology, we study microbial systems. We are in particular concerned with the modelling of protein localizations and oscillations by means of reaction diffusion equations.
The following issues are currently under investigation:
 Mathematical modelling of protein oscillations in Myxococcus Xanthus
 Nonstandard diffusion phenomena in cells
 Mathematical modelling of the formation of flagella
We cooperate with Gert Bange, Lotte SogaardAndersen, Peter Lenz and Peter Graumann.


 Construction of Wavelets
This project is concerned with the construction of smooth and welllocalized wavelets and multiwavelets and their adaptation to intervals and more complicated domains.


