Stephan Dahlke - Current Research Interests

• Adaptive quarklet schemes for operator equations
  

  Quarklets are newly developed dictionaries for function spaces which are constructed by polynomial enrichment of classcial wavelet bases. Numerical methods which are based on these dictionaries can be interpreted as the wavelet analogon of the well-known hp-methods in the realm of finite elements. The overall goal is the construction of adaptive schemes with provable order of convergence, where we particularly strive for an exponential order. This is a joint project with Thorsten Raasch

  
  

• Regularity theory of partial differential equations
  

  This research project is motivated by the following general question: Under which conditions are adaptive numerical algorithms really superior when compared to non-adaptive (uniform) methods? The convergence order that can be achieved by non-adaptive methods is usually determined by the regularity of the exact solution in the classical Sobolev scale. In contrast to this, the best possible convergence rate of adaptive algorithms depends on the regularity in non-classical smoothness spaces, the Besov spaces. Therefore, to ensure that adaptivity really pays, the Besov smoothness of the exact solution has to be investigated.. The reasearch foci are : Regularity theory of non-linear elliptic partial differential equations Regularity theory of parabolic partial differential equations This is a joint project with Winfried Sickel and Cornelia Schneider .

  
  

• Shearlet theory and applications
  

  Shearlets are recently developed affine representation systems that are in particular suitable for the detection of directional information. In contrast to other representaion systems such as curvelets, contourlets, or ridgelets, shearlets have the advantage that they 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 (More information). We are working on: The construction and the analysis of shearlet coorbit spaces The analysis of shearlet groups and their relations to other groups The applications fo the shearlet transformation to problems in image analysis This is a joint project with Gabriele Steidl, Gerd Teschke, Ernesto de Vito, and Filippo de Mari .

  
  

• Coorbit theory
  

  The famous coorbit theory was established by H. Feichtinger and K. Gröchenig. Based on square integrable group representations, this theory gives rise to the construction of new smoothness spaces, where the smoothness is measured by the decay of the associated voice transform. Moreover, in the realm of this theory, it is possible to establish atomic decompositions and Banach frames for these spaces. Two important special cases are the Besov spaces and the modulation spaces which can be interpreted as the coorbit sapces related with the affine group and the Weyl-Heisenberg group, respectively. The following questions are currently in the centre of attraction: Generalization of the coorbit theory to non-integrable representations Coorbit spaces and frames on manifolds. This is a joint project with Gabriele Steidl, Gerd Teschke, Ernesto de Vito, and Filippo de Mari.