Workgroup Numerics

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 non-linear elliptic problems
      • Adaptive methods using tensor wavelets
      • Development of adaptive wavelet method based on quarkonial decompositions
      • Development of adaptive methods for the p-Poisson 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 well-known 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 non-linear 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.
      Poster: Adaptive Wavelet Methods for SPDEs

  • 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 non-adaptive (uniform) methods? The convergence rate of non-adaptive 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 non-classical 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 non-linear elliptic equation (project)
    • Regularity theory for Navier-Stokes 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.
    Poster: Spatial Besov Regularity for SPDEs

  • Shearlets
    Shearlets are recently developed affine representation systems that are well-suited 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 Weyl-Heisenberg 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 Sogaard-Andersen, Peter Lenz and Peter Graumann.
    Poster: SIAM
    Poster: SIAM

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