Space and Time Parallel Multigrid for Optimization and Uncertainty Quantification in PDE Simulations

  • Lars GrasedyckEmail author
  • Christian Löbbert
  • Gabriel Wittum
  • Arne Nägel
  • Volker Schulz
  • Martin Siebenborn
  • Rolf Krause
  • Pietro Benedusi
  • Uwe Küster
  • Björn Dick
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 113)


In this article we present a complete parallelization approach for simulations of PDEs with applications in optimization and uncertainty quantification. The method of choice for linear or nonlinear elliptic or parabolic problems is the geometric multigrid method since it can achieve optimal (linear) complexity in terms of degrees of freedom, and it can be combined with adaptive refinement strategies in order to find the minimal number of degrees of freedom. This optimal solver is parallelized such that weak and strong scaling is possible for extreme scale HPC architectures. For the space parallelization of the multigrid method we use a tree based approach that allows for an adaptive grid refinement and online load balancing. Parallelization in time is achieved by SDC/ISDC or a space-time formulation. As an example we consider the permeation through human skin which serves as a diffusion model problem where aspects of shape optimization, uncertainty quantification as well as sensitivity to geometry and material parameters are studied. All methods are developed and tested in the UG4 library.


Clock Frequency Uncertainty Quantification Shape Gradient Precision Time Protocol Uniform Refinement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



All ten authors gratefully acknowledge support from the DFG (Deutsche Forschungsgemeinschaft) within the DFG priority program on software for exascale computing (SPPEXA), project Exasolvers.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Lars Grasedyck
    • 1
    Email author
  • Christian Löbbert
    • 1
  • Gabriel Wittum
    • 2
  • Arne Nägel
    • 2
  • Volker Schulz
    • 3
  • Martin Siebenborn
    • 3
  • Rolf Krause
    • 4
  • Pietro Benedusi
    • 4
  • Uwe Küster
    • 5
  • Björn Dick
    • 5
  1. 1.IGPMRWTH AachenAachenGermany
  2. 2.G-CSCUniversity of FrankfurtFrankfurtGermany
  3. 3.University of TrierTrierGermany
  4. 4.ICSUniversity of LuganoLuganoGermany
  5. 5.HLRSUniversity of StuttgartStuttgartGermany

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