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Software for Exascale Computing - SPPEXA 2013-2015 pp 507–523Cite as

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

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Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 113))

Abstract

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.

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Notes

  1. 1.

    We use the commonly utilized term “energy consumption” despite the fact that electrical energy is converted to thermal energy.

  2. 2.

    cpufreq_set_frequency()

  3. 3.

    HLRS, Stuttgart, Germany, http://www.hlrs.de/systems/platforms/cray-xe6-hermit/

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Acknowledgements

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

Author information

Authors and Affiliations

  1. IGPM, RWTH Aachen, Aachen, Germany

    Lars Grasedyck & Christian Löbbert

  2. G-CSC, University of Frankfurt, Frankfurt, Germany

    Gabriel Wittum & Arne Nägel

  3. University of Trier, Trier, Germany

    Volker Schulz & Martin Siebenborn

  4. ICS, University of Lugano, Lugano, Germany

    Rolf Krause & Pietro Benedusi

  5. HLRS, University of Stuttgart, Stuttgart, Germany

    Uwe Küster & Björn Dick

Authors
  1. Lars Grasedyck
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  2. Christian Löbbert
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  3. Gabriel Wittum
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  4. Arne Nägel
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  5. Volker Schulz
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  6. Martin Siebenborn
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  7. Rolf Krause
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  8. Pietro Benedusi
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  9. Uwe Küster
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  10. Björn Dick
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Corresponding author

Correspondence to Lars Grasedyck .

Editor information

Editors and Affiliations

  1. Technische Universität München Institut für Informatik, Garching, Bayern, Germany

    Hans-Joachim Bungartz

  2. Technische Universität München Institut für Informatik, Garching, Germany

    Philipp Neumann

  3. Technische Universität Dresden, Dresden, Germany

    Wolfgang E. Nagel

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Grasedyck, L. et al. (2016). Space and Time Parallel Multigrid for Optimization and Uncertainty Quantification in PDE Simulations. In: Bungartz, HJ., Neumann, P., Nagel, W. (eds) Software for Exascale Computing - SPPEXA 2013-2015. Lecture Notes in Computational Science and Engineering, vol 113. Springer, Cham. https://doi.org/10.1007/978-3-319-40528-5_23

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