Scalable Algorithms for the Solution of Higher-Dimensional PDEs

  • Mario HeeneEmail author
  • Dirk Pflüger
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 113)


The solution of higher-dimensional problems, such as the simulation of plasma turbulence in a fusion device as described by the five-dimensional gyrokinetic equations, is a grand challenge for current and future high-performance computing. The sparse grid combination technique is a promising approach to the solution of these problems on large-scale distributed memory systems. The combination technique numerically decomposes a single large problem into multiple moderately-sized partial problems that can be computed in parallel, independently and asynchronously of each other. The ability to efficiently combine the individual partial solutions to a common sparse grid solution is a key to the overall performance of such large-scale computations. In this work, we present new algorithms for the recombination of distributed component grids and demonstrate their scalability to 180, 225 cores on the supercomputer Hazel Hen.


Grid Point Domain Decomposition Process Group Sparse Grid Combine Solution 
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.



This work was supported by the German Research Foundation (DFG) through the Priority Program 1648 “Software for Exascale Computing” (SPPEXA).


  1. 1.
  2. 2.
    Ali, M.M., Strazdins, P.E., Harding, B., Hegland, M., Larson, J.W.: A fault-tolerant gyrokinetic plasma application using the sparse grid combination technique. In: International Conference on High Performance Computing & Simulation (HPCS), Amsterdam, pp. 499–507. IEEE (2015)Google Scholar
  3. 3.
    Brizard, A., Hahm, T.: Foundations of nonlinear gyrokinetic theory. Rev. Mod. Phys. 79, 421–468 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bungartz, H.J., Griebel, M.: Sparse grids. Acta Numer. 13, 147–269 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cappello, F., Geist, A., Gropp, W., Kale, S., Kramer, B., Snir, M.: Toward exascale resilience: 2014 update. Supercomput. Front. Innov. 1 (1), 5–28 (2014)Google Scholar
  6. 6.
    Dannert, T.: Gyrokinetische Simulation von Plasmaturbulenz mit gefangenen Teilchen und elektromagnetischen Effekten. Ph.D. thesis, Technische Universität München (2004)Google Scholar
  7. 7.
    Dannert, T., Görler, T., Jenko, F., Merz, F.: Jülich blue gene/p extreme scaling workshop 2009. Technical report, Jülich Supercomputing Center (2010)Google Scholar
  8. 8.
    Doyle, E.J., Kamada, Y., Osborne, T.H., et al.: Chapter 2: plasma confinement and transport. Nucl. Fusion 47 (6), S18 (2007)CrossRefGoogle Scholar
  9. 9.
    Görler, T., Lapillonne, X., Brunner, S., Dannert, T., Jenko, F., Merz, F., Told, D.: The global version of the gyrokinetic turbulence code GENE. J. Comput. Phys. 230 (18), 7053–7071 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Griebel, M., Huber, W., Rüde, U., Störtkuhl, T.: The combination technique for parallel sparse-grid-preconditioning or -solution of PDEs on workstation networks. In: Parallel Processing: CONPAR 92 VAPP V. LNCS, vol. 634. Springer, Berlin/New York (1992)Google Scholar
  11. 11.
    Griebel, M., Schneider, M., Zenger, C.: A combination technique for the solution of sparse grid problems. In: de Groen, P., Beauwens, R. (eds.) Iterative Methods in Linear Algebra. IMACS, pp. 263–281. Elsevier/North Holland (1992)Google Scholar
  12. 12.
    Heene, M., Pflüger, D.: Efficient and scalable distributed-memory hierarchization algorithms for the sparse grid combination technique. In: Parallel Computing: On the Road to Exascale. Advances in Parallel Computing, vol. 27. IOS Press, Amsterdam (2016)Google Scholar
  13. 13.
    Heene, M., Kowitz, C., Pflüger, D.: Load balancing for massively parallel computations with the sparse grid combination technique. In: Parallel Computing: Accelerating Computational Science and Engineering (CSE). Advances in Parallel Computing, vol. 25, pp. 574–583. IOS Press, Amsterdam (2014)Google Scholar
  14. 14.
    Hegland, M., Garcke, J., Challis, V.: The combination technique and some generalisations. Linear Algebra Appl. 420 (2–3), 249–275 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hegland, M., Harding, B., Kowitz, C., Pflüger, D., Strazdins, P.: Recent developments in the theory and application of the sparse grid combination technique. In: Proceedings of the SPPEXA Symposium 2016, Garching. Lecture Notes in Computational Science and Engineering. Springer (2016)Google Scholar
  16. 16.
    Hupp, P., Jacob, R., Heene, M., et al.: Global communication schemes for the sparse grid combination technique. Par. Comput.: Accel. Comput. Sci. Eng. 25, pp. 564–573 (2014)Google Scholar
  17. 17.
    Hupp, P., Heene, M., Jacob, R., Pflüger, D.: Global communication schemes for the numerical solution of high-dimensional {PDEs}. Parallel Comput. 52, 78–105 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kowitz, C., Hegland, M.: The sparse grid combination technique for computing eigenvalues in linear gyrokinetics. Procedia Comput. Sci. 18 (0), 449–458 (2013). 2013 International Conference on Computational ScienceGoogle Scholar
  19. 19.
    Parra Hinojosa, A., Kowitz, C., Heene, M., Pflüger, D., Bungartz, H.J.: Towards a fault-tolerant, scalable implementation of GENE. In: Proceedings of ICCE 2014, Nara. Lecture Notes in Computational Science and Engineering. Springer (2015)Google Scholar
  20. 20.
    Parra Hinojosa, A., Harding, B., Hegland, M., Bungartz, H.J.: Handling silent data corruption with the sparse grid combination technique. In: Proceedings of the SPPEXA Symposium 2016, Garching. Lecture Notes in Computational Science and Engineering. Springer (2016)Google Scholar
  21. 21.
    Pflüger, D., Bungartz, H.J., Griebel, M., Jenko, F., et al.: EXAHD: an exa-scalable two-level sparse grid approach for higher-dimensional problems in plasma physics and beyond. In: Euro-Par 2014: parallel processing workshops, Porto. Lecture Notes in Computer Science, vol. 8806, pp. 565–576. Springer International Publishing (2014)Google Scholar
  22. 22.
    Thakur, R., Rabenseifner, R., Gropp, W.: Optimization of collective communication operations in MPICH. Int. J. High Perform. C. 19, 49–66 (2005)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute for Parallel and Distributed SystemsUniversity of StuttgartStuttgartGermany

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