Skip to main content
SpringerLink
Log in

Recent Developments in the Theory and Application of the Sparse Grid Combination Technique

  • Conference paper
  • First Online:
Software for Exascale Computing - SPPEXA 2013-2015

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

Abstract

Substantial modifications of both the choice of the grids, the combination coefficients, the parallel data structures and the algorithms used for the combination technique lead to numerical methods which are scalable. This is demonstrated by the provision of error and complexity bounds and in performance studies based on a state of the art code for the solution of the gyrokinetic equations of plasma physics. The key ideas for a new fault-tolerant combination technique are mentioned. New algorithms for both initial- and eigenvalue problems have been developed and are shown to have good performance.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Ali, M.M., Southern, J., Strazdins, P.E., Harding, B.: Application level fault recovery: Using fault-tolerant open MPI in a PDE solver. In: 2014 IEEE International Parallel & Distributed Processing Symposium Workshops, Phoenix, 19–23 May 2014, pp. 1169–1178. IEEE (2014)

    Google Scholar 

  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: Proceedings of the 2015 International Conference on High Performance Computing & Simulation (HPCS 2015), pp. 499–507. IEEE, Amsterdam (2015). Outstanding paper award

    Google Scholar 

  3. Benk, J., Bungartz, H.J., Nagy, A.E., Schraufstetter, S.: Variants of the combination technique for multi-dimensional option pricing. In: Günther, M., Bartel, A., Brunk, M., Schöps, S., Striebel, M. (eds.) Progress in Industrial Mathematics at ECMI 2010, pp. 231–237. Springer, Berlin/Heidelberg (2010)

    Google Scholar 

  4. Benk, J., Pflüger, D.: Hybrid parallel solutions of the Black-Scholes PDE with the truncated combination technique. In: Smari, W.W., Zeljkovic, V. (eds.) 2012 International Conference on High Performance Computing & Simulation, HPCS 2012, Madrid, 2–6 July 2012, pp. 678–683. IEEE (2012)

    Google Scholar 

  5. Bungartz, H.J., Griebel, M., Rüde, U.: Extrapolation, combination, and sparse grid techniques for elliptic boundary value problems. Comput. Method. Appl. M. 116 (1–4), 243–252 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Das, S., Neumaier, A.: Solving overdetermined eigenvalue problems. SIAM J. Sci. Comput. 35 (2), 541–560 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fang, Y.: One dimensional combination technique and its implementation. ANZIAM J. Electron. Suppl. 52 (C), C644–C660 (2010)

    MathSciNet  Google Scholar 

  8. Franz, S., Liu, F., Roos, H.G., Stynes, M., Zhou, A.: The combination technique for a two-dimensional convection-diffusion problem with exponential layers. Appl. Math. 54 (3), 203–223 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Garcke, J.: Regression with the optimised combination technique. In: Proceedings of the 23rd International Conference on Machine Learning (ICML 2006), vol. 2006, pp. 321–328. ACM, New York (2006)

    Google Scholar 

  10. Garcke, J.: A dimension adaptive sparse grid combination technique for machine learning. ANZIAM J. 48 (C), C725–C740 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Garcke, J., Griebel, M.: On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric fields with the sparse grid combination technique. J. Comput. Phys. 165(2), 694–716 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Garcke, J., Hegland, M.: Fitting multidimensional data using gradient penalties and combination techniques. In: Modeling, Simulation and Optimization of Complex Processes, pp. 235–248. Springer, Berlin (2008)

    Google Scholar 

  13. Garcke, J., Hegland, M.: Fitting multidimensional data using gradient penalties and the sparse grid combination technique. Computing 84 (1–2), 1–25 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gene Development Team: GENE. http://www.genecode.org/

  15. Griebel, M.: The combination technique for the sparse grid solution of PDEs on multiprocessor machines. Parallel Process. Lett. 2, 61–70 (1992)

    Article  Google Scholar 

  16. Griebel, M., Harbrecht, H.: On the convergence of the combination technique. Lect. Notes Comput. Sci. 97, 55–74 (2014)

    Article  MATH  Google Scholar 

  17. Griebel, M., Thurner, V.: The efficient solution of fluid dynamics problems by the combination technique. Int. J. Numer. Method. H. 5 (3), 251–269 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Griebel, M., Huber, W., Rüde, U., Störtkuhl, T.: The combination technique for parallel sparse-grid-preconditioning or -solution of PDEs on workstation networks. In: Bougé, L., Cosnard, M., Robert, Y., Trystram, D.(eds.) Parallel Processing: CONPAR 92 – VAPP V, Lecture Notes in Computer Science. vol. 634, pp. 217–228. Springer, Berlin/Heidelberg/London (1992). Proceedings of the Second Joint International Conference on Vector and Parallel Processing, Lyon, 1–4 Sept 1992

    Google Scholar 

  19. Griebel, M., Schneider, M., Zenger, C.: A combination technique for the solution of sparse grid problems. In: Iterative Methods in Linear Algebra (Brussels, 1991), pp. 263–281. North-Holland, Amsterdam (1992)

    Google Scholar 

  20. Harding, B.: Adaptive sparse grids and extrapolation techniques. In: Proceedings of Sparse Grids and Applications 2014. Lecture Notes in Computational Science and Engineering, vol. 109, pp. 79–102. Springer, New York (2015)

    Google Scholar 

  21. Harding, B.: Combination technique coefficients via error splittings. ANZIAM J. 56, C355–C368 (2016). (Online)

    Google Scholar 

  22. Harding, B.: Fault tolerant computation of hyperbolic PDEs with the sparse grid combination technique. Ph.D. thesis, The Australian National University (2016)

    Google Scholar 

  23. Harding, B., Hegland, M.: A robust combination technique. ANZIAM J. Electron. Suppl. 54 (C), C394–C411 (2012)

    MathSciNet  Google Scholar 

  24. Harding, B., Hegland, M.: A parallel fault tolerant combination technique. Adv. Parallel Comput. 25, 584–592 (2014)

    Google Scholar 

  25. Harding, B., Hegland, M.: Robust solutions to PDEs with multiple grids. In: Garcke, J., Pflüger, D. (eds.) Sparse Grids and Applications, Munich 2012. Lecture Notes in Computer Science. vol. 97, pp. 171–193. Springer, Cham (2014)

    Chapter  Google Scholar 

  26. Harding, B., Hegland, M., Larson, J.W., Southern, J.: Fault tolerant computation with the sparse grid combination technique. SIAM J. Sci. Comput. 37 (3), C331–C353 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Heene, M., Pflüger, D.: Scalable algorithms for the solution of higher-dimensional PDEs. In: Proceedings of SPPEXA Symposium 2016. Lecture Notes in Computational Science and Engineering. Springer, Berlin/Heidelberg (2016)

    Google Scholar 

  28. Heene, M., Kowitz, C., Pflüger, D.: Load balancing for massively parallel computations with the sparse grid combination technique. In: Bader, M., Bungartz, H.J., Bode, A., Gerndt, M., Joubert, G.R. (eds.) Parallel Computing: Accelerating Computational Science and Engineering (CSE). pp. 574–583. IOS Press, Amsterdam (2014)

    Google Scholar 

  29. Hegland, M.: Adaptive sparse grids. In: Burrage, K., Sidje, R.B. (eds.) Proceedings of 10th Computational Techniques and Applications Conference CTAC-2001, vol. 44, pp. C335–C353 (2003)

    Google Scholar 

  30. Hegland, M., Garcke, J., Challis, V.: The combination technique and some generalisations. Linear Algebra Appl. 420 (2–3), 249–275 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Hupp, P., Jacob, R., Heene, M., Pflüger, D., Hegland, M.: Global communication schemes for the sparse grid combination technique. Adv. Parallel Comput. 25, 564–573 (2014)

    Google Scholar 

  32. 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)

    Article  MathSciNet  Google Scholar 

  33. Jennings, L.S., Osborne, M.: Generalized eigenvalue problems for rectangular matrices. IMA J. Appl. Math. 20 (4), 443–458 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kowitz, C.: Applying the sparse grid combination technique in Linear Gyrokinetics. Ph.D. thesis, Technische Universität München (2016)

    Google Scholar 

  35. Kowitz, C., Hegland, M.: The sparse grid combination technique for computing eigenvalues in linear gyrokinetics. Procedia Comput. Sci. 18, 449–458 (2013). 2013 International Conference on Computational Science

    Google Scholar 

  36. Kowitz, C., Hegland, M.: An opticom method for computing eigenpairs. In: Garcke, J., Pflüger, D. (eds.) Sparse Grids and Applications, Munich 2012 SE – 10. Lecture Notes in Computer Science. vol. 97, pp. 239–253. Springer, Cham (2014)

    Chapter  Google Scholar 

  37. Kowitz, C., Pflüger, D., Jenko, F., Hegland, M.: The combination technique for the initial value problem in linear gyrokinetics. Lecture Notes in Computer Science, vol. 88, pp. 205–222. Springer, Heidelberg (2013)

    Google Scholar 

  38. Larson, J.W., Hegland, M., Harding, B., Roberts, S., Stals, L., Rendell, A., Strazdins, P., Ali, M.M., Kowitz, C., Nobes, R., Southern, J., Wilson, N., Li, M., Oishi, Y.: Fault-tolerant grid-based solvers: Combining concepts from sparse grids and mapreduce. Procedia Comput. Sci. 18, 130–139 (2013). 2013 International Conference on Computational Science

    Google Scholar 

  39. Larson, J.W., Strazdins, P.E., Hegland, M., Harding, B., Roberts, S.G., Stals, L., Rendell, A.P., Ali, M.M., Southern, J.: Managing complexity in the parallel sparse grid combination technique. In: Bader, M., Bode, A., Bungartz, H.J., Gerndt, M., Joubert, G.R., Peters, F.J. (eds.) PARCO. Advances in Parallel Computing, vol. 25, pp. 593–602. IOS Press, Amsterdam (2013)

    Google Scholar 

  40. Larson, J., Strazdins, P., Hegland, M., Harding, B., Roberts, S., Stals, L., Rendell, A., Ali, M., Southern, J.: Managing complexity in the parallel sparse grid combination technique. Adv. Parallel Comput. 25, 593–602 (2014)

    Google Scholar 

  41. Lastdrager, B., Koren, B., Verwer, J.: The sparse-grid combination technique applied to time-dependent advection problems. In: Multigrid Methods, VI (Gent, 1999). Lecture Notes of Computer Science & Engineering, vol. 14, pp. 143–149. Springer, Berlin (2000)

    Google Scholar 

  42. Osborne, M.R.: A new method for the solution of eigenvalue problems. Comput. J. 7 (3), 228–232 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  43. Parra Hinojosa, A., Kowitz, C., Heene, M., Pflüger, D., Bungartz, H.J.: Towards a fault-tolerant, scalable implementation of GENE. In: Recent Trends in Computation Engineering – CE2014. Lecture Notes in Computer Science, vol. 105, pp. 47–65. Springer, Cham (2015)

    Google Scholar 

  44. Pflaum, C.: Convergence of the combination technique for second-order elliptic differential equations. SIAM J. Numer. Anal. 34 (6), 2431–2455 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  45. Pflaum, C., Zhou, A.: Error analysis of the combination technique. Numer. Math. 84 (2), 327–350 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  46. Pflüger, D., Bungartz, H.J., Griebel, M., Jenko, F., Dannert, T., Heene, M., Kowitz, C., Parra Hinojosa, A., Zaspel, P.: 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, Cham (2014)

    Google Scholar 

  47. Reisinger, C.: Analysis of linear difference schemes in the sparse grid combination technique. IMA J. Numer. Anal. 33 (2), 544–581 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  48. Strazdins, P.E., Ali, M.M., Harding, B.: Highly scalable algorithms for the sparse grid combination technique. In: IPDPS Workshops, Hyderabad, pp. 941–950. IEEE (2015)

    Google Scholar 

  49. Strazdins, P.E., Ali, M.M., Harding, B.: The design and analysis of two highly scalable sparse grid combination algorithms (2015, under review)

    Google Scholar 

  50. Wilkinson, J.H.: Rounding Errors in Algebraic Processes. Her Majesty’s Stationery Office, London (1963)

    MATH  Google Scholar 

  51. Wong, M., Hegland, M.: Maximum a posteriori density estimation and the sparse grid combination technique. ANZIAM J. Electron. Suppl. 54 (C), C508–C522 (2012)

    MathSciNet  Google Scholar 

  52. Wong, M., Hegland, M.: Opticom and the iterative combination technique for convex minimisation. Lect. Notes Comput. Sci. 97, 317–336 (2014)

    Article  MATH  Google Scholar 

  53. Zenger, C.: Sparse grids. In: Hackbusch, W. (ed.) Parallel Algorithms for Partial Differential Equations. Notes on Numerical Fluid Mechanics, vol. 31, pp. 241–251. Vieweg, Braunschweig (1991)

    Google Scholar 

Download references

Acknowledgements

The work presented here reviews some results of a German-Australian collaboration going over several years which was supported by grants from the German DFG (SPP-1648 SPPEXA: EXAHD) and the Australian ARC (LP110200410), contributions by Fujitsu Laboratories of Europe (FLE) and involved researchers from the ANU, FLE, TUM, and the Universities of Stuttgart and Bonn. Contributors to this research included Stephen Roberts, Jay Larson, Moshin Ali, Ross Nobes, James Southern, Nick Wilson, Hans-Joachim Bungartz, Valeriy Khakhutskyy, Alfredo Hinojosa, Mario Heene, Michael Griebel, Jochen Garcke, Rico Jacob, Philip Hupp, Yuan Fang, Matthias Wong, Vivien Challis and several others.

Author information

Authors and Affiliations

  1. Mathematical Sciences Institute, The Australian National University, Canberra, Australia

    Markus Hegland & Brendan Harding

  2. Chair of Scientific Computing, Technische Universität München, Munich, Germany

    Christoph Kowitz

  3. Institute for Parallel and Distributed Systems, University of Stuttgart, Stuttgart, Germany

    Dirk Pflüger

  4. Engineering and Computer Science, The Australian National University, Canberra, Australia

    Peter Strazdins

Authors
  1. Markus Hegland
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Brendan Harding
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christoph Kowitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Dirk Pflüger
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Peter Strazdins
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Markus Hegland .

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

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Hegland, M., Harding, B., Kowitz, C., Pflüger, D., Strazdins, P. (2016). Recent Developments in the Theory and Application of the Sparse Grid Combination Technique. 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_7

Download citation

Publish with us

Policies and ethics

Search

Navigation