Skip to main content
SpringerLink
Log in

Efficient Task Scheduling in the Parallel Result-Verifying Solution of Nonlinear Systems

  • Published:
Reliable Computing

Abstract

Nonlinear systems occur in diverse applications, e.g., in the steady state analysis of chemical processes. If safety concerns require the results to be provably correct then result-verifying algorithms relying on interval arithmetic should be used for solving these systems. Since such algorithms are very computationally intensive, the coarse-grained inter-box parallelism should be exploited to make them feasible in practice. In this paper we briefly describe our framework SONIC for the verified solution of nonlinear systems and give detailed information about its parallelization with OpenMP and MPI. Our numerical results show that the implemented parallelization schemes are indeed successful. The more sophisticated MPI implementation seems to be superior to the easy-to-implement OpenMP version and shows almost linear speedup up to a large number of processors.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. An Mey, D.: Two OpenMP Programming Patterns, in: An Mey, D. (ed.), Proc. EWOMP '03, September 22–26, 2003, Aachen, Germany, Aachen, 2004, pp. 51–60.

  2. Babichev, A. B., Kashevarova, T. P., Leshchenko, A. S., and Semenov, A. L.: UniCalc: An Intelligent Solver for Mathematical Problems, in: Proc. of the East-West Conference on Artificial Intelligence: From Theory to Practice (EWAIC'93), Moscow, 1993, pp. 257–260.

  3. Beelitz, T., Bischof, C. H., and Lang, B.: A Hybrid Subdivision Strategy for Result-Verifying Nonlinear Solvers, in: Proc. Applied Mathematics and Mechanics (PAMM), Vol. 4, 2004, pp. 632–633.

    Google Scholar 

  4. Beelitz, T., Bischof, C. H., and Lang, B.: Intervals and OpenMP: Towards an Efficient Parallel Result-Verifying Nonlinear Solver, in: An Mey, D. (ed.), Proc. EWOMP '03, September 22–26, 2003, Aachen, Germany, Aachen, 2003, pp. 119–125.

  5. Beelitz, T., Bischof, C. H., Lang, B., and Willems, P.: SONIC—A Framework for the Rigorous Solution of Nonlinear Problems, Preprint BUW-SC 04/7, 2004.

  6. Bischof, C. H., Lang, B., Marquardt, W., and Mönnigmann, M.: Verified Determination of Singularities in Chemical Processes, in: Krämer, W. and Wolff von Gudenberg, J. (eds): Scientific Computing, Validated Numerics, Interval Methods, Kluwer Academic/Plenum Publishers, New York, 2001, pp. 305–316.

    Google Scholar 

  7. GlobSol home page: http://www.mcsu.mu.edu/~globsol.

  8. Golubitsky, M. and Schaeffer, D. G.: Singularities and Groups in Bifurcation Theory, Volume I, Springer-Verlag, New York, 1985.

    Google Scholar 

  9. Hammer, R., Hocks, M., Kulisch, U., and Ratz, D.: C++ Toolbox for Verified Computing—Basic Numerical Problems, Springer-Verlag, Heidelberg, 1995.

  10. Hofschuster, W. and Krämer, W.: C-XSC 2.0—A C++ Library for Extended Scientific Computing, in: Alt, R. et al. (eds): Numerical Software with Result Verification, LNCS 2991, Springer, Heidelberg, pp. 15–35.

  11. Hofschuster, W., Krämer, W., Lerch, M., Tischler, G., and Wolff von Gudenberg, J.: filib-++—A Fast Interval Library Supporting Containment Computations, to appear in ACM Trans. Math. Software, 2006.

  12. Intel Corporation: KAI C++ User's Guide, 2001, http://www.kai.com/kpts/-guide.

  13. Intel Corporation: Intel C++ Compiler User's Guide, 2003, http://developer.intel.com/software/products/compilers.

  14. Kearfott, R. B.: Rigorous Global Search: Continuous Problems, Kluwer Academic Publishers, Dordrecht, 1996.

    Google Scholar 

  15. Lebbah, Y., Michel, C., Rueher, M., Daney, D., and Merlet, J.-R: Efficient and Safe Global Constraints for Handling Numeric Constraint Systems, SIAM J. Numer. Anal. 42 (5) (2005), pp. 2076–2097.

    Article  MathSciNet  Google Scholar 

  16. Message Passing Interface Forum: MPI: A Message-Passing Interface Standard, 1994, http://www.mpi-forum.org.

  17. Message Passing Interface Forum: MPI-2: Extensions to the Message-Passing Interface, 1997, http://www.mpi-forum.org.

  18. Mönnigmann, M., Marquardt, W., Bischof, C. H., Beelitz, T., Lang, B., and Willems, P.: A Hybrid Approach for Efficient Robust Design of Dynamic Systems, Preprint BUW-SC 2004/9, 2004.

  19. Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  20. Sun Microsystems: Sun Studio 9: C++ Interval Arithmetic Programming Reference, http://docs.sun.com/app/docs/doc/817–6705.

  21. Willems, P.: Symbolisch-numerische Techniken zum verifizierten Lösen nichtlinearer Gleichungs- systeme, Diplomarbeit, RWTH Aachen University, 2004.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Applied Computer Science and Scientific Computing, Bergische Universität Wuppertal, D-42097, Wuppertal, Germany

    Thomas Beelitz & Bruno Lang

  2. Institute for Scientific Computing, RWTH Aachen University, D-52056, Aachen, Germany

    Christian H. Bischof

Authors
  1. Thomas Beelitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bruno Lang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christian H. Bischof
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thomas Beelitz.

Additional information

This work was partially funded by VolkswagenStiftung within the project “Konstruktive Meth-oden der Nichtlinearen Dynamik zum Entwurf verfahrenstechnischer Prozesse,” Geschäftszeichen 1/77 058 and 1/79 288.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Beelitz, T., Lang, B. & Bischof, C.H. Efficient Task Scheduling in the Parallel Result-Verifying Solution of Nonlinear Systems. Reliable Comput 12, 141–151 (2006). https://doi.org/10.1007/s11155-006-4872-4

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11155-006-4872-4

Keywords

Search

Navigation