Skip to main content
SpringerLink
Log in

Grand Challenges and Scientific Standards in Interval Analysis

  • Letter to the Editor
  • Published:
Reliable Computing

Abstract

This paper contains a list of “grand challenge” problems in interval analysis, together with some remarks on improved interaction with mainstream mathematics and on raising scientific standards in interval analysis.

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

References

  1. Alefeld, G. and Herzberger, J.: Introduction to Interval Computations, Academic Press, New York, 1983.

    Google Scholar 

  2. Bliek, C., Spellucci, P.,Vicente, L.N., Neumaier, A., Granvilliers, L.,Monfroy, E., Benhamou, F., Huens, E., Van Hentenryck, P., Sam-Haroud, D., and Faltings, B.: Algorithms for Solving Nonlinear Constrained and Optimization Problems: The State of the Art, A progress report of the COCONUT project, 2001, http://www.mat.univie.ac.at/~neum/glopt/ coconut/StArt.html

  3. Eckmann, J.-P., Koch, H., and Wittwer, P.: A Computer-Assisted Proof of Universality for Area-Preserving Maps, Amer. Math. Soc. Memoir 289, AMS, Providence, 1984.

    Google Scholar 

  4. Fefferman, C. L. and Seco, L. A.: Interval Arithmetic in Quantum Mechanics, in: Kearfott, R. B. and Kreinovich, V. (eds), Applications of Interval Computations, Kluwer, Dordrecht, 1996, pp. 145–167, http://www-risklab.erin.utoronto.ca/seco/publ.htm

    Google Scholar 

  5. Frommer, A.: Proving Conjectures by Use of Interval Arithmetic, in: Kulisch, U., Lohner, R., and Facius, A. (eds), Perspective on Enclosure Methods, Springer, Wien, 2001, pp. 1–13.

    Google Scholar 

  6. Hager, W. W.: Condition Estimates, SIAM J. Sci. Statist. Comput. 5 (1984), pp. 311–316.

    Google Scholar 

  7. Hairer, S. E., Norsett, P., and Wanner, G.: Solving Ordinary Differential Equations, Vol. 1, Springer, Berlin, 1987.

    Google Scholar 

  8. Hairer, S. E. and Wanner, G.: Solving Ordinary Differential Equations, Vol. 2, Springer, Berlin, 1991.

    Google Scholar 

  9. Hales, T. C.: The Kepler Conjecture, Manuscript (1998), math.MG/9811071, http://www.math.lsa.umich.edu/~hales/countdown/

  10. Hass, J., Hutchings, M., and Schlafli, R.: Double Bubbles Minimize, Ann. Math. 515 (2) (2000), pp. 459–515, http://math.ucdavis.edu/~hass/bubbles.html

    Google Scholar 

  11. Hoefkens, J., Berz, M., and Makino, K.: Efficient High-Order Methods for ODEs and DAEs, in: Corliss, G. et al. (eds), Automatic Differentiation: From Simulation to Optimization, Springer, New York, 2001, pp. 341–351.

    Google Scholar 

  12. Lohner, R. J.: AWA-Software for the Computation of Guaranteed Bounds for Solutions of Ordinary Initial Value Problems, ftp://ftp.iam.uni-karlsruhe.de/pub/awa/

  13. Lohner, R. J.: Einschliessung der Lösung gewöhnlicher Anfangs-und Randwertaufgaben und Anwendungen, Dissertation, Fakultät für Mathematik, Universität Karlsruhe, 1988.

  14. Mehlhorn, K. and Näher, S.: The LEDA Platform of Combinatorial and Geometric Computing, Cambridge University Press, Cambridge, 1999.

    Google Scholar 

  15. Mischaikow, K. and Mrozek, M.: Chaos in the Lorenz equations: A Computer Assisted Proof. Part II: Details, Math. Comput. 67 (1998), pp. 1023–1046.

    Google Scholar 

  16. Muhanna, R. L. and Mullen, R. L.: Uncertainty in Mechanics Problems-Interval-Based Approach, J. Engin. Mech. 127 (2001), pp. 557–566.

    Google Scholar 

  17. Nedialkov, N. S., Jackson, K. R., and Corliss, G.: Validated Solutions of Initial Value Problems for Ordinary Differential Equations, Appl. Math. Comput. 105 (1999), pp. 21–68.

    Google Scholar 

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

    Google Scholar 

  19. Neumaier, A.: Introduction to Numerical Analysis, Cambridge University Press, Cambridge, 2001.

    Google Scholar 

  20. Rage, T., Neumaier, A., and Schlier, C.: Rigorous Verification of Chaos in a Molecular Model, Phys. Rev. E. 50 (1994), pp. 2682–2688.

    Google Scholar 

  21. Rump, S. M.: INTLAB-INTerval LABoratory, in: Csendes, T. (ed.), Developments in Reliable Computing, Kluwer, Dordrecht, 1999, pp. 77–104, http://www.ti3.tu-harburg.de/ rump/intlab/index.html

    Google Scholar 

  22. Rump, S. M.: Verification Methods for Dense and Sparse Systems of Equations, in: Herzberger, J. (ed.), Topics in Validated Computations-Studies in Computational Mathematics, Elsevier, Amsterdam, 1994, pp. 63–136.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Wien, Strudhofgasse 4, A-1090, Wien, Austria

    Arnold Neumaier

Authors
  1. Arnold Neumaier
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Neumaier, A. Grand Challenges and Scientific Standards in Interval Analysis. Reliable Computing 8, 313–320 (2002). https://doi.org/10.1023/A:1016341317043

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1016341317043

Keywords

Search

Navigation