Reliable Computing

, Volume 8, Issue 4, pp 313–320 | Cite as

Grand Challenges and Scientific Standards in Interval Analysis

  • Arnold Neumaier
Letter to the Editor


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.


Mathematical Modeling Computational Mathematic Industrial Mathematic Interval Analysis Grand Challenge 
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.


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  1. 1.
    Alefeld, G. and Herzberger, J.: Introduction to Interval Computations, Academic Press, New York, 1983.Google Scholar
  2. 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, coconut/StArt.htmlGoogle Scholar
  3. 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. 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, Scholar
  5. 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. 6.
    Hager, W. W.: Condition Estimates, SIAM J. Sci. Statist. Comput. 5 (1984), pp. 311–316.Google Scholar
  7. 7.
    Hairer, S. E., Norsett, P., and Wanner, G.: Solving Ordinary Differential Equations, Vol. 1, Springer, Berlin, 1987.Google Scholar
  8. 8.
    Hairer, S. E. and Wanner, G.: Solving Ordinary Differential Equations, Vol. 2, Springer, Berlin, 1991.Google Scholar
  9. 9.
    Hales, T. C.: The Kepler Conjecture, Manuscript (1998), math.MG/9811071, Scholar
  10. 10.
    Hass, J., Hutchings, M., and Schlafli, R.: Double Bubbles Minimize, Ann. Math. 515 (2) (2000), pp. 459–515, Scholar
  11. 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. 12.
    Lohner, R. J.: AWA-Software for the Computation of Guaranteed Bounds for Solutions of Ordinary Initial Value Problems, Scholar
  13. 13.
    Lohner, R. J.: Einschliessung der Lösung gewöhnlicher Anfangs-und Randwertaufgaben und Anwendungen, Dissertation, Fakultät für Mathematik, Universität Karlsruhe, 1988.Google Scholar
  14. 14.
    Mehlhorn, K. and Näher, S.: The LEDA Platform of Combinatorial and Geometric Computing, Cambridge University Press, Cambridge, 1999.Google Scholar
  15. 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. 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. 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. 18.
    Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.Google Scholar
  19. 19.
    Neumaier, A.: Introduction to Numerical Analysis, Cambridge University Press, Cambridge, 2001.Google Scholar
  20. 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. 21.
    Rump, S. M.: INTLAB-INTerval LABoratory, in: Csendes, T. (ed.), Developments in Reliable Computing, Kluwer, Dordrecht, 1999, pp. 77–104, rump/intlab/index.htmlGoogle Scholar
  22. 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

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Arnold Neumaier
    • 1
  1. 1.Institut für MathematikUniversität WienWienAustria

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