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Guaranteeing the Homotopy Type of a Set Defined by Non-Linear Inequalities

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Reliable Computing

Abstract

This paper provides an effective method to create an abstract simplicial complex homotopy equivalent to a given set \(\mathbb{S}\) described by non-linear inequalities (polynomial or not). To our knowledge, no other numerical algorithm is able to deal with this type of problem. The proposed approach divides \(\mathbb{S}\) into subsets that have been proven to be contractible using interval arithmetic. The method is close to Čech cohomology and uses the nerve theorem. Some examples illustrate the principle of the approach. This algorithm has been implemented.

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References

  1. Basu, S., Pollack, R., and Roy, M.-F.: Computing the First Betti Number and the Connected Components of Semi-Algebraic Sets, in: Proceedings of STOC 2005.

  2. Collins, G. E.: Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition, Lecture Notes in Computer Science 33, Springer-Verlag, Berlin, 1975, pages 134–183.

  3. Delanoue, N., Jaulin, L., and Cottenceau, B.: Using Interval Arithmetic to Prove That a Set Is Path-Connected, Theoretical Computer Science, Special Issue: Real Numbers and Computers, 2004.

  4. Edelsbrunner, H. and Shah, N. R.: Triangulating Topological Spaces, in: Proceedings of the Tenth Annual Symposium on Computational Geometry, United States Publication, Stony Brook, New York, 1994, pp. 285–292.

  5. Jaulin L., Kieffer M., Didrit O., Walter E. (2001). Applied Interval Analysis. Springer-Verlag, London

    MATH  Google Scholar 

  6. Jaulin L., Walter E. (1993) Set Inversion via Interval Analysis for Non-Linear Bounded-Error Estimation. Automatica 29 (4): pp. 1053–1064

    Article  MATH  MathSciNet  Google Scholar 

  7. Hart, J. C. and Stander, B. T.: Guaranteeing the Topology of an Implicit Surface Polygonization for Interactive Modeling, in: Proc. SIGGRAPH 97, 1997, pp. 279–286.

  8. Hatcher, A.: Algebraic Topology, Cambridge University Press, 2002.

  9. Homotopy via Interval Analysis, http://www.istia.univ-angers.fr/~delanoue/.

  10. Lamure, H. and Michelucci, D.: Solving Geometric Constraints by Homotopy, in: Symposium on Solid Modeling and Applications, 1995.

  11. Maunder, C. R. F.: Algebraic Topology, Van Nostrand Reinhold, London, 1970.

  12. Milnor, J. W.: Morse Theory, Princeton University Press, 1963.

  13. Moore, R. E.: Interval Analysis, Prentice Hall, 1966.

  14. Neumaier, A.: Interval Methods for Systems of Equations, in: Encyclopedia of Mathematics and Its Applications 37, Cambridge University Press, Cambridge, 1990.

  15. Sergeraert, F.: http://www-fourier.ujf-grenoble.fr/~sergerar/.

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Authors and Affiliations

  1. LISA, University of Angers, 62 av. Notre Dame du Lac, Angers, 49000, France

    Nicolas Delanoue

  2. E3I2, ENSIETA, 2 rue François Verny, Brest Cedex 09, 29806, France

    Luc Jaulin

  3. LISA, University of Angers, 62 av. Notre Dame du Lac, Angers, 49000, France

    Bertrand Cottenceau

Authors
  1. Nicolas Delanoue
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  2. Luc Jaulin
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  3. Bertrand Cottenceau
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Correspondence to Nicolas Delanoue.

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Delanoue, N., Jaulin, L. & Cottenceau, B. Guaranteeing the Homotopy Type of a Set Defined by Non-Linear Inequalities. Reliable Comput 13, 381–398 (2007). https://doi.org/10.1007/s11155-007-9043-8

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  • DOI: https://doi.org/10.1007/s11155-007-9043-8

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