Advertisement

Reliable Computing

, Volume 12, Issue 3, pp 203–223 | Cite as

Geometric Constructions with Discretized Random Variables

  • Hans-Peter SchröckerEmail author
  • Johannes Wallner
Article

Abstract

We generalize the DEnv (Distribution envelope determination) method for bounding the result of arithmetic operations on random variables with unknown dependence to higher-dimensional settings. In order to minimize both the influence of the coordinate frame and information loss we suggest a nested thicket representation for random variables and a corresponding intersection algorithm.

Keywords

Mathematical Modeling Computational Mathematic Industrial Mathematic Arithmetic Operation Coordinate Frame 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berleant, D. and Goodman-Strauss, C.: Bounding the Results of Arithmetic Operations on Random Variables of Unknown Dependency Using Intervals, Reliable Computing 4 (2) (1998), pp. 147–165.CrossRefMathSciNetGoogle Scholar
  2. 2.
    Berleant, D., Xie, L., and Zhang, J.: Stattool: A Tool for Distribution Envelope Determination (DEnv), an Interval-Based Algorithm for Arithmetic Operations on Random Variables, Reliable Computing 9 (2) (2003), pp. 91–108.CrossRefGoogle Scholar
  3. 3.
    Berleant, D. and Zhang, J.: Representation and Problem Solving with Distribution Envelope Determination, Reliability Engineering and System Safety 85 (1–3) (2004), pp. 153–168.Google Scholar
  4. 4.
    Berleant, D. and Zhang, J.: Using Pearson Correlation to Improve Envelopes around the Distributions of Functions, Reliable Computing 10 (2) (2004), pp. 139–161.CrossRefMathSciNetGoogle Scholar
  5. 5.
    Hu, S.-M. and Wallner, J.: Error Propagation through Geometric Transformations, J. Geom. Graphics 8 (2) (2004), pp. 171–183.MathSciNetGoogle Scholar
  6. 6.
    Karloff, H.: Linear Programming, Birkhäuser, Boston, 1991.Google Scholar
  7. 7.
    Pottmann, H., Odehnal, B., Peternell, M., Wallner, J., and Ait Haddou, R.: On Optimal Tolerancing in Computer-Aided Design, in: Martin, R. and Wang, W. (eds), Geometric Modeling and Processing 2000, IEEE Computer Society, Los Alamitos, 2000, pp. 347–363.Google Scholar
  8. 8.
    Regan, H. A., Ferson, S., and Berleant, D.: Equivalence of Methods for Uncertainty Propagation of Real-Valued Random Variables, Internat. J. Approx. Reason. 36 (2004), pp. 1–30.MathSciNetGoogle Scholar
  9. 9.
    Requicha, A. A. G.: Towards a Theory of Geometric Tolerancing, Int. J. of Robotics Research 2 (1983), pp. 45–60.Google Scholar
  10. 10.
    Walley, P.: Measures of Uncertainty in Expert Systems, Artificial Intelligence 83 (1996), pp. 1–58.CrossRefMathSciNetGoogle Scholar
  11. 11.
    Wallner, J., Krasauskas, R., and Pottmann, H.: Error Propagation in Geometric Constructions, Computer-Aided Design 32 (2000), pp. 631–641.CrossRefGoogle Scholar
  12. 12.
    Wallner, J., Schröcker, H.-P., and Hu, S.-M.: Tolerances in Geometric Constraint Problems, Reliable Computing 11 (3) (2005), pp. 235–251.CrossRefMathSciNetGoogle Scholar
  13. 13.
    Williamson, R. C. and Downs, T.: Probabilistic Arithmetic. I. Numerical Methods for Calculating Convolutions and Dependency Bounds, Internat. J. Approx. Reason. 4 (2) (1990), pp. 89–158.MathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Institut für Technische Mathematik, Geometrie und BauinformatikUniversität InnsbruckInnsbruckAustria
  2. 2.Institut für Diskrete Mathematik und GeometrieTechnische Universität WienWienAustria

Personalised recommendations