Computing expensive multivariate functions of fuzzy numbers using sparse grids

  • Authors:

  • Andreas Klimke;Barbara Wohlmuth

  • Affiliations:

  • Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany;Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2005

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Abstract

Fuzzy arithmetic provides a powerful tool to introduce uncertainty into mathematical models. With Zadeh's extension principle, one can obtain a fuzzy extension of any objective function. Computing expensive multivariate functions of fuzzy numbers, however, often poses a difficult problem due to non-applicability of common fuzzy arithmetic algorithms, severe overestimation, or very high computational complexity. This paper proposes a new approach based on sparse grids, consisting of two parts: First, we compute a surrogate function using sparse grid interpolation. Second, we perform the fuzzy-valued evaluation of the surrogate function by a suitable implementation of the extension principle based on real or interval arithmetic. The new approach gives accurate results and requires only few function evaluations.