Efficient fuzzy arithmetic for nonlinear functions of modest dimension using sparse grids

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. We consider the difficult case of the objective function being an expensive to compute multivariate function of modest dimension (say d up to 16) where only real-valued evaluations of f are permitted. This often poses a difficult problem due to non-applicability of common fuzzy arithmetic algorithms, severe overestimation, or very high computational complexity. Our approach is composed 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.