AN EFFICIENT METHOD FOR UNCERTAINTY PROPAGATION USING FUZZY SETS

In the field of uncertainty quantification (UQ), propagation of uncertainty is one of the most important tasks, as it is essential to almost all UQ analysis. While many numerical methods have been developed for uncertainty propagation in probabilistic framework, much less has been discussed in the nonprobabilistic framework, which is important whenever one does not possess sufficient data or knowledge of the underlying systems. In this paper, we focus on the use of fuzzy sets to model uncertainty and their propagation through physical systems. In particular, we develop a numerical strategy that can efficiently propagate fuzzy sets in complex systems. The method utilizes an accurate approximation model for the solution over the support of the input fuzzy sets and then retrieves the output fuzzy set information via the approximation model. By doing so the method becomes highly efficient, as the only simulation cost is in the construction of the approximation model. In particular, we discuss the use of orthogonal polynomials for the approximation model. This is very similar to the probabilistic method of generalized polynomial chaos. However, no probability is assumed and the method remains nonprobabilistic. We prove that the fuzzy sets propagated by the approximation model can be accurate and establish the convergence via a new distance measure between fuzzy sets. Several numerical examples are provided to demonstrate the effectiveness of the method and verify the convergence analysis.