Forward and Backward Uncertainty Propagation for Discontinuous System Response Using the Padé-legendre Method

The Padé-Legendre method has been introduced as an effective approach to characterize uncertainties in the presence of strongly non-linear or discontinuous system responses—thus, it supports forward propagation. The method is based on the construction of a ratio of polynomials that approximate the available data. Two criteria for the choice of the best approximant are considered and an optimization approach is proposed. Moreover, the approach is applied in a case in which the discontinuity in the system response is due to limited data, to demonstrate how the successive addition of data transforms the rational approximant into a simple polynomial interpolant (the denominator becomes a constant). Finally, the present method is applied to estimate an input parameter characterized by a sharp discontinuity, using Bayesian inference starting from observations of the system response—thus, it also supports backward propagation.