  Global sensitivity analysis aims at determining which uncertain input
parameters of a computational model primarily drives the variance of the output
quantities of interest. Sobol' indices are now routinely applied in this
context when the input parameters are modelled by classical probability theory
using random variables. In many practical applications however, input
parameters are affected by both aleatory and epistemic (so-called polymorphic)
uncertainty, for which imprecise probability representations have become
popular in the last decade. In this paper, we consider that the uncertain input
parameters are modelled by parametric probability boxes (p-boxes). We propose
interval-valued (so-called imprecise) Sobol' indices as an extension of their
classical definition. An original algorithm based on the concepts of augmented
space, isoprobabilistic transforms and sparse polynomial chaos expansions is
devised to allow for the computation of these imprecise Sobol' indices at
extremely low cost. In particular, phantoms points are introduced to build an
experimental design in the augmented space (necessary for the calibration of
the sparse PCE) which leads to a smart reuse of runs of the original
computational model. The approach is illustrated on three analytical and
engineering examples which allows one to validate the proposed algorithms
against brute-force double-loop Monte Carlo simulation.
