A Stepwise Uncertainty Reduction (SUR) strategy aims at constructing a sequence X1(f),X2(f), . . . of evaluation points of a function f : Rd → R in such a way that the residual uncertainty about a quantity of interest S(f) given the information provided by the evaluation results is small. In Bect, Ginsbourger, Li, Picheny and Vazquez, Statistics and Computing, 2011, several SUR approaches have been shown to be particularly efficient for the problem of estimating the volume of an excursion set of a function f above a threshold. Here, we build upon these results and we present fast implementations of some SUR strategies, which are based on two ideas. The first idea is to take advantage of update formulas for kriging. The second idea is to derive closed-form expressions for some integrals that appear in the SUR criteria. We are able to demonstrate significant speed-ups and we illustrate our algorithms on a nuclear safety application.