  We consider an unknown multivariate function representing a system-such as a
complex numerical simulator-taking both deterministic and uncertain inputs. Our
objective is to estimate the set of deterministic inputs leading to outputs
whose probability (with respect to the distribution of the uncertain inputs) of
belonging to a given set is less than a given threshold. This problem, which we
call Quantile Set Inversion (QSI), occurs for instance in the context of robust
(reliability-based) optimization problems, when looking for the set of
solutions that satisfy the constraints with sufficiently large probability. To
solve the QSI problem we propose a Bayesian strategy, based on Gaussian process
modeling and the Stepwise Uncertainty Reduction (SUR) principle, to
sequentially choose the points at which the function should be evaluated to
efficiently approximate the set of interest. We illustrate the performance and
interest of the proposed SUR strategy through several numerical experiments.
