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We consider an unknown multivariate function representing a system-such as acomplex numerical simulator-taking both deterministic and uncertain inputs. Ourobjective is to estimate the set of deterministic inputs leading to outputswhose probability (with respect to the distribution of the uncertain inputs) ofbelonging to a given set is less than a given threshold. This problem, which wecall Quantile Set Inversion (QSI), occurs for instance in the context of robust(reliability-based) optimization problems, when looking for the set ofsolutions that satisfy the constraints with sufficiently large probability. Tosolve the QSI problem, we propose a Bayesian strategy based on Gaussian processmodeling and the Stepwise Uncertainty Reduction (SUR) principle, tosequentially choose the points at which the function should be evaluated toefficiently approximate the set of interest. We illustrate the performance andinterest of the proposed SUR strategy through several numerical experiments.

Bayesian sequential design of computer experiments for quantile set inversion