Bayesian sequential design of computer experiments for quantile set inversion

Romain Ait Abdelmalek-Lomenech (1, 2) , Julien Bect (1, 2) , Vincent Chabridon (3, 2) , Emmanuel Vazquez (1, 2)

1 L2S - Laboratoire des signaux et systèmes
2 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques
3 EDF R&D PRISME - Performance, Risque Industriel, Surveillance pour la Maintenance et l’Exploitation

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.

Mots clés: Gaussian processes; Active learning; Design of computer experiments; Stepwise Uncertainty Reduction; Set inversion; Uncertainty quantification; Gaussian processes Active learning Design of computer experiments Stepwise Uncertainty Reduction Set inversion Uncertainty quantification

Domaines: Machine Learning [stat.ML]; Applications [stat.AP]

Romain Ait Abdelmalek-Lomenech, Julien Bect, Vincent Chabridon, Emmanuel Vazquez. Bayesian sequential design of computer experiments for quantile set inversion. 2023. ⟨hal-03835704v3⟩