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.

翻译：我们考虑一个未知多元函数所表征的系统（如复杂数值模拟器），该系统同时具有确定性与不确定性两类输入。目标在于估计使输出值（依据不确定性输入的概率分布）落入给定集合的概率低于特定阈值的确定性输入集。该问题被称为分位数集反演（Quantile Set Inversion, QSI），常见于鲁棒（基于可靠性的）优化问题中，例如需找出以足够大概率满足约束条件的解集。为求解QSI问题，我们提出一种基于高斯过程建模与逐步不确定性缩减（Stepwise Uncertainty Reduction, SUR）原理的贝叶斯策略，通过序贯选择函数评估点以高效逼近目标集合。通过多项数值实验，我们展示了所提出的SUR策略的性能与优势。