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：我们考虑一个代表系统的未知多元函数（如复杂的数值模拟器），该函数同时包含确定性输入和不确定输入。我们的目标是估计确定性输入集合，使得系统输出（基于不确定输入的分布）属于给定集合的概率小于某个阈值。我们将此问题称为分位数集逆推（Quantile Set Inversion, QSI），它出现在例如稳健（基于可靠性的）优化问题中，用于寻找能够以足够大概率满足约束条件的解集。为解决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）原则的贝叶斯策略，以序贯方式选择函数评估点，从而有效逼近目标集合。通过多个数值实验，我们展示了所提SUR策略的性能和优势。