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：我们考虑一个未知的多元函数，它代表一个系统——例如一个复杂的数值模拟器——同时接受确定性和不确定性输入。我们的目标是估计一组确定性输入，使得输出属于给定集合的概率（相对于不确定性输入的分布）低于给定阈值。这个问题，我们称之为分位数集反演（QSI），例如出现在鲁棒（基于可靠性的）优化问题中，当寻找以足够大概率满足约束的解集时。为解决QSI问题，我们提出一种基于高斯过程建模和逐步不确定性降低（SUR）原则的贝叶斯策略，以序贯选择函数应被评估的点，从而高效逼近目标集。我们通过若干数值实验说明了所提出的SUR策略的性能和意义。