Computing cross-partial derivatives using fewer model runs is relevant in modeling, such as stochastic approximation, derivative-based ANOVA, exploring complex models, and active subspaces. This paper introduces surrogates of all the cross-partial derivatives of functions by evaluating such functions at $N$ randomized points and using a set of $L$ constraints. Randomized points rely on independent, central, and symmetric variables. The associated estimators, based on $NL$ model runs, reach the optimal rates of convergence (i.e., $\mathcal{O}(N^{-1})$), and the biases of our approximations do not suffer from the curse of dimensionality for a wide class of functions. Such results are used for i) computing the main and upper-bounds of sensitivity indices, and ii) deriving emulators of simulators or surrogates of functions thanks to the derivative-based ANOVA. Simulations are presented to show the accuracy of our emulators and estimators of sensitivity indices. The plug-in estimates of indices using the U-statistics of one sample are numerically much stable.

翻译：利用更少的模型运行次数计算交叉偏导数在随机逼近、基于导数的方差分析、复杂模型探索及主动子空间等建模场景中具有重要意义。本文通过评估函数在$N$个随机化点处的取值并利用$L$个约束条件，构建了函数所有交叉偏导数的代理模型。随机化点依赖于独立、中心化且对称分布的随机变量。基于$NL$次模型运行构建的估计器达到了最优收敛速率（即$\mathcal{O}(N^{-1})$），且对于广泛函数类，其近似偏差不受维度灾难影响。相关成果被应用于：i) 计算灵敏度指标的主效应与上界估计；ii) 借助基于导数的方差分析推导模拟器的代理模型或函数的替代模型。数值模拟展示了灵敏度指标估计器与代理模型的精度。基于单样本U统计量的指标插件估计在数值上具有更高的稳定性。