The simplex gradient, a popular numerical differentiation method due to its flexibility, lacks a principled method by which to construct the sample set, specifically the location of function evaluations. Such evaluations, especially from real-world systems, are often noisy and expensive to obtain, making it essential that each evaluation is carefully chosen to reduce cost and increase accuracy. This paper introduces the curvature aligned simplex gradient (CASG), which provably selects the optimal sample set under a mean squared error objective. As CASG requires function-dependent information often not available in practice, we additionally introduce a framework which exploits a history of function evaluations often present in practical applications. Our numerical results, focusing on applications in sensitivity analysis and derivative free optimization, show that our methodology significantly outperforms or matches the performance of the benchmark gradient estimator given by forward differences (FD) which is given exact function-dependent information that is not available in practice. Furthermore, our methodology is comparable to the performance of central differences (CD) that requires twice the number of function evaluations.

翻译：单纯形梯度作为一种因灵活性而广泛应用的数值微分方法，缺乏构建样本集（特别是函数评估位置）的原理性依据。现实系统中的此类评估通常带有噪声且获取成本高昂，因此必须精心选择每次评估以降低成本并提高精度。本文提出曲率对齐单纯形梯度（CASG），该算法可在均方误差目标下可证明地选择最优样本集。由于CASG需要实际应用中通常不可获得的函数相关信息，我们额外引入了一个框架，该框架利用实际应用中常见的函数评估历史记录。我们的数值实验聚焦于敏感性分析和无导数优化应用，结果表明：与使用实际中不可获取的精确函数相关信息的正向差分（FD）基准梯度估计器相比，我们的方法在性能上显著超越或相当。此外，我们的方法与需要两倍函数评估次数的中心差分（CD）相比也具有可比性。