The univariate dimension reduction (UDR) method stands as a way to estimate the statistical moments of the output that is effective in a large class of uncertainty quantification (UQ) problems. UDR's fundamental strategy is to approximate the original function using univariate functions so that the UQ cost only scales linearly with the dimension of the problem. Nonetheless, UDR's effectiveness can diminish when uncertain inputs have high variance, particularly when assessing the output's second and higher-order statistical moments. This paper proposes a new method, gradient-enhanced univariate dimension reduction (GUDR), that enhances the accuracy of UDR by incorporating univariate gradient function terms into the UDR approximation function. Theoretical results indicate that the GUDR approximation is expected to be one order more accurate than UDR in approximating the original function, and it is expected to generate more accurate results in computing the output's second and higher-order statistical moments. Our proposed method uses a computational graph transformation strategy to efficiently evaluate the GUDR approximation function on tensor-grid quadrature inputs, and use the tensor-grid input-output data to compute the statistical moments of the output. With an efficient automatic differentiation method to compute the gradients, our method preserves UDR's linear scaling of computation time with problem dimension. Numerical results show that the GUDR is more accurate than UDR in estimating the standard deviation of the output and has a performance comparable to the method of moments using a third-order Taylor series expansion.

翻译：单变量降维（UDR）方法是估计输出统计矩的有效手段，适用于大量不确定性量化（UQ）问题。其基本策略是利用单变量函数逼近原始函数，从而使UQ成本仅随问题维度线性增长。然而，当不确定输入具有高方差时，UDR的有效性可能降低，尤其在评估输出的二阶及更高阶统计矩时。本文提出一种新方法——梯度增强单变量降维（GUDR），通过将单变量梯度函数项纳入UDR逼近函数来提升精度。理论结果表明，GUDR逼近在近似原始函数时预计比UDR高一个精度阶，并且在计算输出的二阶及更高阶统计矩时能产生更精确的结果。本文方法采用计算图变换策略，高效评估张量网格求积输入下的GUDR逼近函数，并利用张量网格输入-输出数据计算输出的统计矩。借助自动微分方法高效计算梯度，本方法保持了UDR计算时间随问题维度线性增长的特性。数值结果表明，在估计输出的标准差时，GUDR比UDR更精确，且性能与使用三阶泰勒展开的矩方法相当。