  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.
