Graph-accelerated non-intrusive polynomial chaos expansion using partially tensor-structured quadrature rules for uncertainty quantification

Recently, the graph-accelerated non-intrusive polynomial chaos (NIPC) method has been proposed for solving uncertainty quantification (UQ) problems. This method leverages the full-grid integration-based NIPC method to address UQ problems while employing the computational graph transformation approach, AMTC, to accelerate the tensor-grid evaluations. This method exhibits remarkable efficacy on a broad range of low-dimensional (three dimensions or less) UQ problems featuring multidisciplinary models. However, it often does not scale well with problem dimensions due to the exponential increase in the number of quadrature points when using the full-grid quadrature rule. To expand the applicability of this method to a broader range of UQ problems, this paper introduces a new framework for generating a tailored, partially tensor-structured quadrature rule to use with the graph-accelerated NIPC method. This quadrature rule, generated through the designed quadrature approach, possesses a tensor structure that is tailored for the computational model. The selection of the tensor structure is guided by an analysis of the computational graph, ensuring that the quadrature rule effectively capitalizes on the sparsity within the computational graph when paired with the AMTC method. This method has been tested on one 4D and one 6D UQ problem, both originating from aircraft design scenarios and featuring multidisciplinary models. Numerical results show that, when using with graph-accelerated NIPC method, our approach generates a partially tensor-structured quadrature rule that outperforms the full-grid Gauss quadrature and the designed quadrature methods (more than 40% reduction in computational costs) in both of the test problems.