  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.
