  Methods such as non-intrusive polynomial chaos (NIPC), and stochastic
collocation are frequently used for uncertainty propagation problems.
Particularly for low-dimensional problems, these methods often use a
tensor-product grid for sampling the space of uncertain inputs. A limitation of
this approach is that it encounters a significant challenge: the number of
sample points grows exponentially with the increase of uncertain inputs.
Current strategies to mitigate computational costs abandon the tensor structure
of sampling points, with the aim of reducing their overall count.
Contrastingly, our investigation reveals that preserving the tensor structure
of sample points can offer distinct advantages in specific scenarios. Notably,
by manipulating the computational graph of the targeted model, it is feasible
to avoid redundant evaluations at the operation level to significantly reduce
the model evaluation cost on tensor-grid inputs. This paper presents a
pioneering method: Accelerated Model Evaluations on Tensor grids using
Computational graph transformations (AMTC). The core premise of AMTC lies in
the strategic modification of the computational graph of the target model to
algorithmically remove the repeated evaluations on the operation level. We
implemented the AMTC method within the compiler of a new modeling language
called the Computational System Design Language (CSDL). We demonstrate the
effectiveness of AMTC by using it with the full-grid NIPC method to solve four
low-dimensional UQ problems involving an analytical piston model, a
multidisciplinary unmanned aerial vehicle design model, a multi-point air taxi
mission analysis model, and a single-disciplinary rotor model, respectively.
For three of the four test problems, AMTC reduces the model evaluation cost by
between 50% and 90%, making the full-grid NIPC the most efficacious method to
use among the UQ methods implemented.
