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

Bingran Wang, Nicholas C. Orndorff, John T. Hwang

TL;DR

This paper introduces a new framework for generating a tailored, partially tensor-structured quadrature rule to use with the graph-accelerated NIPC method, and shows that this quadrature rule outperforms the full-grid Gauss quadrature and the designed quadrature methods in both of the test problems.

Abstract

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.

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

TL;DR

This paper introduces a new framework for generating a tailored, partially tensor-structured quadrature rule to use with the graph-accelerated NIPC method, and shows that this quadrature rule outperforms the full-grid Gauss quadrature and the designed quadrature methods in both of the test problems.

Abstract

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.
Paper Structure (16 sections, 53 equations, 9 figures, 6 tables, 1 algorithm)

This paper contains 16 sections, 53 equations, 9 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Non-intrusive polynomial chaos
  4. Numerical quadrature rules
  5. Gauss quadrature rule
  6. Designed quadrature method
  7. Other popular numerical quadrature methods
  8. Computational graph transformations for efficient tensor-grid evaluations
  9. Methodology
  10. Generation of tensor-structured quadrature rules
  11. Tensor-grid evaluations with AMTC
  12. Finding a desirable tensor structure option
  13. Numerical Results
  14. Laser-beam powered UAV multidisciplinary model
  15. Air-taxi trajectory design multidisciplinary model
  16. ...and 1 more sections

Figures (9)

  • Figure 1: Computational graphs with data size for full-grid input points evaluation on $f = cos(u_1) + exp(-u_2)$wang2023accelerating
  • Figure 2: Visualizations of quadrature points with different tensorial structure options
  • Figure 3: Computational graphs after AMTC on fully and partially tensor structured input points
  • Figure 4: A flow chart showing how the partially tensor-structured quadrature rule can be used with graph-accelerated NIPC to solve an UQ problem.
  • Figure 5: A circular cruise mission around a ground station
  • ...and 4 more figures