Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Extension of graph-accelerated non-intrusive polynomial chaos to high-dimensional uncertainty quantification through the active subspace method

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

TL;DR

The paper tackles high-dimensional uncertainty quantification for multidisciplinary systems by coupling active-subspace (AS) dimension reduction with integration-based non-intrusive polynomial chaos (NIPC). It introduces AS-NIPC to generate PCE bases in the active subspace and a quadrature rule in original input space, and extends this to AS-AMTC to create tensor-structured quadrature rules that exploit computational-graph sparsity via AMTC. Numerical results on a 7D piston model and an 81D air-taxi trajectory model show AS-NIPC achieves up to 30% lower relative error and AS-AMTC up to 80% lower relative error compared with baseline methods, with significant reductions in required model evaluations. Overall, the approach enables scalable, accurate UQ for high-dimensional, multidisciplinary problems by integrating dimension reduction, integration-based PCE, and graph-transformation acceleration.

Abstract

The recently introduced graph-accelerated non-intrusive polynomial chaos (NIPC) method has shown effectiveness in solving a broad range of uncertainty quantification (UQ) problems with multidisciplinary systems. It uses integration-based NIPC to solve the UQ problem and generates the quadrature rule in a desired tensor structure, so that the model evaluations can be efficiently accelerated through the computational graph transformation method, Accelerated Model evaluations on Tensor grids using Computational graph transformations (AMTC). This method is efficient when the model's computational graph possesses a certain type of sparsity which is commonly the case in multidisciplinary problems. However, it faces limitations in high-dimensional cases due to the curse of dimensionality. To broaden its applicability in high-dimensional UQ problems, we propose AS-AMTC, which integrates the AMTC approach with the active subspace (AS) method, a widely-used dimension reduction technique. In developing this new method, we have also developed AS-NIPC, linking integration-based NIPC with the AS method for solving high-dimensional UQ problems. AS-AMTC incorporates rigorous approaches to generate orthogonal polynomial basis functions for lower-dimensional active variables and efficient quadrature rules to estimate their coefficients. The AS-AMTC method extends AS-NIPC by generating a quadrature rule with a desired tensor structure. This allows the AMTC method to exploit the computational graph sparsity, leading to efficient model evaluations. In an 81-dimensional UQ problem derived from an air-taxi trajectory optimization scenario, AS-NIPC demonstrates a 30% decrease in relative error compared to the existing methods, while AS-AMTC achieves an 80% reduction.

Extension of graph-accelerated non-intrusive polynomial chaos to high-dimensional uncertainty quantification through the active subspace method

TL;DR

The paper tackles high-dimensional uncertainty quantification for multidisciplinary systems by coupling active-subspace (AS) dimension reduction with integration-based non-intrusive polynomial chaos (NIPC). It introduces AS-NIPC to generate PCE bases in the active subspace and a quadrature rule in original input space, and extends this to AS-AMTC to create tensor-structured quadrature rules that exploit computational-graph sparsity via AMTC. Numerical results on a 7D piston model and an 81D air-taxi trajectory model show AS-NIPC achieves up to 30% lower relative error and AS-AMTC up to 80% lower relative error compared with baseline methods, with significant reductions in required model evaluations. Overall, the approach enables scalable, accurate UQ for high-dimensional, multidisciplinary problems by integrating dimension reduction, integration-based PCE, and graph-transformation acceleration.

Abstract

The recently introduced graph-accelerated non-intrusive polynomial chaos (NIPC) method has shown effectiveness in solving a broad range of uncertainty quantification (UQ) problems with multidisciplinary systems. It uses integration-based NIPC to solve the UQ problem and generates the quadrature rule in a desired tensor structure, so that the model evaluations can be efficiently accelerated through the computational graph transformation method, Accelerated Model evaluations on Tensor grids using Computational graph transformations (AMTC). This method is efficient when the model's computational graph possesses a certain type of sparsity which is commonly the case in multidisciplinary problems. However, it faces limitations in high-dimensional cases due to the curse of dimensionality. To broaden its applicability in high-dimensional UQ problems, we propose AS-AMTC, which integrates the AMTC approach with the active subspace (AS) method, a widely-used dimension reduction technique. In developing this new method, we have also developed AS-NIPC, linking integration-based NIPC with the AS method for solving high-dimensional UQ problems. AS-AMTC incorporates rigorous approaches to generate orthogonal polynomial basis functions for lower-dimensional active variables and efficient quadrature rules to estimate their coefficients. The AS-AMTC method extends AS-NIPC by generating a quadrature rule with a desired tensor structure. This allows the AMTC method to exploit the computational graph sparsity, leading to efficient model evaluations. In an 81-dimensional UQ problem derived from an air-taxi trajectory optimization scenario, AS-NIPC demonstrates a 30% decrease in relative error compared to the existing methods, while AS-AMTC achieves an 80% reduction.
Paper Structure (18 sections, 51 equations, 7 figures, 6 tables)

This paper contains 18 sections, 51 equations, 7 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Non-intrusive polynomial chaos
  4. Computational graph transformations to accelerate non-intrusive polynomial chaos
  5. Active subspace
  6. Methodology
  7. AS-based NIPC
  8. Generating PCE basis functions
  9. Generating an efficient quadrature rule
  10. Special case: Gaussian uncertain inputs
  11. AS-NIPC algorithm
  12. AS based NIPC with AMTC
  13. AS-AMTC algorithm
  14. Numerical Results
  15. 7-dimensional analytical piston simulation problem
  16. ...and 3 more sections

Figures (7)

  • Figure 1: Computational graphs with data size for full-grid input points evaluation on $f = cos(u_1) + exp(-u_2)$
  • Figure 2: Demonstration for the AMTC implementation in CSDL
  • Figure 3: UQ results on the piston problem
  • Figure 4: Representation of the lift-plus-cruise electric air taxi orndorff2023air
  • Figure 5: Control inputs with confidence intervals
  • ...and 2 more figures