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S-OPT: A Points Selection Algorithm for Hyper-Reduction in Reduced Order Models

Jessica T. Lauzon, Siu Wun Cheung, Yeonjong Shin, Youngsoo Choi, Dylan Matthew Copeland, Kevin Huynh

TL;DR

This work tackles the computational bottleneck of projection-based reduced order models by integrating a hyper-reduction strategy based on the S-OPT point selection algorithm. S-OPT seeks index sets that maximize the combined column orthogonality and determinant via the metric $\mathcal{S}(\boldsymbol{Z}^T \bm{Q})$, delivering more stable and accurate reduced solutions than oversampled DEIM with fewer indices. The method is demonstrated on four challenging problems (1D Burgers, 2D laminar airfoil, 2D Gresho vortex, 3D Sedov blast), using both linear subspace (LS-ROM) and nonlinear manifold (NM-ROM) PROMs, and shows improved accuracy and stability, albeit with some increased per-iteration cost due to wider neighbor requirements. The findings suggest that S-OPT offers a practical path to reliable, efficient hyper-reduction in PROMs for complex, advection-dominated and nonlinear systems, with potential for broader adoption in design, optimization, and uncertainty quantification contexts.

Abstract

While projection-based reduced order models can reduce the dimension of full order solutions, the resulting reduced models may still contain terms that scale with the full order dimension. Hyper-reduction techniques are sampling-based methods that further reduce this computational complexity by approximating such terms with a much smaller dimension. The goal of this work is to introduce a points selection algorithm developed by Shin and Xiu [SIAM J. Sci. Comput., 38 (2016), pp. A385--A411], as a hyper-reduction method. The selection algorithm is originally proposed as a stochastic collocation method for uncertainty quantification. Since the algorithm aims at maximizing a quantity S that measures both the column orthogonality and the determinant, we refer to the algorithm as S-OPT. Numerical examples are provided to demonstrate the performance of S-OPT and to compare its performance with an over-sampled Discrete Empirical Interpolation (DEIM) algorithm. We found that using the S-OPT algorithm is shown to predict the full order solutions with higher accuracy for a given number of indices.

S-OPT: A Points Selection Algorithm for Hyper-Reduction in Reduced Order Models

TL;DR

This work tackles the computational bottleneck of projection-based reduced order models by integrating a hyper-reduction strategy based on the S-OPT point selection algorithm. S-OPT seeks index sets that maximize the combined column orthogonality and determinant via the metric , delivering more stable and accurate reduced solutions than oversampled DEIM with fewer indices. The method is demonstrated on four challenging problems (1D Burgers, 2D laminar airfoil, 2D Gresho vortex, 3D Sedov blast), using both linear subspace (LS-ROM) and nonlinear manifold (NM-ROM) PROMs, and shows improved accuracy and stability, albeit with some increased per-iteration cost due to wider neighbor requirements. The findings suggest that S-OPT offers a practical path to reliable, efficient hyper-reduction in PROMs for complex, advection-dominated and nonlinear systems, with potential for broader adoption in design, optimization, and uncertainty quantification contexts.

Abstract

While projection-based reduced order models can reduce the dimension of full order solutions, the resulting reduced models may still contain terms that scale with the full order dimension. Hyper-reduction techniques are sampling-based methods that further reduce this computational complexity by approximating such terms with a much smaller dimension. The goal of this work is to introduce a points selection algorithm developed by Shin and Xiu [SIAM J. Sci. Comput., 38 (2016), pp. A385--A411], as a hyper-reduction method. The selection algorithm is originally proposed as a stochastic collocation method for uncertainty quantification. Since the algorithm aims at maximizing a quantity S that measures both the column orthogonality and the determinant, we refer to the algorithm as S-OPT. Numerical examples are provided to demonstrate the performance of S-OPT and to compare its performance with an over-sampled Discrete Empirical Interpolation (DEIM) algorithm. We found that using the S-OPT algorithm is shown to predict the full order solutions with higher accuracy for a given number of indices.
Paper Structure (18 sections, 1 theorem, 47 equations, 15 figures, 2 algorithms)

This paper contains 18 sections, 1 theorem, 47 equations, 15 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Paper organization
  3. Problem Formulation
  4. Full Order Model
  5. Projection-Based Reduced Order Model
  6. Linear Subspace Reduced Order Model
  7. Nonlinear Manifold Reduced Order Model
  8. Hyper-Reduction and Sampling
  9. Sampling Algorithms
  10. Oversampled DEIM Algorithm
  11. S-OPT: A Quasi-Optimal Points Selection Algorithm
  12. Main results
  13. 1D Burgers Equation
  14. 2D Laminar Airfoil
  15. Lagrangian hydrodynamics
  16. ...and 3 more sections

Key Result

Theorem 3.1

\newlabelthm:error-estimate0 Let $\boldsymbol{Z} \in \mathbb{R}^{N\times n}$ be a sampling matrix and $\bm{M} \in \mathbb{R}^{N\times p}$ be a basis matrix of full rank with $p \le n \le N$. Let $\tilde{\bm{a}}({\boldsymbol{Z}}) = \mathop{\mathrm{\arg\!\min}}\limits_{\bm{a}} \|\boldsymbol{Z}^{\top where $\bm{M}=\bm{QR}$ is a QR factorization of $\bm{M}$, $\text{proj}_{\bm{M}^\perp} \bm{b} := (\bm

Figures (15)

  • Figure 1: Initial condition (left) and final-time solution (right) for 1D Burgers equation.
  • Figure 2: Maximum-in-time relative $L^2$ error with varying number of sampling indices in LS-ROM (left) and NM-ROM (right) for 1D Burgers equation.
  • Figure 3: Selected nodes in LS-ROM for 1D Burgers equation.
  • Figure 4: Selected nodes in NM-ROM for 1D Burgers equation.
  • Figure 5: First four POD modes for the density variable for the laminar airfoil.
  • ...and 10 more figures

Theorems & Definitions (4)

  • Theorem 3.1
  • Proof 1
  • Remark 3.2
  • Remark 3.3