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

• Published in: SIAM journal on scientific computing 2024-08, Vol.46 (4), p.B474-B501
• Main Authors: Lauzon, Jessica T., Cheung, Siu Wun, Shin, Yeonjong, Choi, Youngsoo, Copeland, Dylan M., Huynh, Kevin
• Format: Article
• Language: English
• Subjects: Galerkin projection; hyper-reduction; MATHEMATICS AND COMPUTING; nonlinear model reduction; reduced order modeling; sampling algorithm
• ISSN: 1064-8275; 1095-7197
• DOI: 10.1137/22M1484018

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 the points selection algorithm developed by Shin and Xiu as a hyper-reduction method. The selection algorithm was originally proposed as a stochastic collocation method for uncertainty quantification. Since the algorithm aims at maximizing a quantity $\mathcal{S}$ 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 a gappy proper orthogonal decomposition (POD) algorithm. Here, we found that using the S-OPT algorithm is shown to predict the full order solutions with higher accuracy than gappy POD especially when the number of sampling points is small, although we note that S-OPT shows slow asymptotic convergence with respect to the number of samples for some applications, e.g., Lagrangian hydrodynamics.