SNS: A Solution-based Nonlinear Subspace method for time-dependent model order reduction

TL;DR

The paper introduces SNS, a new nonlinear subspace method for model order reduction that avoids the costly snapshot collection of nonlinear terms, leading to more efficient offline computations for large-scale nonlinear dynamical systems.

Contribution

SNS provides a novel approach to hyper-reduction by using solution snapshots instead of nonlinear term snapshots, reducing offline costs and maintaining accuracy.

Findings

Achieves 2-100x speed-up in offline phase.

Maintains comparable accuracy to traditional methods.

Theoretically justified by subspace conditions.

Abstract

Several reduced order models have been developed for nonlinear dynamical systems. To achieve a considerable speed-up, a hyper-reduction step is needed to reduce the computational complexity due to nonlinear terms. Many hyper-reduction techniques require the construction of nonlinear term basis, which introduces a computationally expensive offline phase. A novel way of constructing nonlinear term basis within the hyper-reduction process is introduced. In contrast to the traditional hyper-reduction techniques where the collection of nonlinear term snapshots is required, the SNS method avoids collecting the nonlinear term snapshots. Instead, it uses the solution snapshots that are used for building a solution basis, which enables avoiding an extra data compression of nonlinear term snapshots. As a result, the SNS method provides a more efficient offline strategy than the traditional model order reduction techniques, such as the DEIM, GNAT, and ST-GNAT methods. The SNS method is theoretically justified by the conforming subspace condition and the subspace inclusion relation. It is useful for model order reduction of large-scale nonlinear dynamical problems to reduce the offline cost. It is especially useful for ST-GNAT that has shown promising results, such as a good accuracy with a considerable online speed-up for hyperbolic problems in a recent paper by Choi and Carlberg in SISC 2019, because ST-GNAT involves an expensive offline cost related to collecting nonlinear term snapshots. Numerical results support that the accuracy of the solution from the SNS method is comparable to the traditional methods and a considerable speed-up (i.e., a factor of two to a hundred) is achieved in the offline phase.