SIAM Journal on Scientific Computing, Volume 42, Issue 2, Page A1116-A1146, January 2020.
Several reduced order models have been successfully 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 [SIAM J. Sci. Comput., 41 (2019), pp. A26--A58], because ST-GNAT involves an expensive offline cost related to collecting nonlinear term snapshots. Error analysis for the SNS method is presented. 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.

中文翻译：

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SNS：基于时间的模型降阶的基于解决方案的非线性子空间方法

SIAM科学计算杂志，第42卷，第2期，第A1116-A1146页，2020年1月。
已经针对非线性动力学系统成功开发了几种降阶模型。为了实现显着的加速，由于非线性项，需要超缩减步骤来减少计算复杂性。许多超还原技术需要构造非线性项基础，这会引入计算上昂贵的离线阶段。介绍了一种在超还原过程中构造非线性项基础的新方法。与需要收集非线性项快照的传统超还原技术相反，SNS方法避免了收集非线性项快照。相反，它使用用于构建解决方案基础的解决方案快照，从而可以避免对非线性术语快照进行额外的数据压缩。结果是，与传统的模型降阶技术（例如DEIM，GNAT和ST-GNAT方法）相比，SNS方法提供了更有效的脱机策略。从理论上说，SNS方法是由符合子空间条件和子空间包含关系证明的。对于减少大规模非线性动力学问题的模型阶数以减少离线成本非常有用。Choi和Carlberg在最近的一篇论文中对ST-GNAT尤其有用，它显示了令人鼓舞的结果，例如具有很高的准确性和可观的在线速度以解决双曲线问题[SIAM J. Sci。Comput。，41（2019），pp。A26--A58]，因为ST-GNAT涉及与收集非线性项快照有关的昂贵离线成本。提出了SNS方法的误差分析。