  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.
