  This paper introduces a sparse matrix discrete interpolation method to
effectively compute matrix approximations in the reduced order modeling
framework. The sparse algorithm developed herein relies on the discrete
empirical interpolation method and uses only samples of the nonzero entries of
the matrix series. The proposed approach can approximate very large matrices,
unlike the current matrix discrete empirical interpolation method which is
limited by its large computational memory requirements. The empirical
interpolation indexes obtained by the sparse algorithm slightly differ from the
ones computed by the matrix discrete empirical interpolation method as a
consequence of the singular vectors round-off errors introduced by the economy
or full singular value decomposition (SVD) algorithms when applied to the full
matrix snapshots. When appropriately padded with zeros the economy SVD
factorization of the nonzero elements of the snapshots matrix is a valid
economy SVD for the full snapshots matrix. Numerical experiments are performed
with the 1D Burgers and 2D Shallow Water Equations test problems where the
quadratic reduced nonlinearities are computed via tensorial calculus. The
sparse matrix approximation strategy is compared against five existing methods
for computing reduced Jacobians: a) matrix discrete empirical interpolation
method, b) discrete empirical interpolation method, c) tensorial calculus, d)
full Jacobian projection onto the reduced basis subspace, and e) directional
derivatives of the model along the reduced basis functions. The sparse matrix
method outperforms all other algorithms. The use of traditional matrix discrete
empirical interpolation method is not possible for very large instances due to
its excessive memory requirements.
