  In a number of previous papers, local (coarse grid) multiscale model
reduction techniques are developed using a Generalized Multiscale Finite
Element Method. In these approaches, multiscale basis functions are constructed
using local snapshot spaces, where a snapshot space is a large space that
represents the solution behavior in a coarse block. In a number of applications
(e.g., those discussed in the paper), one may have a sparsity in the snapshot
space for an appropriate choice of a snapshot space. More precisely, the
solution may only involve a portion of the snapshot space. In this case, one
can use sparsity techniques to identify multiscale basis functions. In this
paper, we consider two such sparse local multiscale model reduction approaches.
  In the first approach (which is used for parameter-dependent multiscale
PDEs), we use local minimization techniques, such as sparse POD, to identify
multiscale basis functions, which are sparse in the snapshot space. These
minimization techniques use $l_1$ minimization to find local multiscale basis
functions, which are further used for finding the solution. In the second
approach (which is used for the Helmholtz equation), we directly apply $l_1$
minimization techniques to solve the underlying PDEs. This approach is more
expensive as it involves a large snapshot space; however, in this example, we
can not identify a local minimization principle, such as local generalized SVD.
