  In this paper, we propose a model's sparse representation based on reduced
mixed generalized multiscale finite element (GMsFE) basis methods for elliptic
PDEs with random inputs. Mixed generalized multiscale finite element method
(GMsFEM) is one of the accurate and efficient approaches to solve multiscale
problem in a coarse grid with local mass conservation. When the inputs of the
PDEs are parameterized by the random variables, the GMsFE basis functions
usually depend on the random parameters. This leads to a large number degree of
freedoms for the mixed GMsFEM and substantially impacts on the computation
efficiency. In order to overcome the difficulty, we develop reduced mixed GMsFE
basis methods such that the multiscale basis functions are independent of the
random parameters and span a low-dimensional space. To this end, a greedy
algorithm is used to find a set of optimal samples from a training set
scattered in the parameter space. Reduced mixed GMsFE basis functions are
constructed based on the optimal samples using two optimal sampling strategies:
basis-oriented cross-validation and proper orthogonal decomposition. Although
the dimension of the space spanned by the reduced mixed GMsFE basis functions
is much smaller than the dimension of the original full order model, the online
computation still depends on the number of coarse degree of freedoms. To
significantly improve the online computation, we integrate the reduced mixed
GMsFE basis methods with sparse tensor approximation and obtain a sparse
representation for the model's outputs. The sparse representation is very
efficient for evaluating the model's outputs for many instances of parameters.
To illustrate the efficacy of the proposed methods, we present a few numerical
examples for multsicale problems with random inputs.
