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We introduce a new concept of sparsity for the stochastic elliptic operator$-{\rm div}\left(a(x,\omega)\nabla(\cdot)\right)$, which reflects thecompactness of its inverse operator in the stochastic direction and allows forspatially heterogeneous stochastic structure. This new concept of sparsitymotivates a heterogeneous stochastic finite element method ({\bf HSFEM})framework for linear elliptic equations, which discretizes the equations usingthe heterogeneous coupling of spatial basis with local stochastic basis toexploit the local stochastic structure of the solution space. We also provide asampling method to construct the local stochastic basis for this frameworkusing the randomized range finding techniques. The resulting HSFEM involves twostages and suits the multi-query setting: in the offline stage, the localstochastic structure of the solution space is identified; in the online stage,the equation can be efficiently solved for multiple forcing functions. Anonline error estimation and correction procedure through Monte Carlo samplingis given. Numerical results for several problems with high dimensionalstochastic input are presented to demonstrate the efficiency of the HSFEM inthe online stage.

A Heterogeneous Stochastic FEM Framework for Elliptic PDEs