A heterogeneous stochastic FEM framework for elliptic PDEs
We introduce a new concept of sparsity for the stochastic elliptic operator - div(ɑ(x,ω)∇(•)), which reflects the compactness of its inverse operator in the stochastic direction and allows for spatially heterogeneous stochastic structure. This new concept of sparsity motivates a heterogeneous stochastic finite element method (HSFEM) framework for linear elliptic equations, which discretizes the equations using the heterogeneous coupling of spatial basis with local stochastic basis to exploit the local stochastic structure of the solution space. We also provide a sampling method to construct the local stochastic basis for this framework using the randomized range finding techniques. The resulting HSFEM involves two stages and suits the multi-query setting: in the offline stage, the local stochastic structure of the solution space is identified; in the online stage, the equation can be efficiently solved for multiple forcing functions. An online error estimation and correction procedure through Monte Carlo sampling is given. Numerical results for several problems with high dimensional stochastic input are presented to demonstrate the efficiency of the HSFEM in the online stage.
© 2014 Elsevier Inc. Received 14 April 2014. Received in revised form 12 September 2014. Accepted 9 October 2014. Available online 18 October 2014. We would like to thank the two anonymous reviewers for their constructive comments which help improve the quality of this paper. This research was in part supported by Air Force MURI Grant FA9550-09-1-0613, DOE Grant DE-FG02-06ER257, and NSF Grants Nos.DMS-1318377, DMS-1159138.
Submitted - 1409.3619v1.pdf