A Heterogeneous Stochastic FEM Framework for Elliptic PDEs

Abstract

We introduce a new concept of sparsity for the stochastic elliptic operator $-\mathrm{div}\left(a(x,\omega )\nabla (\cdot )\right)$, 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 ({\bf 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.

Cited by in corpus (5)

• Sparse Generalized Multiscale Finite Element Methods and their applications
• Optimal Local Multi-scale Basis Functions for Linear Elliptic Equations with Rough Coefficient
• A data-driven approach for multiscale elliptic PDEs with random coefficients based on intrinsic dimension reduction
• Random Sampling and Efficient Algorithms for Multiscale PDEs
• A model reduction method for multiscale elliptic PDEs with random coefficients using an optimization approach