A Heterogeneous Stochastic FEM Framework for Elliptic PDEs

TL;DR

This paper develops a heterogeneous stochastic finite element method (HSFEM) for elliptic PDEs that exploits local stochastic structures, enabling efficient multi-query solutions with high-dimensional randomness.

Contribution

It introduces a new sparsity concept for stochastic elliptic operators and constructs a HSFEM framework with randomized basis sampling for efficient solutions.

Findings

Efficient multi-query solutions for high-dimensional stochastic elliptic PDEs.

The HSFEM framework outperforms traditional methods in computational efficiency.

Numerical results demonstrate the effectiveness of the local stochastic basis construction.

Abstract

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