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Title: A heterogeneous stochastic FEM framework for elliptic PDEs

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Journal Article: Publisher's Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 281; Journal Issue: C; Related Information: CHORUS Timestamp: 2016-09-04 18:28:57; Journal ID: ISSN 0021-9991
Country of Publication:
United States

Citation Formats

Hou, Thomas Y., and Liu, Pengfei. A heterogeneous stochastic FEM framework for elliptic PDEs. United States: N. p., 2015. Web. doi:10.1016/
Hou, Thomas Y., & Liu, Pengfei. A heterogeneous stochastic FEM framework for elliptic PDEs. United States. doi:10.1016/
Hou, Thomas Y., and Liu, Pengfei. 2015. "A heterogeneous stochastic FEM framework for elliptic PDEs". United States. doi:10.1016/
title = {A heterogeneous stochastic FEM framework for elliptic PDEs},
author = {Hou, Thomas Y. and Liu, Pengfei},
abstractNote = {},
doi = {10.1016/},
journal = {Journal of Computational Physics},
number = C,
volume = 281,
place = {United States},
year = 2015,
month = 1

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record at 10.1016/

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Cited by: 4works
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  • We introduce a new concept of sparsity for the stochastic elliptic operator −div(a(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 involvesmore » 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.« less
  • Flow through porous media is ubiquitous, occurring from large geological scales down to the microscopic scales. Several critical engineering phenomena like contaminant spread, nuclear waste disposal and oil recovery rely on accurate analysis and prediction of these multiscale phenomena. Such analysis is complicated by inherent uncertainties as well as the limited information available to characterize the system. Any realistic modeling of these transport phenomena has to resolve two key issues: (i) the multi-length scale variations in permeability that these systems exhibit, and (ii) the inherently limited information available to quantify these property variations that necessitates posing these phenomena as stochasticmore » processes. A stochastic variational multiscale formulation is developed to incorporate uncertain multiscale features. A stochastic analogue to a mixed multiscale finite element framework is used to formulate the physical stochastic multiscale process. Recent developments in linear and non-linear model reduction techniques are used to convert the limited information available about the permeability variation into a viable stochastic input model. An adaptive sparse grid collocation strategy is used to efficiently solve the resulting stochastic partial differential equations (SPDEs). The framework is applied to analyze flow through random heterogeneous media when only limited statistics about the permeability variation are given.« less
  • A local relaxation method for solving linear elliptic PDEs with O(N) processors and O(..sqrt..N) computation time is proposed. The authors first examine the implementation of traditional relaxation algorithms for solving elliptic PDEs on mesh-connected processor arrays, which require O(N) processors and O(N) computation time. The disadvantage of these implementations is that the determination of the acceleration factors requires some global communication at each iteration. The high communication cost increases the computation time per iteration significantly. Therefore, a local relaxation scheme is proposed to achieve the acceleration effect with very little global communication in the loading stage. They use a Fouriermore » analysis approach to analyze the local relaxation method and also show its convergence. The convergence rate of the local relaxation method is studied by computer simulation.« less
  • The problem of approximating the solution to a class of (PDEs) posed on unbounded domains using finite domain approximations is considered. A finite-element method is formulated for the approximation on the finite subregions, and a domain extension strategy that balances the finite-element error and domain-truncation error is developed. It is shown that this scheme yields asymptotically optimal finite-element approximation properties to the solution on the unbounded domain as the grid is extended. Error estimates for adaptive refinement and domain truncation are developed.