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Title: Spectral Upscaling for Graph Laplacian Problems with Application to Reservoir Simulation

Abstract

Here, we consider coarsening procedures for graph Laplacian problems written in a mixed saddle-point form. In that form, in addition to the original (vertex) degrees of freedom (dofs), we also have edge degrees of freedom. We extend previously developed aggregation-based coarsening procedures applied to both sets of dofs to now allow more than one coarse vertex dof per aggregate. Those dofs are selected as certain eigenvectors of local graph Laplacians associated with each aggregate. Additionally, we coarsen the edge dofs by using traces of the discrete gradients of the already constructed coarse vertex dofs. These traces are defined on the interface edges that connect any two adjacent aggregates. The overall procedure is a modification of the spectral upscaling procedure developed in for the mixed finite element discretization of diffusion type PDEs which has the important property of maintaining inf-sup stability on coarse levels and having provable approximation properties. We consider applications to partitioning a general graph and to a finite volume discretization interpreted as a graph Laplacian, developing consistent and accurate coarse-scale models of a fine-scale problem.

Authors:
 [1];  [1];  [2]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Portland State Univ., Portland, OR (United States)
Publication Date:
Research Org.:
Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1438732
Report Number(s):
LLNL-JRNL-693123
Journal ID: ISSN 1064-8275
Grant/Contract Number:  
AC52-07NA27344
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 39; Journal Issue: 5; Journal ID: ISSN 1064-8275
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; 58 GEOSCIENCES; graph Laplacian; finite volume methods; numerical upscaling; algebraic multigrid; reservoir simulation

Citation Formats

Barker, Andrew T., Lee, Chak S., and Vassilevski, Panayot S.. Spectral Upscaling for Graph Laplacian Problems with Application to Reservoir Simulation. United States: N. p., 2017. Web. doi:10.1137/16M1077581.
Barker, Andrew T., Lee, Chak S., & Vassilevski, Panayot S.. Spectral Upscaling for Graph Laplacian Problems with Application to Reservoir Simulation. United States. doi:10.1137/16M1077581.
Barker, Andrew T., Lee, Chak S., and Vassilevski, Panayot S.. Thu . "Spectral Upscaling for Graph Laplacian Problems with Application to Reservoir Simulation". United States. doi:10.1137/16M1077581.
@article{osti_1438732,
title = {Spectral Upscaling for Graph Laplacian Problems with Application to Reservoir Simulation},
author = {Barker, Andrew T. and Lee, Chak S. and Vassilevski, Panayot S.},
abstractNote = {Here, we consider coarsening procedures for graph Laplacian problems written in a mixed saddle-point form. In that form, in addition to the original (vertex) degrees of freedom (dofs), we also have edge degrees of freedom. We extend previously developed aggregation-based coarsening procedures applied to both sets of dofs to now allow more than one coarse vertex dof per aggregate. Those dofs are selected as certain eigenvectors of local graph Laplacians associated with each aggregate. Additionally, we coarsen the edge dofs by using traces of the discrete gradients of the already constructed coarse vertex dofs. These traces are defined on the interface edges that connect any two adjacent aggregates. The overall procedure is a modification of the spectral upscaling procedure developed in for the mixed finite element discretization of diffusion type PDEs which has the important property of maintaining inf-sup stability on coarse levels and having provable approximation properties. We consider applications to partitioning a general graph and to a finite volume discretization interpreted as a graph Laplacian, developing consistent and accurate coarse-scale models of a fine-scale problem.},
doi = {10.1137/16M1077581},
journal = {SIAM Journal on Scientific Computing},
number = 5,
volume = 39,
place = {United States},
year = {Thu Oct 26 00:00:00 EDT 2017},
month = {Thu Oct 26 00:00:00 EDT 2017}
}

Journal Article:
Free Publicly Available Full Text
This content will become publicly available on October 26, 2018
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