skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

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:
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. https://doi.org/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. https://doi.org/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. https://doi.org/10.1137/16M1077581. https://www.osti.gov/servlets/purl/1438732.
@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 = {2017},
month = {10}
}

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

Save / Share:

Works referenced in this record:

Multiscale solvers and systematic upscaling in computational physics
journal, July 2005


Algebraic Multilevel Preconditioners for the Graph Laplacian Based on Matching in Graphs
journal, January 2013

  • Brannick, J.; Chen, Y.; Kraus, J.
  • SIAM Journal on Numerical Analysis, Vol. 51, Issue 3
  • DOI: 10.1137/120876083

Adaptive AMG with coarsening based on compatible weighted matching
journal, April 2013

  • D’Ambra, Pasqua; Vassilevski, Panayot S.
  • Computing and Visualization in Science, Vol. 16, Issue 2
  • DOI: 10.1007/s00791-014-0224-9

Upscaling: a review
journal, January 2002

  • Farmer, C. L.
  • International Journal for Numerical Methods in Fluids, Vol. 40, Issue 1-2
  • DOI: 10.1002/fld.267

A two-grid SA-AMG convergence bound that improves when increasing the polynomial degree: IMPROVING TG CONVERGENCE WITH INCREASING SMOOTHING STEPS
journal, June 2016

  • Hu, Xiaozhe; Vassilevski, Panayot S.; Xu, Jinchao
  • Numerical Linear Algebra with Applications, Vol. 23, Issue 4
  • DOI: 10.1002/nla.2053

The egg model - a geological ensemble for reservoir simulation
journal, November 2014

  • Jansen, J. D.; Fonseca, R. M.; Kahrobaei, S.
  • Geoscience Data Journal, Vol. 1, Issue 2
  • DOI: 10.1002/gdj3.21

Multi-scale finite-volume method for elliptic problems in subsurface flow simulation
journal, May 2003


Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media
journal, September 2006


Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method
journal, January 2016

  • Kalchev, D. Z.; Lee, C. S.; Villa, U.
  • SIAM Journal on Scientific Computing, Vol. 38, Issue 5
  • DOI: 10.1137/15M1036683

A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
journal, January 1998


Element agglomeration coarse Raviart-Thomas spaces with improved approximation properties: COARSE RAVIART-THOMAS SPACES WITH IMPROVED APPROXIMATION PROPERTIES
journal, January 2012

  • Lashuk, I. V.; Vassilevski, P. S.
  • Numerical Linear Algebra with Applications, Vol. 19, Issue 2
  • DOI: 10.1002/nla.1819

The Construction of the Coarse de Rham Complexes with Improved Approximation Properties
journal, January 2014

  • Lashuk, Ilya V.; Vassilevski, Panayot S.
  • Computational Methods in Applied Mathematics, Vol. 14, Issue 2
  • DOI: 10.1515/cmam-2014-0004

Multiscale finite-volume formulation for multiphase flow in porous media: black oil formulation of compressible, three-phase flow with gravity
journal, January 2008


Lean Algebraic Multigrid (LAMG): Fast Graph Laplacian Linear Solver
journal, January 2012

  • Livne, Oren E.; Brandt, Achi
  • SIAM Journal on Scientific Computing, Vol. 34, Issue 4
  • DOI: 10.1137/110843563

Multiscale finite-volume method for compressible multiphase flow in porous media
journal, August 2006


Preconditioning discretizations of systems of partial differential equations
journal, April 2010

  • Mardal, Kent-Andre; Winther, Ragnar
  • Numerical Linear Algebra with Applications, Vol. 18, Issue 1
  • DOI: 10.1002/nla.716

Exact de Rham Sequences of Spaces Defined on Macro-Elements in Two and Three Spatial Dimensions
journal, January 2008

  • Pasciak, Joseph E.; Vassilevski, Panayot S.
  • SIAM Journal on Scientific Computing, Vol. 30, Issue 5
  • DOI: 10.1137/070698178

Adaptive Multiscale Finite-Volume Framework for Reservoir Simulation
journal, June 2007

  • Tchelepi, Hamdi A.; Jenny, Patrick; Lee, Seong Hee
  • SPE Journal, Vol. 12, Issue 02
  • DOI: 10.2118/93395-PA

Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems
journal, September 1996


Sparse matrix element topology with application to AMG(e) and preconditioning
journal, January 2002

  • Vassilevski, Panayot S.
  • Numerical Linear Algebra with Applications, Vol. 9, Issue 6-7
  • DOI: 10.1002/nla.300

Coarse Spaces by Algebraic Multigrid: Multigrid Convergence and Upscaling Error Estimates
journal, April 2011


Commuting projections on graphs: COMMUTING PROJECTIONS ON GRAPHS
journal, February 2013

  • Vassilevski, Panayot S.; Zikatanov, Ludmil T.
  • Numerical Linear Algebra with Applications, Vol. 21, Issue 3
  • DOI: 10.1002/nla.1872