Multilevel Spectral Coarsening for Graph Laplacian Problems with Application to Reservoir Simulation
Abstract
We extend previously developed two-level coarsening procedures for graph Laplacian problems written in a mixed saddle point form to the fully recursive multilevel case. The resulting hierarchy of discretizations gives rise to a hierarchy of upscaled models, in the sense that they provide approximation in the natural norms (in the mixed setting). This property enables us to utilize them in three applications: (i) as an accurate reduced model, (ii) as a tool in multilevel Monte Carlo simulations (in application to finite volume discretizations), and (iii) for providing a sequence of nonlinear operators in a full approximation scheme for solving nonlinear pressure equations discretized by the conservative two-point flux approximation. Finally, we illustrate the potential of the proposed multilevel technique in all three applications on a number of popular benchmark problems used in reservoir simulation.
- Authors:
-
[1];
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
- Portland State Univ., OR (United States)
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Portland State Univ., OR (United States)
- Publication Date:
- Research Org.:
- Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1824746
- Report Number(s):
- LLNL-JRNL-795643
Journal ID: ISSN 1064-8275; 996879
- Grant/Contract Number:
- AC52-07NA27344
- Resource Type:
- Accepted Manuscript
- Journal Name:
- SIAM Journal on Scientific Computing
- Additional Journal Information:
- Journal Volume: 43; Journal Issue: 4; Journal ID: ISSN 1064-8275
- Publisher:
- Society for Industrial and Applied Mathematics (SIAM)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 58 GEOSCIENCES; 97 MATHEMATICS AND COMPUTING; graph Laplacian; algebraic multigrid; finite volume methods; numerical upscaling; multilevel Monte Carlo; full approximation scheme
Citation Formats
Barker, Andrew T., Gelever, Stephan V., Lee, Chak S., Osborn, Sarah V., and Vassilevski, Panayot S. Multilevel Spectral Coarsening for Graph Laplacian Problems with Application to Reservoir Simulation. United States: N. p., 2021.
Web. doi:10.1137/19m1296343.
Barker, Andrew T., Gelever, Stephan V., Lee, Chak S., Osborn, Sarah V., & Vassilevski, Panayot S. Multilevel Spectral Coarsening for Graph Laplacian Problems with Application to Reservoir Simulation. United States. https://doi.org/10.1137/19m1296343
Barker, Andrew T., Gelever, Stephan V., Lee, Chak S., Osborn, Sarah V., and Vassilevski, Panayot S. Wed .
"Multilevel Spectral Coarsening for Graph Laplacian Problems with Application to Reservoir Simulation". United States. https://doi.org/10.1137/19m1296343. https://www.osti.gov/servlets/purl/1824746.
@article{osti_1824746,
title = {Multilevel Spectral Coarsening for Graph Laplacian Problems with Application to Reservoir Simulation},
author = {Barker, Andrew T. and Gelever, Stephan V. and Lee, Chak S. and Osborn, Sarah V. and Vassilevski, Panayot S.},
abstractNote = {We extend previously developed two-level coarsening procedures for graph Laplacian problems written in a mixed saddle point form to the fully recursive multilevel case. The resulting hierarchy of discretizations gives rise to a hierarchy of upscaled models, in the sense that they provide approximation in the natural norms (in the mixed setting). This property enables us to utilize them in three applications: (i) as an accurate reduced model, (ii) as a tool in multilevel Monte Carlo simulations (in application to finite volume discretizations), and (iii) for providing a sequence of nonlinear operators in a full approximation scheme for solving nonlinear pressure equations discretized by the conservative two-point flux approximation. Finally, we illustrate the potential of the proposed multilevel technique in all three applications on a number of popular benchmark problems used in reservoir simulation.},
doi = {10.1137/19m1296343},
journal = {SIAM Journal on Scientific Computing},
number = 4,
volume = 43,
place = {United States},
year = {Wed Aug 04 00:00:00 EDT 2021},
month = {Wed Aug 04 00:00:00 EDT 2021}
}
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Works referenced in this record:
A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
journal, January 1998
- Karypis, George; Kumar, Vipin
- SIAM Journal on Scientific Computing, Vol. 20, Issue 1
Think continuous: Markovian Gaussian models in spatial statistics
journal, May 2012
- Simpson, Daniel; Lindgren, Finn; Rue, Håvard
- Spatial Statistics, Vol. 1
A Multiscale Mortar Mixed Finite Element Method
journal, January 2007
- Arbogast, Todd; Pencheva, Gergina; Wheeler, Mary F.
- Multiscale Modeling & Simulation, Vol. 6, Issue 1
Multi-scale finite-volume method for elliptic problems in subsurface flow simulation
journal, May 2003
- Jenny, P.; Lee, S. H.; Tchelepi, H. A.
- Journal of Computational Physics, Vol. 187, Issue 1
On some versions of the element agglomeration AMGe method
journal, September 2008
- Lashuk, Ilya; Vassilevski, Panayot S.
- Numerical Linear Algebra with Applications, Vol. 15, Issue 7
Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients
journal, January 2011
- Cliffe, K. A.; Giles, M. B.; Scheichl, R.
- Computing and Visualization in Science, Vol. 14, Issue 1
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
Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media
journal, September 2006
- Jenny, P.; Lee, S. H.; Tchelepi, H. A.
- Journal of Computational Physics, Vol. 217, Issue 2
Nonlinear multigrid solvers exploiting AMGe coarse spaces with approximation properties
journal, October 2018
- Christensen, Max la Cour; Vassilevski, Panayot S.; Villa, Umberto
- Journal of Computational and Applied Mathematics, Vol. 340
A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields
journal, January 2017
- Osborn, Sarah; Vassilevski, Panayot S.; Villa, Umberto
- SIAM Journal on Scientific Computing, Vol. 39, Issue 5
Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps
journal, August 2013
- Spillane, N.; Dolean, V.; Hauret, P.
- Numerische Mathematik, Vol. 126, Issue 4
Upscaling hydraulic conductivities in heterogeneous media: An overview
journal, August 1996
- Wen, Xian-Huan; Gómez-Hernández, J. Jaime
- Journal of Hydrology, Vol. 183, Issue 1-2
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
A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media
journal, June 1997
- Hou, Thomas Y.; Wu, Xiao-Hui
- Journal of Computational Physics, Vol. 134, Issue 1
Coarse Spaces by Algebraic Multigrid: Multigrid Convergence and Upscaling Error Estimates
journal, April 2011
- Vassilevski, Panayot S.
- Advances in Adaptive Data Analysis, Vol. 03, Issue 01n02
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
Spectral Upscaling for Graph Laplacian Problems with Application to Reservoir Simulation
journal, January 2017
- Barker, Andrew T.; Lee, Chak S.; Vassilevski, Panayot S.
- SIAM Journal on Scientific Computing, Vol. 39, Issue 5
Multigrid methods nonlinear problems: an overview
conference, July 2003
- Henson, Van E.
- Electronic Imaging 2003, SPIE Proceedings
Algebraic Hybridization and Static Condensation with Application to Scalable $H$(div) Preconditioning
journal, January 2019
- Dobrev, V.; Kolev, T.; Lee, C. S.
- SIAM Journal on Scientific Computing, Vol. 41, Issue 3
Multilevel upscaling through variational coarsening: MULTILEVEL VARIATIONAL UPSCALING
journal, February 2006
- MacLachlan, Scott P.; Moulton, J. David
- Water Resources Research, Vol. 42, Issue 2
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
An Efficient Multigrid Method for Graph Laplacian Systems II: Robust Aggregation
journal, January 2017
- Napov, Artem; Notay, Yvan
- SIAM Journal on Scientific Computing, Vol. 39, Issue 5
Mixed and nonconforming finite element methods : implementation, postprocessing and error estimates
journal, January 1985
- Arnold, D. N.; Brezzi, F.
- ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 19, Issue 1
Generalized multiscale finite element methods (GMsFEM)
journal, October 2013
- Efendiev, Yalchin; Galvis, Juan; Hou, Thomas Y.
- Journal of Computational Physics, Vol. 251
The heterogeneous multiscale method
journal, April 2012
- Abdulle, Assyr; Weinan, E.; Engquist, Björn
- Acta Numerica, Vol. 21
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
Spectral AMGe ($\rho$AMGe)
journal, January 2003
- Chartier, T.; Falgout, R. D.; Henson, V. E.
- SIAM Journal on Scientific Computing, Vol. 25, Issue 1
An explicit link between Gaussian fields and Gaussian Markov random fields: the stochastic partial differential equation approach: Link between Gaussian Fields and Gaussian Markov Random Fields
journal, August 2011
- Lindgren, Finn; Rue, Håvard; Lindström, Johan
- Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 73, Issue 4
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
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
Nonlinear multigrid based on local spectral coarsening for heterogeneous diffusion problems
journal, December 2020
- Lee, Chak Shing; Hamon, François; Castelletto, Nicola
- Computer Methods in Applied Mechanics and Engineering, Vol. 372
Numerical Multilevel Upscaling for Incompressible Flow in Reservoir Simulation: An Element-Based Algebraic Multigrid (AMGe) Approach
journal, January 2017
- la Cour Christensen, Max; Villa, Umberto; Engsig-Karup, Allan P.
- SIAM Journal on Scientific Computing, Vol. 39, Issue 1
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
On some versions of the element agglomeration AMGe method
journal, September 2008
- Lashuk, Ilya; Vassilevski, Panayot S.
- Numerical Linear Algebra with Applications, Vol. 15, Issue 7
On propagation of acoustic waves in the medium consisting of a fluid and an elastic material
journal, July 2007
- Shamaev, A. S.; Samarin, V. A.
- Journal of Mathematical Sciences, Vol. 144, Issue 4
The Heterognous Multiscale Methods
journal, January 2003
- E., Weinan; Engquist, Bjorn
- Communications in Mathematical Sciences, Vol. 1, Issue 1
A Multilevel, Hierarchical Sampling Technique for Spatially Correlated Random Fields
preprint, January 2017
- Osborn, Sarah; Vassilevski, Panayot; Villa, Umberto
- arXiv