Commuting projections on graphs
Abstract
For a given (connected) graph, we consider vector spaces of (discrete) functions defined on its vertices and its edges. These two spaces are related by a discrete gradient operator, Grad and its adjoint, ₋Div, referred to as (negative) discrete divergence. We also consider a coarse graph obtained by aggregation of vertices of the original one. Then a coarse vertex space is identified with the subspace of piecewise constant functions over the aggregates. We consider the ℓ2-projection QH onto the space of these piecewise constants. In the present paper, our main result is the construction of a projection π H from the original edge-space onto a properly constructed coarse edge-space associated with the edges of the coarse graph. The projections π H and QH commute with the discrete divergence operator, i.e., we have div π H = QH div. The respective pair of coarse edge-space and coarse vertexspace offer the potential to construct two-level, and by recursion, multilevel methods for the mixed formulation of the graph Laplacian which utilizes the discrete divergence operator. The performance of one two-level method with overlapping Schwarz smoothing and correction based on the constructed coarse spaces for solving such mixed graph Laplacian systems is illustrated onmore »
- Authors:
-
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing
- Pennsylvania State Univ., University Park, PA (United States). Dept. of Mathematics
- Publication Date:
- Research Org.:
- Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1228024
- Report Number(s):
- LLNL-JRNL-556851
Journal ID: ISSN 1070-5325
- DOE Contract Number:
- AC52-07NA27344
- Resource Type:
- Journal Article
- Journal Name:
- Numerical Linear Algebra with Applications
- Additional Journal Information:
- Journal Volume: 21; Journal Issue: 3; Journal ID: ISSN 1070-5325
- Publisher:
- Wiley
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; graph Laplacian; mix ed formulation; commuting projections; coarsening
Citation Formats
Vassilevski, Panayot S., and Zikatanov, Ludmil T. Commuting projections on graphs. United States: N. p., 2013.
Web. doi:10.1002/nla.1872.
Vassilevski, Panayot S., & Zikatanov, Ludmil T. Commuting projections on graphs. United States. https://doi.org/10.1002/nla.1872
Vassilevski, Panayot S., and Zikatanov, Ludmil T. 2013.
"Commuting projections on graphs". United States. https://doi.org/10.1002/nla.1872. https://www.osti.gov/servlets/purl/1228024.
@article{osti_1228024,
title = {Commuting projections on graphs},
author = {Vassilevski, Panayot S. and Zikatanov, Ludmil T.},
abstractNote = {For a given (connected) graph, we consider vector spaces of (discrete) functions defined on its vertices and its edges. These two spaces are related by a discrete gradient operator, Grad and its adjoint, ₋Div, referred to as (negative) discrete divergence. We also consider a coarse graph obtained by aggregation of vertices of the original one. Then a coarse vertex space is identified with the subspace of piecewise constant functions over the aggregates. We consider the ℓ2-projection QH onto the space of these piecewise constants. In the present paper, our main result is the construction of a projection π H from the original edge-space onto a properly constructed coarse edge-space associated with the edges of the coarse graph. The projections π H and QH commute with the discrete divergence operator, i.e., we have div π H = QH div. The respective pair of coarse edge-space and coarse vertexspace offer the potential to construct two-level, and by recursion, multilevel methods for the mixed formulation of the graph Laplacian which utilizes the discrete divergence operator. The performance of one two-level method with overlapping Schwarz smoothing and correction based on the constructed coarse spaces for solving such mixed graph Laplacian systems is illustrated on a number of graph examples.},
doi = {10.1002/nla.1872},
url = {https://www.osti.gov/biblio/1228024},
journal = {Numerical Linear Algebra with Applications},
issn = {1070-5325},
number = 3,
volume = 21,
place = {United States},
year = {Tue Feb 19 00:00:00 EST 2013},
month = {Tue Feb 19 00:00:00 EST 2013}
}
Works referenced in this record:
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
Difference equations, isoperimetric inequality and transience of certain random walks
journal, February 1984
- Dodziuk, Jozef
- Transactions of the American Mathematical Society, Vol. 284, Issue 2
Sobolev spaces on graphs
journal, December 2005
- Ostrovskii, M. I.
- Quaestiones Mathematicae, Vol. 28, Issue 4
A multigrid method based on graph matching for convection-diffusion equations
journal, January 2002
- Kim, HwanHo; Xu, Jinchao; Zikatanov, Ludmil
- Numerical Linear Algebra with Applications, Vol. 10, Issue 1-2
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
An algebraic multilevel method for anisotropic elliptic equations based on subgraph matching: AN AMG METHOD FOR ANISOTROPIC EQUATIONS BASED ON SUBGRAPH MATCHING
journal, January 2012
- Brannick, James; Chen, Yao; Zikatanov, Ludmil
- Numerical Linear Algebra with Applications, Vol. 19, Issue 2
On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers
journal, January 1974
- Brezzi, F.
- Revue française d'automatique, informatique, recherche opérationnelle. Analyse numérique, Vol. 8, Issue R2
Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems
journal, September 1996
- Vaněk, P.; Mandel, J.; Brezina, M.
- Computing, Vol. 56, Issue 3
Algorithmic Aspects of Vertex Elimination on Graphs
journal, June 1976
- Rose, Donald J.; Tarjan, R. Endre; Lueker, George S.
- SIAM Journal on Computing, Vol. 5, Issue 2
Convergence estimates for multigrid algorithms without regularity assumptions
journal, September 1991
- Bramble, James H.; Pasciak, Joseph E.; Wang, Jun Ping
- Mathematics of Computation, Vol. 57, Issue 195