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Title: 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 » a number of graph examples.« less

Authors:
 [1];  [2]
  1. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Center for Applied Scientific Computing
  2. 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}
}

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