Algebraic multilevel preconditioners for the graph laplacian based on matching in graphs

J. Brannick, Y. Chen, J. Kraus, L. Zikatanov

Research output: Contribution to journalArticle

10 Scopus citations

Abstract

This paper presents estimates of the convergence rate and complexity of an algebraic multilevel preconditioner based on piecewise constant coarse vector spaces applied to the graph Laplacian. A bound is derived on the energy norm of the projection operator onto any piecewise constant vector space, which results in an estimate of the two-level convergence rate where the coarse level graph is obtained by matching. The two-level convergence of the method is then used to establish the convergence of an algebraic multilevel iteration that uses the two-level scheme recursively. On structured grids, the method is proved to have convergence rate ≈ (1 - 1/log n) and O(n log n) complexity for each cycle, where n denotes the number of unknowns in the given problem. Numerical results of the algorithm applied to various graph Laplacians are reported. It is also shown that all the theoretical estimates derived for matching can be generalized to the case of aggregates containing more than two vertices.

Original languageEnglish (US)
Pages (from-to)1805-1827
Number of pages23
JournalSIAM Journal on Numerical Analysis
Volume51
Issue number3
DOIs
StatePublished - Oct 9 2013

All Science Journal Classification (ASJC) codes

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

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