Paper Info
Title
Algebraic Multilevel Preconditioners for the Graph Laplacian Based on Matching in Graphs.
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 approximate to (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.
Year
DOI
Venue
2013
10.1137/120876083
SIAM JOURNAL ON NUMERICAL ANALYSIS
Keywords
Field
DocType
multilevel preconditioning,graph Laplacian,matching in graphs,aggregation
Mathematical analysis,Rate of convergence,Voltage graph,Laplacian matrix,Discrete mathematics,Mathematical optimization,Combinatorics,Line graph,Coxeter graph,Algebraic connectivity,Factor-critical graph,Distance-regular graph,Mathematics
Journal
Volume
Issue
ISSN
51
3
0036-1429
Citations 
PageRank 
References 
6
0.45
13
Authors
4
Name
Order
Citations
PageRank
James J. Brannick1332.74
Yao Chen2120.99
Johannes Kraus3162.91
Ludmil Zikatanov418925.89