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In this paper, we discuss the convergence of an Algebraic MultiGrid (AMG)method for general symmetric positive-definite matrices. The method relies onan aggregation algorithm, named \emph{coarsening based on compatible weightedmatching}, which exploits the interplay between the principle of compatiblerelaxation and the maximum product matching in undirected weighted graphs. Theresults are based on a general convergence analysis theory applied to the classof AMG methods employing unsmoothed aggregation and identifying a qualitymeasure for the coarsening; similar quality measures were originally introducedand applied to other methods as tools to obtain good quality aggregates leadingto optimal convergence for M-matrices. The analysis, as well as the coarseningprocedure, is purely algebraic and, in our case, allows an \emph{a posteriori}evaluation of the quality of the aggregation procedure which we apply toanalyze the impact of approximate algorithms for matching computation and thedefinition of graph edge weights. We also explore the connection between thechoice of the aggregates and the compatible relaxation convergence, confirmingthe consistency between theories for designing coarsening procedures in purelyalgebraic multigrid methods and the effectiveness of the coarsening based oncompatible weighted matching. We discuss various completely automaticalgorithmic approaches to obtain aggregates for which good convergenceproperties are achieved on various test cases.

Automatic coarsening in Algebraic Multigrid utilizing quality measures for matching-based aggregations