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Accelerated multigrid for graph Laplacian operators

Author

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  • Dell’Acqua, Pietro
  • Frangioni, Antonio
  • Serra-Capizzano, Stefano

Abstract

We consider multigrid type techniques for the numerical solution of large linear systems, whose coefficient matrices show the structure of (weighted) graph Laplacian operators. We combine ad hoc coarser-grid operators with iterative techniques used as smoothers. Empirical tests suggest that the most effective smoothers have to be of Krylov type with subgraph preconditioners, while the projectors, which define the coarser-grid operators, have to be designed for maintaining as much as possible the graph structure of the projected matrix at the inner levels. The main theoretical contribution of the paper is the characterization of necessary and sufficient conditions for preserving the graph structure. In this framework it is possible to explain why the classical projectors inherited from differential equations are good in the differential context and why they may behave unsatisfactorily for unstructured graphs. Furthermore, we report and discuss several numerical experiments, showing that our approach is effective even in very difficult cases where the known approaches are rather slow. As a conclusion, the main advantage of the proposed approach is the robustness, since our multigrid type technique behaves uniformly well in all cases, without requiring either the setting or the knowledge of critical parameters, as it happens when using the best known preconditioned Krylov methods.

Suggested Citation

  • Dell’Acqua, Pietro & Frangioni, Antonio & Serra-Capizzano, Stefano, 2015. "Accelerated multigrid for graph Laplacian operators," Applied Mathematics and Computation, Elsevier, vol. 270(C), pages 193-215.
    • Handle: RePEc:eee:apmaco:v:270:y:2015:i:c:p:193-215
    DOI: 10.1016/j.amc.2015.08.033
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    References listed on IDEAS

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    1. Antonio Frangioni & Giorgio Gallo, 1999. "A Bundle Type Dual-Ascent Approach to Linear Multicommodity Min-Cost Flow Problems," INFORMS Journal on Computing, INFORMS, vol. 11(4), pages 370-393, November.
    2. Antonio Frangioni & Antonio Manca, 2006. "A Computational Study of Cost Reoptimization for Min-Cost Flow Problems," INFORMS Journal on Computing, INFORMS, vol. 18(1), pages 61-70, February.
    3. Bocanegra, Silvana & Castro, Jordi & Oliveira, Aurelio R.L., 2013. "Improving an interior-point approach for large block-angular problems by hybrid preconditioners," European Journal of Operational Research, Elsevier, vol. 231(2), pages 263-273.
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