Efficient and Practical Tree Preconditioning for Solving Laplacian Systems

  • Luca Castelli Aleardi
  • Alexandre Nolin
  • Maks Ovsjanikov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9125)


We consider the problem of designing efficient iterative methods for solving linear systems. In its full generality, this is one of the oldest problems in numerical analysis with a tremendous number of practical applications. We focus on a particular type of linear systems, associated with Laplacian matrices of undirected graphs, and study a class of iterative methods for which it is possible to speed up the convergence through combinatorial preconditioning. We consider a class of preconditioners, known as tree preconditioners, introduced by Vaidya, that have been shown to lead to asymptotic speed-up in certain cases. Rather than trying to improve the structure of the trees used in preconditioning, we propose a very simple modification to the basic tree preconditioner, which can significantly improve the performance of the iterative linear solvers in practice. We show that our modification leads to better conditioning for some special graphs, and provide extensive experimental evidence for the decrease in the complexity of the preconditioned conjugate gradient method for several graphs, including 3D meshes and complex networks.


Span Tree Condition Number Conjugate Gradient Spectral Cluster Laplacian Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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This work is supported by the ANR EGOS 12 JS02 002 01, a Google Faculty Research Award, the Marie Curie grant CIG-334283-HRGP, and a CNRS chaire dexcellence, Jean Marjoulet professorial chair.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Luca Castelli Aleardi
    • 1
  • Alexandre Nolin
    • 1
  • Maks Ovsjanikov
    • 1
  1. 1.LIX - École PolytechniquePalaiseauFrance

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