Fast multilevel methods for Markov chains

Hans De Sterck, Killian Miller, Eran Treister, Irad Yavneh

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10 Citations (Scopus)


This paper describes multilevel methods for the calculation of the stationary probability vector of large, sparse, irreducible Markov chains. In particular, several recently proposed significant improvements to the multilevel aggregation method of Horton and Leutenegger are described and compared. Furthermore, we propose a very simple improvement of that method using an over-correction mechanism. We also compare with more traditional iterative methods for Markov chains such as weighted Jacobi, two-level aggregation/disaggregation, and preconditioned stabilized biconjugate gradient and generalized minimal residual method. Numerical experiments confirm that our improvements lead to significant speedup, and result in multilevel methods that are competitive with leading iterative solvers for Markov chains.

Original languageEnglish
Pages (from-to)961-980
Number of pages20
JournalNumerical Linear Algebra with Applications
Issue number6
Publication statusPublished - Nov 2011
Externally publishedYes


  • Markov chain
  • Multigrid
  • Multilevel aggregation
  • Over-correction
  • Stationary probability vector

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