A parallel solver for large-scale Markov chains

Authors:
Michele Benzi;Miroslav Tůma
Affiliations:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA;Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodárenskou vezí 2, 182 07 Prague 8, Czech Republic
Venue:
Applied Numerical Mathematics - Developments and trends in iterative methods for large systems of equations—in memoriam Rüdiger Weiss
Year:
2002

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Abstract

We consider the parallel computation of the stationary probability distribution vector of ergodic Markov chains with large state spaces by preconditioned Krylov subspace methods. The parallel preconditioner is obtained as an explicit approximation, in factorized form, of a particular generalized inverse of the generator matrix of the Markov process. Graph partitioning is used to parallelize the whole algorithm, resulting in a two-level method.Conditions that guarantee the existence of the preconditioner are given, and the results of a parallel implementation are presented. Our results indicate that this method is well suited for problems in which the generator matrix can be explicitly formed and stored.