Abstract
This paper presents a class of new accelerated restarted GMRES method for calculating the stationary probability vector of an irreducible Markov chain. We focus on the mechanism of this new hybrid method by showing how to periodically combine the GMRES and vector extrapolation method into a much efficient one for improving the convergence rate in Markov chain problems. Numerical experiments are carried out to demonstrate the efficiency of our new algorithm on several typical Markov chain problems.
Highlights
The Markov chain is a very robust tool for studying stochastic systems overtime and is in a wide range of applications including queueing systems [1, 2], computer and communication systems [3], information retrieval, and Web ranking [4,5,6,7]
Three typical Markov chain problems have been used in our experiments
All the iterations are terminated when ‖Axk − xk‖1/‖xk‖1 < tol, where xk are the approximations obtained by the current iteration
Summary
The Markov chain is a very robust tool for studying stochastic systems overtime and is in a wide range of applications including queueing systems [1, 2], computer and communication systems [3], information retrieval, and Web ranking [4,5,6,7]. We consider a class of Krylov subspace methods, restarted GMRES (GMRES(m)) method, for the numerical solutions of the stationary probability vector of an irreducible Markov chain. We consider the restarted GMRES method and propose a new way to accelerate its numerical calculation by use of the polynomial-type vector extrapolation methods. The proposed extrapolation-accelerated GMRES(m) methods are tested on several Markov chain problems, and the experimental results show its effectiveness.
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