Abstract
Abstract Based on the quadratic extrapolation method and its generalization, this paper presents an accelerated two-level multigrid method for speeding up the numerical computation of the stationary probability vector of an irreducible Markov chain. It shows how to combine these vector extrapolation methods with the two-level multigrid method on the coarse level in detail. Numerical results on two Markov chain problems are provided to illustrate the effectiveness of our proposed method in terms of reducing the iteration counts and computing time.
Highlights
The use of Markov chains is of interest in a wide range of applications
There are large amounts of works have devoted to solving the linear system (2)
Since using iterative methods like Gauss-Seidel method given in (5) to calculate the coarselevel linear system Acxc = 0 may require a very long time to converge to the desired solution
Summary
The use of Markov chains is of interest in a wide range of applications. For example, the web ranking and information retrieval [1,2,3], queuing systems [4,5,6,7], stochastic automata networks [8,9], manufacturing systems and inventory control [10] and communication systems [11,12] and so on. The two-level multigrid method is briefly introduced to solve the stationary probability distribution of Markov chains. With the initial approximation xc obtained at step 4 of Algorithm 1, Gauss-Seidel method given in (5) modifies this approximation such that it becomes closer and closer to the true solution at each iteration.
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