Abstract
Smoothed aggregation multigrid method is considered for computing stationary distributions of Markov chains. A judgement which determines whether to implement the whole aggregation procedure is proposed. Through this strategy, a large amount of time in the aggregation procedure is saved without affecting the convergence behavior. Besides this, we explain the shortage and irrationality of the Neighborhood-Based aggregation which is commonly used in multigrid methods. Then a modified version is presented to remedy and improve it. Numerical experiments on some typical Markov chain problems are reported to illustrate the performance of these methods.
Highlights
Markov chains have a large number of applications for scientific research and technological optimization in many areas, including queuing systems and statistical automata networks, web page ranking [1, 2] and gene ranking [3], risk management, and customer relationship analysis
The smoothed aggregation multigrid for Markov chains described in Algorithm 1 is one type of adaptive algebraic multigrid algorithm
Since we focus on the convergence rate affected by original Neighborhood-Based aggregation and modified Neighborhood-Based aggregation and there is no difference between CESAM and smoothed aggregation multigrid (SAM), only the CESAM with these two aggregation methods are implemented for this example
Summary
Markov chains have a large number of applications for scientific research and technological optimization in many areas, including queuing systems and statistical automata networks, web page ranking [1, 2] and gene ranking [3], risk management, and customer relationship analysis. Adaptive algebraic multigrid methods whose aggregates are formed algebraically based on the strength of connection in the problem matrix column-scaled by the current iterate were developed in [14, 15], and for Markov chains [16]. This technique is implemented before the coarse-level probability equation; refer to [17] for details This smoothed aggregation multigrid (SAM) method for Markov chain can be formulated as Algorithm 1.
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