Convergence analysis of an iterative aggregation/disaggregation method for computing stationary probability vectors of stochastic matrices

Abstract

An aggregation/disaggregation iterative algorithm for computing stationary probability vectors of stochastic matrices is analysed. Two convergence results are presented. First, it is shown that fast, global convergence can be achieved provided that a sufficiently high number of relaxations is performed on the fine level. Second, local convergence is shown to take place with just one relaxation performed on the fine level. The convergence proofs are general and require no assumptions on the magnitude of off‐diagonal elements (blocks). Furthermore, a relationship between the errors on the fine and on the coarse level is described. To illustrate the theory, the results of some numerical experiments are presented. © 1998 John Wiley & Sons, Ltd.

Fast multilevel methods for Markov chains

“…Although theoretical convergence results are very difficult to obtain, especially for general problems with nonsymmetric sparsity structure, in many cases empirical evidence has demonstrated good convergence properties and robustness of multilevel methods for Markov chains [2,3,5,[23][24][25][26][27]. We note that although theoretical convergence results do exist for certain classes of two-level methods [13,15,[17][18][19], these results typically deal with only local convergence and do not extend easily (if at all) to multilevel methods. In fact, theoretical results for algebraic multigrid (AMG) solvers applied to nonsymmetric problems are quite rare, with advances having only recently been made (see [28] and the references therein).…”

“…Markov chain literature [12][13][14][15][16][17][18][19]. These methods have been shown to be particularly effective for NCD Markov chains, where the known structure of the problem can be exploited by the aggregation process, which typically results in fast convergence.…”