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mcamg - Submitted to SIAM Journal on Scientic Computing...

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Submitted to SIAM Journal on Scientific Computing, March 2009 ALGEBRAIC MULTIGRID FOR MARKOV CHAINS H. DE STERCK ∗‡ , T.A. MANTEUFFEL †§ , S.F. MCCORMICK †¶ , K. MILLER ∗‡‡ , J. RUGE †bardbl , AND G. SANDERS †∗∗ Abstract. An algebraic multigrid (AMG) method is presented for the calculation of the sta- tionary probability vector of an irreducible Markov chain. The method is based on standard AMG for nonsingular linear systems, but in a multiplicative, adaptive setting. A modified AMG interpo- lation formula is proposed that produces a nonnegative interpolation operator with unit row sums. It is shown how the adoption of a previously described lumping technique maintains the irreducible singular M-matrix character of the coarse-level operators on all levels. Together, these properties are sufficient to guarantee the well-posedness of the algorithm. Numerical results show how it leads to nearly optimal multigrid efficiency for a representative set of test problems. Key words. multilevel method, Markov chain, stationary probability vector, algebraic multigrid AMS subject classifications. 65C40 Computational Markov chains, 60J22 Computational methods in Markov chains, 65F10 Iterative methods for linear systems, 65F15 Eigenvalues, eigenvec- tors 1. Introduction. This paper describes an algebraic multigrid (AMG) method for computing the stationary probability vector of large, sparse, irreducible Markov transition matrices. While multigrid methods of aggregation type have been considered before for Markov chains [13, 10, 9], our present approach is based on standard AMG for non- singular linear systems, but in a multiplicative, adaptive setting. The current method is, in fact, an extension to non-variational coarsening of the variational adaptive AMG scheme originally developed in the early stages of the AMG project by A. Brandt, S. McCormick, and J. Ruge [3] (described earlier in [18]). One of the features of the earlier approach is that it constructed interpolation to exactly match the minimal eigenvector of the matrix. A closely related technique called the Exact Interpola- tion Scheme (EIS) was proposed by Brandt and Ron [4]. The EIS has been applied to eigenvalue problems, for example, as a multigrid solver for one-dimensional Helmholtz eigenvalue problems [14]. Moreover, the current method also incorporates some as- pects of early work on aggregation multigrid for Markov chains. In particular, it uses a multiplicative correction form of the coarse-grid correction process that is similar to the two-level aggregated equations proposed in [21], and its framework is similar to the two-level iterative aggregation/disaggregation method for Markov chains pio- neered in [25] and since used and analyzed extensively (see [22] and [9] for references). The so-called square-and-stretch multigrid algorithm by Treister and Yavneh [26] is a recent aggregation-based multigrid method for Markov chains that is also related to our approach.
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