Iterative Methods for Computing Stationary Distributions of Nearly Completely Decomposable Markov Chains

Koury, J. R.; McAllister, David F.; Stewart, William J.

doi:10.1137/0605019

Cited by 91 publications

(50 citation statements)

References 22 publications

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“…For this application, smoothed V-cycles with distance-one aggregation perform better than cycles with distance-two aggregation. Table 6.10 compares one-level relaxation (with two relexations on the fine level per iteration), twolevel A-AM aggregation (with two relaxations on the coarse level; this is essentially equivalent to the aggregation/disaggregation methods of [7,8,9,10,11,12,13,14]), and unsmoothed, multilevel A-AM (which is the same as MAA in [6], and similar to the multilevel aggregation methods of [4,5]). It can be observed that traditional twolevel and multilevel aggregation methods hardly manage to accelerate convergence for this problem.…”

Section: Uniform 2d Latticementioning

confidence: 99%

“…These methods are multilevel generalizations of the large class of twolevel methods of iterative aggregation/disaggregation type for Markov chains (e.g, [7,8,9,10,11,12,13,14,3]). They are of a multiplicative nature, and include both geometric versions, in which aggregates are chosen based on the a-priori known topology (or connectivity) of the chain, and algebraic versions, in which aggregates are chosen based on strength of connection in the problem matrix.…”

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confidence: 99%

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Smoothed Aggregation Multigrid for Markov Chains

Sterck¹,

Manteuffel²,

McCormick

et al. 2010

SIAM J. Sci. Comput.

147

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Abstract. A smoothed aggregation multigrid method is presented for the numerical calculation of the stationary probability vector of an irreducible sparse Markov chain. It is shown how smoothing the interpolation and restriction operators can dramatically increase the efficiency of aggregation multigrid methods for Markov chains that have been proposed in the literature. The proposed smoothing approach is inspired by smoothed aggregation multigrid for linear systems, supplemented with a new lumping technique that assures well-posedness of the coarse-level problems: the coarse-level operators are singular M-matrices on all levels, resulting in strictly positive coarse-level corrections on all levels. Numerical results show how these methods lead to nearly optimal multigrid efficiency for an extensive set of test problems, both when geometric and algebraic aggregation strategies are used.

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Section: Uniform 2d Latticementioning

confidence: 99%

mentioning

confidence: 99%

Smoothed Aggregation Multigrid for Markov Chains

Sterck¹,

Manteuffel²,

McCormick

et al. 2010

SIAM J. Sci. Comput.

147

View full text Add to dashboard Cite

show abstract

“…We can use this method to generate a starting guess for those algorithms known generally as Iterative Aggregation/'Disaggregation (IAD). These are efficient multi-grid like methods and include the well-known KMS [1] and Takahashi [5] algorithms. We can also develop an adaptive variation by using the algorithm of section 5 to solve for 7To first.…”

Section: A More General Formulationmentioning

confidence: 99%

An Algorithm for Computing the Stationary Distribution of a Discrete-Time Birth-and-Death Process with Banded Infinitesimal Generator.

Borges¹,

Peters²

1995

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“…Iteration on the fine level is accelerated by iteration on the coarse level using the coarse-level transition matrix, followed by multiplicative correction on the fine level. The earliest work along these lines is Takahashi's two-level iterative aggregation/disaggregation method for Markov chains [4], and two-level aggregation/disaggregation has been studied extensively ever since [5,6,7,8,9,10,11,3]. Convergence proofs are given for two-level aggregation/disaggregation methods in [9,11].…”

mentioning

confidence: 99%

Recursively Accelerated Multilevel Aggregation for Markov Chains

Sterck¹,

Miller²,

Sanders³

et al. 2010

SIAM J. Sci. Comput.

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Abstract.A recursive acceleration method is proposed for multiplicative multilevel aggregation algorithms that calculate the stationary probability vector of large, sparse, and irreducible Markov chains. Pairs of consecutive iterates at all branches and levels of a multigrid W cycle with simple, nonoverlapping aggregation are recombined to produce improved iterates at those levels. This is achieved by solving quadratic programming problems with inequality constraints: the linear combination of the two iterates is sought that has a minimal two-norm residual, under the constraint that all vector components are nonnegative. It is shown how the two-dimensional quadratic programming problems can be solved explicitly in an efficient way. The method is further enhanced by windowed top-level acceleration of the W cycles using the same constrained quadratic programming approach. Recursive acceleration is an attractive alternative to smoothing the restriction and interpolation operators, since the operator complexity is better controlled and the probabilistic interpretation of coarse-level operators is maintained on all levels. Numerical results are presented showing that the resulting recursively accelerated multilevel aggregation cycles for Markov chains, combined with top-level acceleration, converge significantly faster than W cycles and lead to close-to-linear computational complexity for challenging test problems.

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Iterative Methods for Computing Stationary Distributions of Nearly Completely Decomposable Markov Chains

Cited by 91 publications

References 22 publications

Smoothed Aggregation Multigrid for Markov Chains

Smoothed Aggregation Multigrid for Markov Chains

An Algorithm for Computing the Stationary Distribution of a Discrete-Time Birth-and-Death Process with Banded Infinitesimal Generator.

Recursively Accelerated Multilevel Aggregation for Markov Chains

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