An iterative aggregation-disaggregation algorithm for solving linear equations
Applied Mathematics and Computation
Algebraic multigrid theory: The symmetric case
Applied Mathematics and Computation - Second Copper Mountain conference on Multigrid methods Copper Mountain, Colorado
Numerical methods in Markov chain modeling
Operations Research
A multi-level solution algorithm for steady-state Markov chains
SIGMETRICS '94 Proceedings of the 1994 ACM SIGMETRICS conference on Measurement and modeling of computer systems
Comparison of Partitioning Techniques for Two-Level Iterative Solvers on Large, Sparse Markov Chains
SIAM Journal on Scientific Computing
Robustness and Scalability of Algebraic Multigrid
SIAM Journal on Scientific Computing
A multigrid tutorial (2nd ed.)
A multigrid tutorial (2nd ed.)
SIAM Journal on Scientific Computing
Google's PageRank and Beyond: The Science of Search Engine Rankings
Google's PageRank and Beyond: The Science of Search Engine Rankings
An Algebraic Multigrid Preconditioner for a Class of Singular M-Matrices
SIAM Journal on Scientific Computing
Analysis of Aggregation-Based Multigrid
SIAM Journal on Scientific Computing
Multilevel Adaptive Aggregation for Markov Chains, with Application to Web Ranking
SIAM Journal on Scientific Computing
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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.