Matrix analysis
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
A multigrid tutorial: second edition
A multigrid tutorial: second edition
Multigrid
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
SIAM Journal on Scientific Computing
Reducing Complexity in Parallel Algebraic Multigrid Preconditioners
SIAM Journal on Matrix Analysis and Applications
Performance Analysis Using Stochastic Petri Nets
IEEE Transactions on Computers
An Algebraic Multigrid Preconditioner for a Class of Singular M-Matrices
SIAM Journal on Scientific Computing
Multilevel Adaptive Aggregation for Markov Chains, with Application to Web Ranking
SIAM Journal on Scientific Computing
Smoothed Aggregation Multigrid for Markov Chains
SIAM Journal on Scientific Computing
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An algebraic multigrid (AMG) method is presented for the calculation of the stationary 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 interpolation formula is proposed that produces a nonnegative interpolation operator with unit row sums. We show 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.