An iterative aggregation-disaggregation algorithm for solving linear equations
Applied Mathematics and Computation
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
Krylov Subspace Acceleration of Nonlinear Multigrid with Application to Recirculating Flows
SIAM Journal on Scientific Computing
Comparison of Partitioning Techniques for Two-Level Iterative Solvers on Large, Sparse Markov Chains
SIAM Journal on Scientific Computing
A multigrid tutorial (2nd ed.)
A multigrid tutorial (2nd ed.)
Algebraic Multigrid on Unstructured Meshes
Algebraic Multigrid on Unstructured Meshes
SIAM Journal on Scientific Computing
Performance Analysis Using Stochastic Petri Nets
IEEE Transactions on Computers
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
Algebraic Multigrid for Markov Chains
SIAM Journal on Scientific Computing
Advances in Computational Mathematics
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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.