GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
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
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 (2nd ed.)
A multigrid tutorial (2nd ed.)
Multigrid
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
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
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
Algebraic Multigrid for Markov Chains
SIAM Journal on Scientific Computing
Recursively Accelerated Multilevel Aggregation for Markov Chains
SIAM Journal on Scientific Computing
Recursively Accelerated Multilevel Aggregation for Markov Chains
SIAM Journal on Scientific Computing
On-the-Fly Adaptive Smoothed Aggregation Multigrid for Markov Chains
SIAM Journal on Scientific Computing
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In many application areas, including information retrieval and networking systems, finding the steady-state distribution vector of an irreducible Markov chain is of interest and it is often difficult to compute efficiently. The steady-state vector is the solution to a nonsymmetric eigenproblem with known eigenvalue, B x驴=驴x, subject to probability constraints $ \Vert{\bf x}\Vert _1 = 1$ and x驴驴0, where B is column stochastic, that is, B驴驴驴O and 1 t B驴=驴1 t . Recently, scalable methods involving Smoothed Aggregation (SA) and Algebraic Multigrid (AMG) were proposed to solve such eigenvalue problems. These methods use multiplicative iterate updates versus the additive error corrections that are typically used in nonsingular linear solvers. This paper discusses an outer iteration that accelerates convergence of multiplicative update methods, similar in principle to a preconditioned flexible Krylov wrapper applied to an additive iteration for a nonsingular linear problem. The acceleration is performed by selecting a linear combination of old iterates to minimize a functional within the space of probability vectors. Two different implementations of this idea are considered and the effectiveness of these approaches is demonstrated with representative examples.