Recursively Accelerated Multilevel Aggregation for Markov Chains

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 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 toplevel 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 (RAMA) 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.