Iteration at Different Levels: Multi-Level Methods fro Structured Markov Chains

For the stationary analysis of large Markov chains in continuous and discrete time a wide variety of solution techniques has been applied in the past. Empirical comparisons show that in particular so called multi-level approaches that perform iterations at different levels are the most efficient solvers for a wide class of Markov chains. The methods combine ideas from aggregation disaggregation methods and algebraic multigrid. The talk gives an overview of the basic ideas of multi level approaches and shows which design alternatives for the algorithms exist. In particular it considers different forms of defining levels, available alternatives to realize prolongation and interpolation operations, different cycle types and different stopping criteria for the smoothing operations at each level. The last part of the talk is devoted to implementation issues and data structures that are necessary for an efficient realization of multi-level methods.