Comparison of Multilevel Methods for Kronecker-based Markovian Representations

Abstract.

The paper presents a class of numerical methods to compute the stationary distribution of Markov chains (MCs) with large and structured state spaces. A popular way of dealing with large state spaces in Markovian modeling and analysis is to employ Kronecker-based representations for the generator matrix and to exploit this matrix structure in numerical analysis methods. This paper presents various multilevel (ML) methods for a broad class of MCs with a hierarchcial Kronecker structure of the generator matrix. The particular ML methods are inspired by multigrid and aggregation-disaggregation techniques, and differ among each other by the type of multigrid cycle, the type of smoother, and the order of component aggregation they use. Numerical experiments demonstrate that so far ML methods with successive over-relaxation as smoother provide the most effective solvers for considerably large Markov chains modeled as HMMs with multiple macrostates.