Browsing by Subject "Multigrid"

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    Comparison of multilevel methods for kronecker-based Markovian representations
    (Springer, 2004) Buchholz, P.; Dayar T.
    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.
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    Decompositional analysis of Kronecker structured Markov chains
    (Kent State University, 2008) Bao, Y.; Bozkur, I. N.; Dayar, T.; Sun, X.; Trivedi, K. S.
    This contribution proposes a decompositional iterative method with low memory requirements for the steadystate analysis ofKronecker structured Markov chains. The Markovian system is formed by a composition of subsystems using the Kronecker sum operator for local transitions and the Kronecker product operator for synchronized transitions. Even though the interactions among subsystems, which are captured by synchronized transitions, need not be weak, numerical experiments indicate that the solver benefits considerably from weak interactions among subsystems, and is to be recommended specifically in this case. © 2008, Kent State University.
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    On the convergence of a class of multilevel methods for large sparse Markov chains
    (Society for Industrial and Applied Mathematics, 2007) Buchholz, P.; Dayar, T.
    This paper investigates the theory behind the steady state analysis of large sparse Markov chains with a recently proposed class of multilevel methods using concepts from algebraic multigrid and iterative aggregation- disaggregation. The motivation is to better understand the convergence characteristics of the class of multilevel methods and to have a clearer formulation that will aid their implementation. In doing this, restriction (or aggregation) and prolongation (or disaggregation) operators of multigrid are used, and the Kronecker-based approach for hierarchical Markovian models is employed, since it suggests a natural and compact definition of grids (or levels). However, the formalism used to describe the class of multilevel methods for large sparse Markov chains has no influence on the theoretical results derived. © 2007 Society for Industrial and Applied Mathematics.