Multigrid Methods for Tensor Structured Markov Chains with Low Rank Approximation

Tensor structured Markov chains are part of stochastic models of many practical applications, e.g., in the description of complex production or telephone networks. The most interesting question in Markov chain models is the determination of the stationary distribution as a description of the long term behavior of the system. This involves the computation of the eigenvector corresponding to the dominant eigenvalue or equivalently the solution of a singular linear system of equations. Due to the tensor structure of the models the dimension of the operators grows rapidly and a direct solution without exploiting the tensor structure becomes infeasible. Algebraic multigrid methods have proven to be efficient when dealing with Markov chains without using tensor structure. In this work we present an approach to adapt the algebraic multigrid framework to the tensor frame, not only using the tensor structure in matrix-vector multiplications, but also tensor structured coarse-grid operators and tensor representations of the solution vector.