Product preconditioning for Markov chain problems

Michele Benzi and Bora Uçar

Abstract. We consider preconditioned Krylov subspace methods for computing the stationary probability distribution vector of irreducible Markov chains. We propose preconditioners constructed as the product of two fairly simple preconditioners. Theoretical properties of the proposed product preconditioners are briefly discussed. We use graph partitioning tools to partition the coefficient matrix in order to build the preconditioner matrices, and we investigate the effect of the partitioning on the proposed preconditioners. Numerical experiments with GMRES on various Markov chain problems generated with the MARCA software package demonstrate that the proposed preconditioners are effective in reducing the number of iterations to convergence. Furthermore, the experimental results show that the number of partitions does not severely affect the number of iterations.

Key words. preconditioning, discrete Markov chains, iterative methods, graph partitioning