  This work concerns the development of an Algebraic Multilevel method for
computing stationary vectors of Markov chains. We present an efficient
Bootstrap Algebraic Multilevel method for this task. In our proposed approach,
we employ a multilevel eigensolver, with interpolation built using ideas based
on compatible relaxation, algebraic distances, and least squares fitting of
test vectors. Our adaptive variational strategy for computation of the state
vector of a given Markov chain is then a combination of this multilevel
eigensolver and associated multilevel preconditioned GMRES iterations. We show
that the Bootstrap AMG eigensolver by itself can efficiently compute accurate
approximations to the state vector. An additional benefit of the Bootstrap
approach is that it yields an accurate interpolation operator for many other
eigenmodes. This in turn allows for the use of the resulting AMG hierarchy to
accelerate the MLE steps using standard multigrid correction steps. The
proposed approach is applied to a range of test problems, involving
non-symmetric stochastic M-matrices, showing promising results for all problems
considered.
