Top-level acceleration of adaptive algebraic multilevel methods for steady-state solution to Markov chains

Abstract

In many application areas, including information retrieval and networking systems, finding the steady-state distribution vector of an irreducible Markov chain is of interest and it is often difficult to compute efficiently. The steady-state vector is the solution to a nonsymmetric eigenproblem with known eigenvalue, B x = x, subject to probability constraints $\Vert{\bf x}\Vert _1 = 1$ and x > 0, where B is column stochastic, that is, B ≥ O and 1 ^t B = 1 ^t. Recently, scalable methods involving Smoothed Aggregation (SA) and Algebraic Multigrid (AMG) were proposed to solve such eigenvalue problems. These methods use multiplicative iterate updates versus the additive error corrections that are typically used in nonsingular linear solvers. This paper discusses an outer iteration that accelerates convergence of multiplicative update methods, similar in principle to a preconditioned flexible Krylov wrapper applied to an additive iteration for a nonsingular linear problem. The acceleration is performed by selecting a linear combination of old iterates to minimize a functional within the space of probability vectors. Two different implementations of this idea are considered and the effectiveness of these approaches is demonstrated with representative examples.