A Bootstrap Algebraic Multilevel method for Markov Chains

TL;DR

This paper introduces a Bootstrap Algebraic Multilevel method that efficiently computes stationary vectors of Markov chains, leveraging multilevel eigensolvers and adaptive strategies for improved accuracy and speed.

Contribution

It develops a novel Bootstrap AMG eigensolver combining compatible relaxation, algebraic distances, and least squares fitting for Markov chain analysis.

Findings

Efficiently computes accurate stationary vectors.

Yields an interpolation operator for multiple eigenmodes.

Accelerates multilevel eigensolver computations.

Abstract

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.