An Adaptively Constructed Algebraic Multigrid Preconditioner for Irreducible Markov Chains

TL;DR

This paper enhances algebraic multigrid methods for efficiently computing stationary distributions of Markov chains with non-symmetric matrices by using singular vectors in the bootstrap setup, leading to faster convergence.

Contribution

It introduces a novel bootstrap algebraic multigrid framework that employs singular vectors, improving convergence speed for solving non-symmetric linear systems in Markov chain analysis.

Findings

Fast convergence demonstrated on test problems

Favorable scaling behavior observed

Theoretical convergence speed results provided

Abstract

The computation of stationary distributions of Markov chains is an important task in the simulation of stochastic models. The linear systems arising in such applications involve non-symmetric M-matrices, making algebraic multigrid methods a natural choice for solving these systems. In this paper we investigate extensions and improvements of the bootstrap algebraic multigrid framework for solving these systems. This is achieved by reworking the bootstrap setup process to use singular vectors instead of eigenvectors in constructing interpolation and restriction. We formulate a result concerning the convergence speed of GMRES for singular systems and experimentally justify why rapid convergence of the proposed method can be expected. We demonstrate its fast convergence and the favorable scaling behavior for various test problems.