Algebraic Multigrid for Markov Chains

Abstract. An algebraic multigrid (AMG) method is presented for the calculation of the stationary probability vector of an irreducible Markov chain. The method is based on standard AMG for nonsingular linear systems, but in a multiplicative, adaptive setting. A modified AMG interpolation formula is proposed that produces a nonnegative interpolation operator with unit row sums. It is shown how the adoption of a previously described lumping technique maintains the irreducible singular M-matrix character of the coarse-level operators on all levels. Together, these properties are sufficient to guarantee the well-posedness of the algorithm. Numerical results show how it leads to nearly optimal multigrid efficiency for a representative set of test problems. Key words. multilevel method, Markov chain, stationary probability vector, algebraic multigrid AMS subject classifications. 65C40 Computational Markov chains, 60J22 Computational methods in Markov chains, 65F10 Iterative methods for line...

• Type: Journal
• Year: 2010
• Where: SIAMSC
• Authors: Hans De Sterck, Thomas A. Manteuffel, Stephen F. McCormick, K. Miller, John Ruge, G. Sanders
• Keywords: Algebraic Multigrid, Markov Chains, SIAMSC 2010, Stationary Probability Vector