Smoothed Aggregation Multigrid for Markov Chains

Abstract

Abstract. A smoothed aggregation multigrid method is presented for the numerical calculation of the stationary probability vector of an irreducible sparse Markov chain. It is shown how smoothing the interpolation and restriction operators can dramatically increase the efficiency of aggregation multigrid methods for Markov chains that have been proposed in the literature. The proposed smoothing approach is inspired by smoothed aggregation multigrid for linear systems, supplemented with a new lumping technique that assures well-posedness of the coarse-level problems: the coarse-level operators are singular M-matrices on all levels, resulting in strictly positive coarse-level corrections on all levels. Numerical results show how these methods lead to nearly optimal multigrid efficiency for an extensive set of test problems, both when geometric and algebraic aggregation strategies are used.

An Accelerated Two-level Multigrid Method for Markov Chains

“…Here, in Algorithm 2, we adopt a strength-based aggregation procedure that proposed by De Sterck et al as our aggregation method, since it is able to improve an algebraically smooth error that varies slowly in a local neighborhood by scaling the original problem matrix A [24,25]. Otherwise go to step 2.…”

“…In Algorithm 3, we set the letter k as 2,3, 4,5,6,7, = k and then the corresponding methods in Algorithm 4 are denoted as TLE (2), TLE(3), TLE(4), TLE(5), TLE (6) and TLE (7), respectively. Here two Markov chain problems studied in [9,25] are considered in our experiments. Now, some special sets of parameters are supplied in this paragraph from [24,25].…”

“…Here two Markov chain problems studied in [9,25] are considered in our experiments. Now, some special sets of parameters are supplied in this paragraph from [24,25]. As mentioned in the previous sections, the weighted Jacobi method has been used as the pre-and post-smoothing approaches in Algorithm 1. in our experiments since this value works well for all tests that we have considered, even though it is likely to be problem-dependent.…”

“…For instance, the matrix splitting iterative methods [13,[15][16][17] Krylov, subspace methods [18][19][20][21] and some preconditioning techniques [6,9,17] and so on. What is more, based on the aggregation of Markov states, multigrid methods have been studied in the literature [22][23][24][25][26][27]. However, with the size of the Markov chains becomes large, the cost of multigrid methods is likely to have an increase.…”

Blending randomness in closed queueing network models

“…However, in intermediate situations both types of approximations tend to be loose. While numerical iterative methods exist to address this type of issue [16], they are unable to cope with state spaces of the scale considered in large queueing network models. This calls for developing approximation techniques for queueing network analysis in random environments that are more robust than average-environment and decomposition approximations.…”