A Bootstrap Algebraic Multilevel Method for Markov Chains

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. Further, we mention that our method, unlike other existing multilevel methods for Markov Chains, does not employ any special processing of the coarse-level systems to ensure that stochastic properties of the fine-level system are maintained there. The proposed approach is applied to a range of test problems, involving non-symmetric stochastic M-matrices, showing promising results for all problems considered.

Lean Algebraic Multigrid (LAMG): Fast Graph Laplacian Linear Solver

“…No parameter tuning should be required. In particular, we apply caliber-1 interpolation between levels, constructed using relaxed Test Vectors (TVs), but without bootstrapping them as in the papers [5,13]. Fast asymptotic convergence is achieved by the following three ideas.…”

“…Note that the nodal energy (3.9a) is a quadratic in x u and {x v } v∈Eu . Define The numerator is the local energy after a temporary relaxation step is performed at u (since the coarse-level correction is executed on a relaxed iterate during the cycle, this is the energy it aims to approximate; the papers [5,18] use a similar idea). The denominator is the energy obtained when x…”

Algebraic multigrid for directed graph Laplacian linear systems (NS‐LAMG)

“…The NS-LAMG algorithm described in this paper is meant as a building block for further study of AMG solvers for directed graphs. There has been an extensive study of MC-AMG algorithms (e.g., other works [21][22][23][24][25][26]28,29,35 ), many of which could easily be used in the setup phase. A few suggestions for future work include the following.…”

“…Thus, in order to solve , we must first solve . There exist a number of AMG algorithms for Markov chain stationary distribution systems (e.g., other works). We propose a simple modification to stationary‐aggregation MC‐AMG solvers that also uses low‐degree elimination, which will be discussed in the following section, to solve for the right null‐space vector.…”

“…Variations of multigrid algorithms for symmetric positive definite systems have been adapted and generalized for Markov chains. [22][23][24][25][26][27] Furthermore, a square-stretch multigrid method has been studied in the works of Treister et al 28,29 Many Markov chain techniques used in multigrid algorithms for Markov chain stationary distribution systems (MC-AMG) can be applied in conjunction with ideas from LAMG to create a multigrid solver for directed graphs.…”