Square and stretch multigrid for stochastic matrix eigenproblems

Abstract

SUMMARYA novel multigrid algorithm for computing the principal eigenvector of column-stochastic matrices is developed. The method is based on an approach originally introduced by Horton and Leutenegger (Perform. Eval. Rev. 1994; 22:191-200) whereby the coarse-grid problem is adapted to yield a better and better coarse representation of the original problem. A special feature of the present approach is the squaring of the stochastic matrix-followed by a stretching of its spectrum-just prior to the coarse-grid correction process. This procedure is shown to yield good convergence properties, even though a cheap and simple aggregation is used for the restriction and prolongation matrices, which is important for maintaining competitive computational costs. A second special feature is a bottom-up procedure for defining coarse-grid aggregates.

Algebraic Two-Level Convergence Theory for Singular Systems

“…However, the results obtained in this paper suggest that, at least regarding the theoretical foundations, this extension should raise no particular difficulties. From that viewpoint the present theoretical study also complements the works where multilevel algorithms have been successfully applied to solve singular problems (e.g., [3,10,11,12,21,35,36,39]). …”

“…In many cases, the systems are large and badly conditioned. In such a context, multilevel algorithms are often methods of choice [31,37,38], and these have effectively been successfully applied to singular systems in a number of works; see, e.g., [3,10,11,12,21,35,36,39]. For general iterative methods, theoretical issues associated with singularities have been studied for a long time and are now fairly well understood.…”

Recursively Accelerated Multilevel Aggregation for Markov Chains

“…Our RAMA method forms an alternative to other recently proposed ways of overcoming the slow convergence of simple nonoverlapping multilevel aggregation methods for Markov chains [18,23,24]. As in [19,20] for linear systems arising from elliptic PDE discretization, the recursively accelerated method does for many problems not lead to smaller iteration counts than competing approaches (and thus does not necessarily perform better for simple problems), but its value lies in its conceptual simplicity, probabilistic interpretation, and operator complexity, which may be lower for difficult problems.…”

“…Our recursive acceleration improves the convergence of the multilevel aggregation approach of [13,14,15] significantly, maintaining the probabilistic interpretation of the coarse-level operators. Finally, we also want to mention some recent different approaches for accelerating the multiplicative aggregation method for Markov chains, including the so-called square-and-stretch multigrid algorithm by Treister and Yavneh, which has shown good performance in a variety of test problems [24].…”

A Cost-Effective Smoothed Multigrid with Modified Neighborhood-Based Aggregation for Markov Chains

“…Here, we consider the aggregation methods which determine aggregates based on strength of connection in matrices. There exist some corresponding aggregation methods like distance-one aggregation, distance-two aggregation [16,17], pairwise aggregation, double pairwise aggregation [22], Neighborhood-Based aggregation [23], bottom-up aggregation [24], and some other types of aggregation. Serving as the aggregation method in [7,18,19], Neighborhood-Based aggregation has some advantages over others.…”