Iterative Aggregation-disaggregation Method Research Articles

Error propogation formula of multi-level aggregation-disaggregation methods for non-symmetric problems

Iterative aggregation-disaggregation methods for numerical computing of stationary probability distribution vectors of stochastic matrices are studied. The methods can use arbitrary numbers of levels and of smoothing steps. A formula for the error propagation is derived. Using this formula, some asymptotic convergence properties of these methods for non-symmetric problems are demonstrated.

Physiology and pathology of iterative aggregation–disaggregation methods

SUMMARYWe found characteristics that can help us predict the convergence or divergence of iterative aggregation–disaggregation methods. We provided two results for spectral radii of asymptotic error propagation matrices: (i) the spectral radius is bounded by unity for symmetric Markov chains and (ii) the spectral radius can be arbitrarily large for a certain class of sparse Markov chains. Surprisingly, permuting states of cyclic Markov chains by algorithms usually used for reducing bandwidth of matrices leads to the latter case. We proposed a sorting method that prevents divergence for this class of Markov chains.Copyright © 2011 John Wiley & Sons, Ltd.

Convergence of multi-level iterative aggregation–disaggregation methods

This paper introduces an error propagation formula of a certain class of multi-level iterative aggregation–disaggregation (IAD) methods for numerical solutions of stationary probability vectors of discrete finite Markov chains. The formula can be used to investigate convergence by computing the spectral radius of the error propagation matrix for specific Markov chains. Numerical experiments indicate that the same type of the formula could be used for a wider class of the multi-level IAD methods. Using the formula we show that for given data there is no relation between convergence of two-level and of multi-level IAD methods.

Necessary and sufficient local convergence condition of one class of iterative aggregation–disaggregation methods

AbstractThis paper concludes one part of the local convergence analysis of a certain class of iterative aggregation–disaggregation methods for computing a stationary probability distribution vector of an irreducible stochastic matrix B. We show that the local convergence of the algorithm is determined only by the sparsity pattern of the matrix and by the choice of the aggregation groups. We introduce the asymptotic convergence rates of the normalized components of approximations corresponding to particular aggregation groups and we also specify an upper bound on the rates. Copyright © 2008 John Wiley & Sons, Ltd.

Local convergence analysis of iterative aggregation–disaggregation methods with polynomial correction

The paper introduces some new results on local convergence analysis of one class of iterative aggregation–disaggregation methods for computing a stationary probability distribution vector of an irreducible stochastic matrix. We focus on methods, where the basic iteration on the fine level corresponds to a multiplication by a polynomial of order one with nonnegative coefficients in the original matrix. We show that this process is locally convergent for matrices with positive diagonals or when the coefficients of the polynomial are positive. On the other hand there are examples for which the process may diverge in a local sense for higher degree polynomials even if it converges for a polynomial of a lower degree for the same matrix.

A note on local and global convergence analysis of iterative aggregation–disaggregation methods

The purpose of the paper is to present some convergence properties of the iterative aggregation–disaggregation method for computing a stationary probability distribution vector of a column stochastic matrix. A sufficient condition for the local convergence property and the corresponding rate of convergence are established. Some global convergence considerations are presented. Several illustrative examples are included.

Iterative aggregation/disaggregation methods for computing some characteristics of Markov chains. II. Fast convergence

A class of iterative aggregation/disaggregation methods (IAD) for computation of some important characteristics of Markov chains such as stationary probability vectors and mean first passage times matrices is presented and convergence properties of the corresponding algorithms are analyzed. Particular attention is focused on the fast convergence. Some classes of problems are identified for which the IAD methods return exact solutions after one single iteration sweap.

Aggregation/Disaggregation Iterative Methods Applied to Leontev Systems and Markov Chains

The paper surveys some recent results on iterative aggregation/disaggregation methods (IAD) for computing stationary probability vectors of stochastic matrices and solutions of Leontev linear systems. A particular attention is paid to fast IAD methods.

On a two-level multigrid solution method for finite Markov chains

We study a two-level algebraic multigrid scheme for computing the stationary distribution of a homogeneous Markov chain with a finite state space. We reveal its relationship to an iterative aggregation-disaggregation method, provide an error analysis, and prove its local convergence. Moreover, we discuss the stochastic meaning of a block SOR procedure that can be used as iterative smoother of the multigrid scheme.

A new aggregation-disaggregation algorithm

This paper proposes a new algorithm for solving the Kolmogoroff equations for large Markov chains. The algorithm consists of two stages. At the first stage, it finds the optimal partition of states into blocks which makes the Markov chain as completely lumpable or decomposable as possible. We define an objective function for evaluating the lumpability or decomposability of a Markov chain with respect to a partition. A lower bound of the objective function is presented and the Branch-and-Bound method is proposed for obtaining the optimal partition. At the second stage, the Kolmogoroff equations are solved by the iterative aggregation-disaggregation method with the optimal partition. Numerical results are presented in order to demonstrate the effectiveness of the proposed algorithm.