• Corpus ID: 119312991

Algebraic Multilevel Methods for Markov Chains

@article{Polthier2017AlgebraicMM,
  title={Algebraic Multilevel Methods for Markov Chains},
  author={Lukas Polthier},
  journal={arXiv: Numerical Analysis},
  year={2017}
}
  • Lukas Polthier
  • Published 12 November 2017
  • Computer Science
  • arXiv: Numerical Analysis
A new algebraic multilevel algorithm for computing the second eigenvector of a column-stochastic matrix is presented. The method is based on a deflation approach in a multilevel aggregation framework. In particular a square and stretch approach, first introduced by Treister and Yavneh, is applied. The method is shown to yield good convergence properties for typical example problems. 

References

SHOWING 1-10 OF 23 REFERENCES

Numerical Methods in Markov Chain Modeling

TLDR
This paper describes and compares several methods for computing stationary probability distributions of Markov chains based on combinations of Krylov subspace techniques, single vector power iteration/relaxation procedures and acceleration techniques.

A Bootstrap Algebraic Multilevel Method for Markov Chains

TLDR
This work presents an efficient bootstrap algebraic multigrid (AMG) method for computing state vectors of Markov chains, and shows that the bootstrap AMG eigensolver by itself can efficiently compute accurate approximations to the steady state vector.

A multi-level solution algorithm for steady-state Markov chains

A new iterative algorithm, the multi-level algorithm, for the numerical solution of steady state Markov chains is presented. The method utilizes a set of recursively coarsened representations of the

Introduction to the numerical solution of Markov Chains

TLDR
This document discusses Markov Chains, Direct Methods, Iterative Methods, and Projection Methods for Stochastic Automata Networks, as well as some of the techniques used to design and implement these systems.

An Algebraic Multigrid Preconditioner for a Class of Singular M-Matrices

  • Elena Virnik
  • Computer Science, Mathematics
    SIAM J. Sci. Comput.
  • 2007
TLDR
Algebraic multigrid (AMG) is applied as a preconditioner for solving large singular linear systems of the type $(I-T^T x=0) with GMRES with adequate adaptation.

Smoothed Aggregation Multigrid for Markov Chains

TLDR
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.

Square and stretch multigrid for stochastic matrix eigenproblems

TLDR
A novel multigrid algorithm for computing the principal eigenvector of column‐stochastic matrices is developed based on an approach originally introduced by Horton and Leutenegger whereby the coarse‐grid problem is adapted to yield a better and better coarse representation of the original problem.

Numerical methods for large eigenvalue problems

TLDR
Over the past decade considerable progress has been made towards the numerical solution of large-scale eigenvalue problems, particularly for nonsymmetric matrices, and the methods and software that have led to these advances are surveyed.

Iterative methods for sparse linear systems

TLDR
This chapter discusses methods related to the normal equations of linear algebra, and some of the techniques used in this chapter were derived from previous chapters of this book.

Nonnegative Matrices in the Mathematical Sciences

1. Matrices which leave a cone invariant 2. Nonnegative matrices 3. Semigroups of nonnegative matrices 4. Symmetric nonnegative matrices 5. Generalized inverse- Positivity 6. M-matrices 7. Iterative