Aggregation/disaggregation iterative methods applied to Leontev systems and Markov chains

Ivo Marek; Petr Mayer

Applications of Mathematics (2002)

  • Volume: 47, Issue: 2, page 139-156
  • ISSN: 0862-7940

Abstract

top
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.

How to cite

top

Marek, Ivo, and Mayer, Petr. "Aggregation/disaggregation iterative methods applied to Leontev systems and Markov chains." Applications of Mathematics 47.2 (2002): 139-156. <http://eudml.org/doc/33109>.

@article{Marek2002,
abstract = {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.},
author = {Marek, Ivo, Mayer, Petr},
journal = {Applications of Mathematics},
keywords = {Leontev model; Markov chain; stochastic matrix; aggregation; stationary probability vector; Leontev model; Markov chain; stochastic matrix; aggregation; stationary probability vector},
language = {eng},
number = {2},
pages = {139-156},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Aggregation/disaggregation iterative methods applied to Leontev systems and Markov chains},
url = {http://eudml.org/doc/33109},
volume = {47},
year = {2002},
}

TY - JOUR
AU - Marek, Ivo
AU - Mayer, Petr
TI - Aggregation/disaggregation iterative methods applied to Leontev systems and Markov chains
JO - Applications of Mathematics
PY - 2002
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 47
IS - 2
SP - 139
EP - 156
AB - 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.
LA - eng
KW - Leontev model; Markov chain; stochastic matrix; aggregation; stationary probability vector; Leontev model; Markov chain; stochastic matrix; aggregation; stationary probability vector
UR - http://eudml.org/doc/33109
ER -

References

top
  1. 10.1145/3828.214137, J.  Assoc. Comput. Mach. 32 (1985), 702–719. (1985) MR0796209DOI10.1145/3828.214137
  2. Application of threshold partitioning of sparse matrices to Markov chains, In: Proceedings of the IEEE International Computer Performance and Dependability Symposium IPDS’96, Urbana, 1996, pp. 158–165. (1996) 
  3. Comparison of partitioning techniques for two-level iterative solvers on large, sparse Markov chains, Tech. Rep. BU-CEIS-9805, Department of Computer Engineering and Information Science, Bilkent University, Ankara, 1998. (1998) 
  4. An implementation of Tarjan’s algorithm for the block triangularization of a matrix, ACM Trans. Math. Software 4 (1978), 337–147. (1978) 
  5. The Theory of Matrices, Gos. Izd. Lit., Moscow, 1954. (Russian) (1954) 
  6. Reliability modelling of safety equipments, In: Proceedings of Programs and Algorithms of Numerical Mathematics, Libverda, June 2000, Mathematical Institute of the Academy of Sciences of the Czech Republic, Prague, 2000, pp. 78–84. (Czech) (2000) 
  7. Some aspects of modelling railway safety, Proc. SANM’99, Nečtiny, I. Marek (ed.), University of West Bohemia, Plzeň, 1999, pp. 135–140. (1999) 
  8. Aggregation/disaggregation method for safety models, Appl. Math. 47 (2002), . (2002) MR1894665
  9. 10.1137/0605019, SIAM J. Alg. Disc. Meth. 5 (1984), 164–186. (1984) MR0745437DOI10.1137/0605019
  10. On iterative aggregation/disaggregation methods for finite Markov chains, Preprint Deutsche Bundespost Telekom, Research and Technology Centre, 1990. (1990) 
  11. Numerical solution methods for large finite Markov chains, In: Performance and Reliability Evaluation, K. Hare et al. (eds.), R. Oldenbourg, Wien, 1994, pp. 267–318. (1994) 
  12. On the utility of the multi-level algorithm for the solution of nearly completely decomposable Markov chains, In: Proceedings of the Second Internationl Workshop on the Numerical Solution of Markov Chains, W. J. Stewart (ed.), Kluwer, Boston, 1995, pp. 425–442. (1995) 
  13. 10.1137/0119060, SIAM J.  Appl. Math. 19 (1970), 607–628. (1970) Zbl0219.47022MR0415405DOI10.1137/0119060
  14. 10.1002/(SICI)1099-1506(199807/08)5:4<253::AID-NLA124>3.0.CO;2-B, Numer. Linear Algebra Appl. 5 (1998), 253–274. (1998) MR1640726DOI10.1002/(SICI)1099-1506(199807/08)5:4<253::AID-NLA124>3.0.CO;2-B
  15. Convergence theory of a class of iterative aggregation/disaggregation methods for computing stationary probability vectors of stochastic matrices, Linear Algebra Appl. (2001), Submitted. (2001) MR1969068
  16. Iterative aggregation/disaggregation methods for computing stationary probability vectors of stochastic matrices can be finitely terminating, International Journal of Differential Equations 3 (2001), 301–313. (2001) MR1848180
  17. Matrix Market. A repository of test matrices of the National Institute of Standards and Technology, http://www math.nist.gov/MatrixMarket. 
  18. Convex Structures and Economic Theory, Academic Press, New York-London, 1968. (1968) Zbl0172.44502MR0277233
  19. Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. (1970) MR0273810
  20. A two stage iteration for solving nearly uncoupled Markov chains, In: IMA Volumes in Mathematics and Applications 60: Recent Advances in Iterative Methods, G. H. Golub, A. Greenbaum and M. Luskin (eds.), Springer Verlag, New York, 1994, pp. 201–216. (1994) 
  21. Introduction to the Numerical Solution of Markov Chains, Princeton University Press, Princeton, 1994. (1994) Zbl0821.65099MR1312831
  22. A lumping method for numerical calculations of stationary distributions of Markov chains, Res. Rep. B-18, Dept. of Inf. Sci. Tokyo Inst. of Tech., Tokyo, Japan, June 1975. (June 1975) 
  23. The error aggregation. A contribution to the theory of decomposable systems and applications, PhD Thesis, Faculty of Applied Sciences, Louvain Catholic University, Louvain-la Neuve, Belgium, 1981. (1981) 
  24. Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, 1962. (1962) MR0158502

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.