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.
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W. L. Cao, W. J. Stewart: Iterative aggregation/disaggregation techniques for nearly uncoupled Markov chains. J. Assoc. Comput. Mach. 32 (1985), 702–719.
H. Choi, D. Szyld: 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.
T. Dayar, W. J. Stewart: 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.
I. S. Duff, J.K. Reid: An implementation of Tarjan's algorithm for the block triangularization of a matrix. ACM Trans. Math. Software 4 (1978), 337–147.
F. R. Gantmacher: The Theory of Matrices. Gos. Izd. Lit., Moscow, 1954 (In Russian.); English translation: Applications of the Theory of Matrices. Interscience, New York, 1959.
Š. Klapka, P. Mayer: 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. (In Czech.)
Š. Klapka, P. Mayer: Some aspects of modelling railway safety. Proc. SANM'99, Neůtiny (I. Marek, ed.). University of West Bohemia, Plzeů, 1999, pp. 135–140.
Š. Klapka, P. Mayer: Aggregation/disaggregation method for safety models. Appl. Math. 47 (2002).
R. Koury, D. F. McAllister and W. J. Stewart: Methods for computing stationary distributions of nearly completely decomposable Markov chains. SIAM J. Alg. Disc. Meth. 5 (1984), 164–186.
U. Krieger: On iterative aggregation/disaggregation methods for finite Markov chains. Preprint Deutsche Bundespost Telekom. Research and Technology Centre, 1990.
U. Krieger: 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.
S. T. Leutenegger, G. H. Horton: 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.
I. Marek: Frobenius theory of positive operators. Comparison theorems and applications. SIAM J. Appl. Math. 19 (1970), 607–628.
I. Marek, P. Mayer: Convergence analysis of an aggregation/disaggregation iterative method for computation stationary probability vectors of stochastic matrices. Numer. Linear Algebra Appl. 5 (1998), 253–274.
I. Marek, P. Mayer: Convergence theory of a class of iterative aggregation/disaggregation methods for computing stationary probability vectors of stochastic matrices. Linear Algebra Appl. (2001). Submitted.
I. Marek, P. Mayer: 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.
Matrix Market. A repository of test matrices of the National Institute of Standards and Technology. http://www math.nist.gov/MatrixMarket.
H. Nikaido: Convex Structures and Economic Theory. Academic Press, New York-London, 1968.
J. Ortega, W. Rheinboldt: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970.
G. W. Stewart, W. J. Stewart and D. F. McAllister: 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.
W. J. Stewart: Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton, 1994.
Y. Takahashi: 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.
H. Vantilborgh: 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.
R. S. Varga: Matrix Iterative Analysis. Prentice-Hall, Englewood Cliffs, 1962.
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Marek, I., Mayer, P. Aggregation/Disaggregation Iterative Methods Applied to Leontev Systems and Markov Chains. Applications of Mathematics 47, 139–156 (2002). https://doi.org/10.1023/A:1021785102298
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DOI: https://doi.org/10.1023/A:1021785102298