Additive Schwarz iterations for Markov chains

SIAM Journal on Matrix Analysis and Applications

Volumn 27, Issue 2, 2006, Pages 445-458

  Bru, Rafael ^a   Pedroche, Francisco ^a   Szyld, Daniel B ^b  

^a UNIVERSITAT POLITÈCNICA DE VALÈNCIA   (Spain)

^b Temple University   (United States)

Author keywords

Additive Schwarz iterations, overlap; Markov chains; Singular linear systems

Indexed keywords

ADDITIVE SCHWARZ ITERATIONS, OVERLAP; DIAGONAL BLOCKS; SINGULAR LINEAR SYSTEMS;

CONVERGENCE OF NUMERICAL METHODS; LINEAR SYSTEMS; MATRIX ALGEBRA; PARTIAL DIFFERENTIAL EQUATIONS;

MARKOV PROCESSES;

EID: 33646271332     PISSN: 08954798     EISSN: 10957162     Source Type: Journal    
DOI: 10.1137/040616541     Document Type: Article

References (25)

1. G. ALEFELD AND N. SCHNEIDER, On square roots of M-matrices, Linear Algebra Appl., 42 (1982), pp. 119-132.
2. M. BENZI, A. FROMMER, R. NABBEN, AND D. B. SZYLD, Algebraic theory of multiplicative Schwarz methods, Numer. Math., 89 (2001), pp. 605-639.
3. M. BENZI AND D. B. SZYLD, Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods, Numer. Math., 76 (1997), pp. 309-321.
4. A. BERMAN AND R. J. PLEMMONS, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
5. A. BERMAN AND R. J. PLEMMONS, Nonnegative Matrices in the Mathematical Sciences, Classics Appl. Math. 9, SIAM, Philadelphia, 1994.
6. E. BOHL AND I. MAREK, A model of amplification, J. Comput. Appl. Math., 63 (1995), pp. 27-47.
7. R. BRU, F. PEDROCHE, AND D. B. SZYLD, Overlapping additive and multiplicative Schwarz iterations for H-matrices, Linear Algebra Appl., 393 (2004), pp. 91-105.
8. H. CHOI AND D. B. 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-Champaign, IL, 1996, IEEE Computer Society Press, Los Alamitos, CA, pp. 158-165.
9. T. DAYAR AND W. J. STEWART, Comparison of partitioning techniques for two-level iterative solvers on large, sparse Markov chains, SIAM J. Sci. Comput., 21 (2000), pp. 1691-1705.
10. A. FROMMER AND D. B. SZYLD, Weighted max norms, splittings, and overlapping additive Schwarz iterations, Numer. Math., 83 (1999), pp. 259-278.
11. I. MAREK AND D. B. SZYLD, Comparison theorems for the convergence factor of iterative methods for singular matrices, Linear Algebra Appl., 316 (2000), pp. 67-87.
12. I. MAREK AND D. B. SZYLD, Comparison of convergence of general stationary iterative methods for singular matrices, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 68-77.
13. I. MAREK AND D. B. SZYLD, Algebraic Schwarz methods for the numerical solution of Markov chains, Linear Algebra Appl., 386 (2004), pp. 67-81.
14. T. P. MATHEW, Uniform convergence of the Schwarz alternating method for solving singularly perturbed advection-diffusion equations, SIAM J. Numer. Anal., 35 (1998), pp. 1663-1683.
15. V. MIGALLÓN, J. PENADÉS, AND D. B. SZYLD, Experimental study of parallel iterative solutions of Markov chains with block partitions, in Numerical Solution of Markov Chains (NSMC'99), B. Plateau, W. J. Stewart, and M. Silva, eds., Prensas Universitarias de Zaragoza, Zaragoza, Spain, 1999, pp. 96-110.
16. B. PHILIPPE, Y. SAAD, AND W. J. STEWART, Numerical methods in Markov chain modeling, Oper. Res., 40 (1992), pp. 1156-1179.
17. A. QUARTERONI AND A. VALLI, Domain Decomposition Methods for Partial Differential Equations, Oxford Science Publications, Clarendon Press, Oxford, UK, 1999.
18. B. F. SMITH, P. E. BJøRSTAD, AND W. D. GROPP, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, Cambridge, UK, 1996.
19. W. J. STEWART, MARCA Models: A Collection of Markov Chain Models, http://www.csc.ncsu.edu/faculty/stewart/MARCA_Models/MARCA_Models.html.
20. W. J. STEWART, Introduction to the Numerical Solution of Markov Chains, Princeton University Press, Princeton, NJ, 1994.
21. W. J. STEWART AND W. WU, Numerical experiments with iteration and aggregation for Markov chains, ORSA J. Comput., 4 (1992), pp. 336-350.
22. D. B. SZYLD, Equivalence of convergence conditions for iterative methods for singular equations, Numer. Linear Algebra Appl., 1 (1994), pp. 151-154.
23. A. TOSELLI AND O. B. WIDLUND, Domain Decomposition - Algorithms and Theory, Springer Ser. Comput. Math. 34, Springer, Berlin, 2005.
24. R. S. VARGA, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962.
25. R. S. VARGA, Matrix Iterative Analysis, Springer Ser. Comput. Math. 27, 2nd ed., Springer, Berlin, 2000.