A bootstrap algebraic multilevel method for markov chains

SIAM Journal on Scientific Computing

Volumn 33, Issue 6, 2011, Pages 3425-3446

  Bolten, Matthias ^a   Brandt, Achi ^b   Brannick, James ^c   Frommer, Andreas ^a   Kahl, Karsten ^a   Livshits, Ira ^d  

^a UNIVERSITY OF WUPPERTAL   (Germany)

^b WEIZMANN INSTITUTE OF SCIENCE   (Israel)

^c PENNSYLVANIA STATE UNIVERSITY   (United States)

^d BALL STATE UNIVERSITY   (United States)

Author keywords

Adaptive methods; Algebraic multigrid; Bootstrap; Least squares prolongation; Markov chains; Preconditioned GMRES iteration

Indexed keywords

CHAINS; INTERPOLATION; ITERATIVE METHODS; MARKOV PROCESSES; MATRIX ALGEBRA; STOCHASTIC SYSTEMS; VECTORS;

ADAPTIVE METHODS; ALGEBRAIC MULTIGRIDS; BOOTSTRAP; GENERALIZED MINIMAL RESIDUALS; LEAST SQUARE; LEAST SQUARE PROLONGATION; MULTILEVEL METHOD; MULTILEVELS; PRECONDITIONED GENERALIZED MINIMAL RESIDUAL ITERATION; STATE VECTOR;

LEAST SQUARES APPROXIMATIONS;

EID: 84856525409     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/100791816     Document Type: Article

References (35)

1. A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics Appl. Math. 9, SIAM, Philadelphia, 1994.
2. A. Brandt, General highly accurate algebraic coarsening, Electron. Trans. Numer. Anal., 10 (2000), pp. 1-20. (Pubitemid 38816963)
3. A. Brandt, Multiscale scientific computation: Review 2001, in Multiscale and Multiresolution Methods: Theory and Applications, T. Barth, T. Chan, and R. Haimes, eds., Springer-Verlag, Heidelberg, 2001, pp. 1-96. Also available at www.wisdom.weizmann.ac.il/~achi/review00.ps.
4. A. Brandt, Principles of systematic upscaling, in Bridging the Scales in Science and Engineering, J. Fish, ed., Oxford University Press, Oxford, UK, 2008, pp. 193-215.
5. A. Brandt, J. Brannick, K. Kahl, and I. Livshits, Bootstrap AMG, SIAM J. Sci. Comput., 33 (2011), pp. 612-632.
6. J. Brannick, Adaptive Algebraic Multigrid Coarsening Strategies, Ph.D. thesis, Department of Applied Mathematics, University of Colorado at Boulder, Boulder, CO, 2005.
7. J. J. Brannick and R. D. Falgout, Compatible relaxation and coarsening in algebraic multigrid, SIAM J. Sci. Comput., 32 (2010), pp. 1393-1416.
8. M. Brezina, R. Falgout, S. MacLachlan, T. Manteuffel, S. McCormick, and J. Ruge, Adaptive smoothed aggregation (aSA) multigrid, SIAM Rev., 47 (2005), pp. 317-346. (Pubitemid 40889969)
9. P. Buchholz, Multilevel solutions for structured Markov chains, SIAM J. Matrix Anal. Appl., 22 (2000), pp. 342-357. (Pubitemid 33256058)
10. P. Buchholz and T. Dayar, Comparison of multilevel methods for Kronecker-based Markovian representations, Computing, 73 (2004), pp. 349-371.
11. P. Buchholz and T. Dayar, On the convergence of a class of multilevel methods for large, sparse Markov chains, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 1025-1049.
12. F. Chatelin andW. Miranker, Acceleration by aggregation of successive approximation methods, Linear Algebra Appl., 43 (1982), pp. 17-47.
13. H. De Sterck, T. A. Manteuffel, S. F. McCormick, K. Miller, J. Pearson, J. Ruge, and G. Sanders, Smoothed aggregation multigrid for Markov chains, SIAM J. Sci. Comput., 32 (2010), pp. 40-61.
14. H. De Sterck, T. A. Manteuffel, S. F. McCormick, K. Miller, J. Ruge, and G. Sanders, Algebraic multigrid for Markov chains, SIAM J. Sci. Comput., 32 (2010), pp. 544-562.
15. H. De Sterck, T. A. Manteuffel, S. F. McCormick, Q. Nguyen, and J. Ruge, Multilevel adaptive aggregation for Markov chains with application to web ranking, SIAM J. Sci. Comput., 30 (2008), pp. 2235-2262.
16. H. De Sterck, K. Miller, G. Sanders, and M. Winlaw, Recursively accelerated multilevel aggregation for Markov chains, SIAM J. Sci. Comput., 32 (2010), pp. 1652-1671.
17. R. Falgout and P. S. Vassilevski, On generalizing the algebraic multigrid framework, SIAM J. Numer. Anal., 42 (2004), pp. 1669-1693. (Pubitemid 41366170)
18. R. Falgout, P. Vassilevski, and L. Zikatanov, On two-grid convergence estimates, Numer. Linear Algebra Appl., 12 (2005), pp. 471-494. (Pubitemid 40977207)
19. M. Haviv, Aggregation/disaggregation methods for computing the stationary distribution of a Markov chain, SIAM J. Numer. Anal., 24 (1987), pp. 952-966. (Pubitemid 17637140)
20. K. Hayami and M. Sugihara, A geometric view of Krylov subspace methods on singular systems, Numer. Linear Algebra Appl., 18 (2001), pp. 449-469.
21. S. Kaczmarz, Angenäherte Auf lösung von Systemen linearer Gleichungen, Bull. Internat. Acad. Polon. Sci. Lettres A, (1937), pp. 335-357.
22. U. Krieger, B. Mueller-Clostermann, and M. Sczittnick, Modeling and analysis of communication systems based on computational methods for Markov chains, IEEE J. Selected Areas in Communications, 8 (1990), pp. 1630-1648.
23. D. Kushnir, M. Galun, and A. Brandt, Efficient multilevel eigensolvers with applications to data analysis tasks, IEEE Trans. Pattern Anal. Machine Intell., 32 (2010), pp. 1377-1391.
24. O. Livne, Coarsening by compatible relaxation, Numer. Linear Algebra Appl., 11 (2004), pp. 205-227. (Pubitemid 39604610)
25. I. Livshits, One-dimensional algorithm for finding eigenbasis of the Schrödinger operator, SIAM J. Sci. Comput., 30 (2008), pp. 416-440.
26. J. Mandel and B. Sekerka, A local convergence proof for the iterative aggregation method, Linear Algebra Appl., 51 (1983), pp. 163-172.
27. I. Marek and I. Pultarova, A note on local and global convergence analysis of iterative aggregation/disaggregation methods, Linear Algebra Appl., 413 (2006), pp. 327-341. (Pubitemid 43152396)
28. Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856-869.
29. E. Seneta, Non-negative Matrices and Markov Chains, 2nd rev. ed., Springer Ser. Statist., Springer, Heidelberg, New York, 1981.
30. H. Simon and A. Ando, Aggregation of variables in dynamic systems, Econometrica, 29 (1961), pp. 111-138.
31. W. J. Stewart, Introduction to the Numerical Solution of Markov Chains, Princeton University Press, Princeton, NJ, 1994.
32. Y. Takahashi, A Lumping Method for Numerical Calculations of Stationary Distributions of Markov Chains, Tech. Report B-18, Department of Information Sciences, Tokyo Institute of Technology, Tokyo, 1975.
33. U. Trottenberg, C. Oosterlee, and A. Schüller, Multigrid, Academic Press, San Diego, 2001.
34. E. Virnik, An algebraic multigrid preconditioner for a class of singular M-matrices, SIAM J. Sci. Comput., 29 (2007), pp. 1982-1991.
35. P. N. Brown and H. F. Walker, GMRES on (nearly) singular systems, SIAM J. Matrix Anal. Appl., 18 (1997), pp. 37-51.