Abstract
Recently, it was shown how the convergence of a class of multigrid methods for computing the stationary distribution of sparse, irreducible Markov chains can be accelerated by the addition of an outer iteration based on iterant recombination. The acceleration was performed by selecting a linear combination of previous fine-level iterates with probability constraints to minimize the two-norm of the residual using a quadratic programming method. In this paper we investigate the alternative of minimizing the one-norm of the residual. This gives rise to a nonlinear convex program which must be solved at each acceleration step. To solve this minimization problem we propose to use a deep-cuts ellipsoid method for nonlinear convex programs. The main purpose of this paper is to investigate whether an iterant recombination approach can be obtained in this way that is competitive in terms of execution time and robustness. We derive formulas for subgradients of the one-norm objective function and the constraint functions, and show how an initial ellipsoid can be constructed that is guaranteed to contain the exact solution and give conditions for its existence. We also investigate using the ellipsoid method to minimize the two-norm. Numerical tests show that the one-norm and two-norm acceleration procedures yield a similar reduction in the number of multigrid cycles. The tests also indicate that one-norm ellipsoid acceleration is competitive with two-norm quadratic programming acceleration in terms of running time with improved robustness.
Similar content being viewed by others
References
Bause F., Kritzinger P.: Stochastic Petri Nets. Springer, Germany (1996)
Bazaraa M.S., Sherali H.D., Shetty C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, New Jersey (2006)
Berman A., Plemmons R.J.: Nonnegative Matrices in the Mathematical Sciences. SIAM, Philadelphia, PA (1987)
Bertsimas D., Tsitisklis J.N.: Introduction to Linear Optimization. Athena Scientific, Belmont, MA (1997)
Bland R.G., Goldfarb D., Todd M.J.: The ellipsoid method: a survey. Oper. Res. 29(6), 1039–1091 (1981)
Brandt A., Mikulinsky V.: On recombining iterants in multigrid algorithms and problems with small islands. SIAM J. Sci. Comput. 16, 20–28 (1995)
Briggs W.L., Henson V.E., McCormick S.F.: A Multigrid Tutorial, 2nd edn. SIAM, Philadelphia, PA (2000)
Buchholz P.: Multilevel solutions for structured Markov chains. SIAM J. Matrix Anal. Appl. 22(2), 342–357 (2000)
Cao W.L., Stewart W.J.: Iterative aggregation/disaggregation techniques for nearly uncoupled Markov chains. JACM 32(3), 702–719 (1985)
Chatelin F., Miranker W.L.: Acceleration by aggregation of successive approximation methods. Linear Algebra Appl. 43, 17–47 (1982)
Sterck H., Manteuffel T., McCormick S.F., Miller K., Pearson J., Ruge J., Sanders G.: Smoothed aggregation multigrid for Markov chains. SIAM J. Sci. Comput. 32, 40–61 (2010)
Sterck H., Manteuffel T., McCormick S.F., Miller K., Ruge J., Sanders G.: Algebraic multigrid for Markov chains. SIAM J. Sci. Comput. 32, 544–562 (2010)
Sterck H., Manteuffel T., McCormick S.F., Nguyen Q., Ruge J.: Multilevel adaptive aggregation for Markov chains, with application to web ranking. SIAM J. Sci. Comput. 30, 2235–2262 (2008)
De Sterck H., Manteuffel T., Miller K., Sanders G.: Top-level acceleration of adaptive algebraic multilevel methods for steady-state solution to Markov chains. Adv. Comput. Math. 35, 375–403 (2010)
Sterck H., Miller K., Sanders G., Winlaw M.: Recursively accelerated multilevel aggregation for Markov chains. SIAM J. Sci. Comput. 32(3), 1652–1671 (2010)
Dziuban S.T., Ecker J.G., Kupferschmid M.: Using deep cuts in an ellipsoid algorithm for nonlinear programming. Math. Program. Stud. 25, 93–107 (1985)
Ecker J.G., Kupferschmid M.: An ellipsoid algorithm for nonlinear programming. Math. Program. 27, 83–106 (1983)
Frenk J.B.G., Gromicho J., Zhang S.: A deep cut ellipsoid algorithm for convex programming: theory and applications. Math. Program. 63, 83–108 (1994)
Goldfarb D., Todd M.J.: Modifications and implementation of the ellipsoid algorithm for linear programming. Math. Program. 23, 1–19 (1982)
Grassmann W., Taksar M., Heyman D.: Regenerative analysis and steady-state distributions for Markov chains. Oper. Res. 33(5), 1107–1116 (1985)
Haviv M.: Aggregation/disaggregation methods for computing the stationary distribution of Markov chains. SIAM J. Numer. Anal. 24(4), 952–966 (1987)
Horn R.A., Johnson C.R.: Matrix Analysis. Cambridge University Press, New York, NY (1985)
Horton, G., Leutenegger, S.T.: A multi-level solution algorithm for steady-state Markov chains. In: Proceedings of the 1994 ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems, pp. 191–200 (1994)
Iudin D.B., Nemirovskii A.S.: Informational complexity and effective methods of solution for convex extremal problems. Matekon Transl. Russ. East Eur. Math. Econ. 13, 3–25 (1976)
Khachiyan L.G.: A polynomial algorithm in linear programming. Sov. Math. Doklady 20, 191–194 (1976)
Koury J.R., McAllister D.F., Stewart W.J.: Iterative methods for computing stationary distributions of nearly completely decomposable Markov chains. SIAM J. Alg. Disc. Meth. 5(2), 164–186 (1984)
Krieger U.R.: Numerical solution of large finite Markov chains by algebraic multigrid techniques. In: Stewart, W. (eds) Numerical Solution of Markov Chains, pp. 403–424. Kluwer, Dordrecht (1995)
Krieger U.R.: On a two-level multigrid solution method for Markov chains. Linear Algebra Appl. 223–224, 415–438 (1995)
Leutenegger, S.T., Horton, G.: On the utility of the multi-level algorithm for the solution of nearly completely decomposable Markov chains. Tech. Rep. 94-44, ICASE (1994)
Lüthi H.-J.: On the solution of variational inequalities by the ellipsoid method. Math. Oper. Res. 10(3), 515–522 (1985)
Mandel J., Sekerka B.: A local convergence proof for the iterative aggregation method. Linear Algebra Appl. 51, 163–172 (1983)
Marek I., Mayer P.: Convergence analysis of an iterative aggregation/disaggregation method for computing stationary probability vectors of stochastic matrices. Numer. Linear Algebra Appl. 5, 253–274 (1998)
Marek I., Mayer P.: Convergence theory of some classes of iterative aggregation/disaggregation methods for computing stationary probability vectors of stochastic matrices. Linear Algebra Appl. 363, 177–200 (2003)
Molloy M.K.: IEEE Trans. Comput. C-31, 913–917 (1982)
Philippe B., Saad Y., Stewart W.J.: Numerical methods in Markov chain modeling. Oper. Res. 40(6), 1156–1179 (1992)
Rockafellar R.T.: Convex Analysis. Princeton University Press, New Jersey (1970)
Shor N.Z.: Cut-off method with space extension in convex programming problems. Cybernetics 13, 94–96 (1977)
Simon H.A., Ando A.: Aggregation of variables in dynamic systems. Econometrica 29, 111–138 (1961)
Stewart W.J.: An Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton, NJ (1994)
Takahashi, Y.: A lumping method for numerical calculations of stationary distributions of Markov chains. Tech. Rep. B-18, Department of Information Sciences, Tokyo Institute of Technology (1975)
Treister E., Yavneh I.: On-the-fly adaptive smoothed aggregation for Markov chains. SISC 33, 2927–2949 (2011)
Treister E., Yavneh I.: Square and stretch multigrid for stochastic matrix eigenproblems. Numer. Linear Algebr. 17, 229–251 (2010)
Trottenberg U., Oosterlee C.W., Schüller A.: Multigrid. Elsevier Academic Press, San Diego, California (2001)
Washio T., Oosterlee C.W.: Krylov subspace acceleration for nonlinear multigrid schemes. Electron. Trans. Numer. Anal. 6, 271–290 (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. W. Oosterlee and A. Borzi Pl.
Rights and permissions
About this article
Cite this article
De Sterck, H., Miller, K. & Sanders, G. Iterant recombination with one-norm minimization for multilevel Markov chain algorithms via the ellipsoid method. Comput. Visual Sci. 14, 51–65 (2011). https://doi.org/10.1007/s00791-011-0163-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00791-011-0163-7