×

Finite approximations of Markov operators. (English) Zbl 1013.65140

Summary: We survey some recent developments in the numerical analysis of Markov operators, and in particular Frobenius-Perron operators associated with chaotic discrete dynamical systems.

MSC:

65P20 Numerical chaos
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beck, C.; Schlögl, F., Thermodynamics of Chaotic Systems (1993), Cambridge University Press: Cambridge University Press Cambridge
[2] Bose, C.; Murray, R., The exact rate of approximation in Ulam’s method, Discrete Control Dynamical Systems, 7, 1, 219-235 (2001) · Zbl 1013.28013
[3] Boyarsky, A.; Góra, P., Laws of Chaos (1997), Birkhäuser: Birkhäuser Basel
[4] Boyarsky, A.; Gora, P.; Lou, Y. S., Constructive approximations to the invariant densities of higher-dimensional transformations, Constr. Approx., 10, 1-13 (1994) · Zbl 0796.28012
[5] Chiu, C.; Du, Q.; Li, T. Y., Error estimates of the Markov finite approximation of the Frobenius-Perron operator, Nonlinear Anal. TMA, 19, 5, 291-308 (1992) · Zbl 0788.28009
[6] Dellnitz, M.; Junge, O., On the approximation of complicated dynamical behavior, SIAM J. Numer. Anal., 36, 2, 491-515 (1999) · Zbl 0916.58021
[7] Deuflhard, P.; Dellnitz, M.; Junge, O.; Schütte, Ch., Computation of essential molecular dynamics by subdivision techniques I: basic concept, Computers and Molecular Dynamics, Lecturer Notes, Comput. Sci. Engng., 4, 98-115 (1998)
[8] Ding, J., Computing invariant measures for piecewise convex transformations, J. Statist. Phys., 83, 3/4, 623-635 (1996) · Zbl 1081.37500
[9] Ding, J.; Du, Q.; Li, T. Y., High order approximations of the Frobenius-Perron operator, Appl. Math. Comput., 53, 151-171 (1993) · Zbl 0769.65025
[10] Ding, J.; Hunt, F., Error estimates for quasi-compact Markov operators, Nonlinear Anal. TMA, 42, 85-95 (2000) · Zbl 0968.60077
[11] Ding, J.; Li, T. Y., Markov finite approximation of Frobenius-Perron operator, Nonlinear Anal. TMA, 17, 8, 759-772 (1991) · Zbl 0758.28014
[12] Ding, J.; Li, T. Y., Projection solutions of Frobenius-Perron operator equations, Internat. J. Math. Math. Sci., 16, 3, 465-484 (1993) · Zbl 0779.41016
[13] Ding, J.; Li, T. Y., A convergence rate analysis for Markov finite approximations to a class of Frobenius-Perron operators, Nonlinear Anal. TMA, 31, 5/6, 765-777 (1998) · Zbl 1003.47026
[14] J. Ding, Z. Wang, Parallel computation of invariant measures, Ann. Oper. Res., to appear.; J. Ding, Z. Wang, Parallel computation of invariant measures, Ann. Oper. Res., to appear. · Zbl 0997.65007
[15] Ding, J.; Zhou, A., The projection method for computing multidimensional absolutely continuous invariant measures, J. Statist. Phys., 77, 3/4, 899-908 (1994) · Zbl 0972.28501
[16] Ding, J.; Zhou, A., Piecewise linear Markov approximations of Frobenius-Perron operators associated with multi-dimensional transformations, Nonlinear Anal. TMA, 25, 4, 399-408 (1995) · Zbl 0907.47025
[17] Ding, J.; Zhou, A., Finite approximations of Frobenius-Perron operators, a solution to Ulam’s conjecture for multi-dimensional transformations, Physica D, 92, 61-68 (1996) · Zbl 0890.58035
[18] Ding, J.; Zhou, A., Absolutely continuous invariant measures for piecewise \(C^2\) and expanding mappings in higher dimensions, Discrete Control Dynamical Systems, 6, 451-458 (2000) · Zbl 1009.37012
[19] Ding, J.; Zhou, A., Constructive approximations of Markov operators, J. Statist. Phys., 105, 5/6, 863-878 (2001) · Zbl 0991.37004
[20] Froyland, G., Approximating physical invariant measures of mixing dynamical systems in higher dimensions, Nonlinear Anal. TMA, 32, 7, 831-860 (1998) · Zbl 0973.37013
[21] Froyland, G., Using Ulam’s method to calculate entropy and other dynamical invariants, Nonlinearity, 12, 79-101 (1999) · Zbl 0917.58019
[22] Froyland, G., Ulam’s method for random interval maps, Nonlinearity, 12, 1029-1052 (1999) · Zbl 0989.37049
[23] Giusti, E., Minimal Surfaces and Functions of Bounded Variation (1984), Birkhäuser: Birkhäuser Basel · Zbl 0545.49018
[24] Góra, P.; Boyarsky, A., Absolutely continuous invariant measures for piecewise expanding \(C^2\) transformations in \(R^N\), Israel J. Math., 67, 3, 272-286 (1989) · Zbl 0691.28004
[25] Hunt, F., A Monte Carlo approach to the approximation of invariant measures, Random & Comput. Dynamical, 2, 1, 111-133 (1994) · Zbl 0804.58033
[26] Hunt, F.; Miller, W., On the approximation of invariant measures, J. Statist. Phys., 66, 535-548 (1992) · Zbl 0892.58048
[27] Jablonski, M., On invariant measures for piecewise \(C^2\)-transformations of the \(n\)-dimensional cube, Ann. Polon. Math., XLIII, 185-195 (1983) · Zbl 0591.28014
[28] Keane, M.; Murray, R.; Young, L.-S., Computing invariant measures for expanding circle maps, Nonlinearity, 11, 27-46 (1998) · Zbl 0903.58019
[29] Keller, G., Stochastic stability in some chaotic dynamical systems, Monograph. Math., 94, 313-353 (1982) · Zbl 0496.58010
[30] Kohda, T.; Murao, K., Piecewise polynomial Galerkin approximation to invariant densities of one-dimensional difference equations, Electron. Comm. Japan, 65A, 6, 1-11 (1982)
[31] Krzyzewski, K.; Szlenk, W., On invariant measures for expanding differential mappings, Stud. Math., 33, 83-92 (1969) · Zbl 0176.00901
[32] Lasota, A.; Mackey, M., Chaos, Fractals and Noises (1994), Springer: Springer Berlin
[33] Lasota, A.; Yorke, J., On the existence of invariant measures for piecewise monotonic transformations, Trans. Amer. Math. Soc., 186, 481-488 (1973) · Zbl 0298.28015
[34] Li, T. Y., Finite approximation for the Frobenius-Perron operator, a solution to Ulam’s conjecture, J. Approx. Theory, 17, 177-186 (1976) · Zbl 0357.41011
[35] Miller, W., Stability and approximation of invariant measures for a class of non-expanding transformations, Nonlinear Anal. TMA, 23, 1013-1025 (1994) · Zbl 0822.28009
[36] Murray, R., Approximation error for invariant density calculations, Discrete Control Dynamics Systems, 4, 3, 535-557 (1998) · Zbl 0988.37009
[37] Ch. Schütte, Conformational dynamics: modeling, theory, algorithm, and application to biomoleculaes, Habilitation Thesis, Freie Universität Berlin, 1999.; Ch. Schütte, Conformational dynamics: modeling, theory, algorithm, and application to biomoleculaes, Habilitation Thesis, Freie Universität Berlin, 1999.
[38] S.M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics Vol. 8, Interscience, New York, 1960.; S.M. Ulam, A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics Vol. 8, Interscience, New York, 1960. · Zbl 0086.24101
[39] Z. Wang, Parallel Monte Carlo computation of invariant measures, Ph.D. Dissertation, Program in Scientific Computing, The University of Southern Mississippi, Mississippi, 2000.; Z. Wang, Parallel Monte Carlo computation of invariant measures, Ph.D. Dissertation, Program in Scientific Computing, The University of Southern Mississippi, Mississippi, 2000.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.