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2013, 3(2): 235-245. doi: 10.3934/naco.2013.3.235

A unified maximum entropy method via spline functions for Frobenius-Perron operators

1. 

Department of Mathematics, University of Southern Mississippi, Hattiesburg, MS 39406-5045

2. 

Department of mathematics and Statistics, The University of Missouri - Kansas City, Kansas City, MO 64110-2499, United States

Received  February 2012 Revised  January 2013 Published  April 2013

We present a general frame of finite element maximum entropy methods for the computation of a stationary density of Frobenius-Perron operators associated with one dimensional transformations, based on spline function approximations. This gives a unified numerical approach to the density recovery for this class of positive operators by combining the principle of maximum entropy with the idea of finite elements. The norm convergence of the method is proved and the numerical results with the piecewise cubic method show its fast convergence.
Citation: Jiu Ding, Noah H. Rhee. A unified maximum entropy method via spline functions for Frobenius-Perron operators. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 235-245. doi: 10.3934/naco.2013.3.235
References:
[1]

C. Beck and Schlögl, "Thermodynamics of Chaotic Systems, an Introduction,", Cambridge University Press, (1993).

[2]

P. Biswas, H. Shimoyama and L. Mead, Lyaponov exponent and natural invariant density determination of chaostic maps: An iterative maximum entropy ansatz,, J. Phys., 43 (2010). doi: 10.1088/1751-8113/43/12/125103.

[3]

C. Bose and R. Murray, The exact rate of approximation in Ulam's method,, Disc. Cont. Dynam. Sys., 7 (2001), 219.

[4]

A. Boyarsky and P. Góra, "Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension,", Birkhäuser, (1997).

[5]

C. de Boor, "A Practical Guide to Splines,", Revised edition, (2001).

[6]

J. M. Borwein and A. S. Lewis, On the convergence of moment problems,, Trans. Amer. Math. Soc., 325 (1991), 249. doi: 10.1090/S0002-9947-1991-1008695-8.

[7]

J. M. Borwein and A. S. Lewis, Convergence of the best entropy estimates,, SIAM J. Optimi., 1 (1991), 191. doi: 10.1137/0801014.

[8]

J. Ding, A maximum entropy method for solving Frobenius-Perron operator equations,, Appl. Math. Comput., 93 (1998), 155. doi: 10.1016/S0096-3003(97)10061-3.

[9]

J. Ding, Q. Du and T.-Y. Li, High order approximation of the Frobenius-Perron operator,, Appl. Math. Comput., 53 (1993), 151. doi: 10.1016/0096-3003(93)90099-Z.

[10]

J. Ding, C. Jin, N. Rhee and A. Zhou, A maximum entropy method based on piecewise linear functions for the recovery of a stationary density of interval mappings,, J. Stat. Phys., 145 (2011), 1620. doi: 10.1007/s10955-011-0366-9.

[11]

J. Ding and T.-Y. Li, Markov approximations of Frobenius-Perron operator,, Nonlinear Anal. TMA, 17 (1991), 759. doi: 10.1016/0362-546X(91)90211-I.

[12]

J. Ding and T.-Y. Li, Projection solutions of Frobenius-Perron operator equations,, Inter. J. Math. Math. Sci., 16 (1993), 465. doi: 10.1155/S0161171293000584.

[13]

J. Ding and L. Mead, Maximum entropy approximation for Lyaponov exponents of chaotic maps,, J. Math. Phys., 43 (2002), 2518. doi: 10.1063/1.1465100.

[14]

J. Ding and N. Rhee, A modified piecewise linear Markov approximation of Markov operators,, Applied Math. Comput., 174 (2006), 236.

[15]

J. Ding and N. Rhee, A maximum entropy method based on orthogonal polynomials for Frobenius-Perron operators,, Adv. Applied Math. Mech., 3 (2011), 204.

[16]

J. Ding and N. Rhee, Birkhoff's ergodic theorem and the piecewise constant maximum entropy method for Frobenius-Perron operators,, Inter. J. Computer Math., 89 (2012), 1083. doi: 10.1080/00207160.2012.680446.

[17]

J. Ding and N. Rhee, On the norm convergence of a piecewise linear least squares method for Frobenius-Perron operators,, J. Math. Anal. Appl., 386 (2012), 91. doi: 10.1016/j.jmaa.2011.07.053.

[18]

J. Ding and A. Zhou, "Statistical Properties of Deterministic Systems,", Springer, (2009).

[19]

G. Froyland, Ulam's method for random interval maps,, Nonlinearity, 12 (1999), 1029. doi: 10.1088/0951-7715/12/4/318.

[20]

M. Keane, R. Murrary and L.-S. Young, Computing invariant measures for expanding circle maps,, Nonlinearity, 11 (1998), 27. doi: 10.1088/0951-7715/11/1/004.

[21]

A. Lasota and M. Mackey, "Chaos, Fractals, and Noises,", 2nd Edition, (1994).

[22]

T.-Y. Li, Finite approximation for the Frobenius-Perron operator, a solution to Ulam's conjecture,, J. Approx. Theory, 17 (1976), 177. doi: 10.1016/0021-9045(76)90037-X.

[23]

C. Liverani, Rigorous numerical investigation of the statistical properties of piecewise expanding maps. A feasibility study,, Nonlinearity, 14 (2001), 463. doi: 10.1088/0951-7715/14/3/303.

[24]

L. R. Mead and N. Papanicolaou, Maximum entropy in the problem of moments,, J. Math. Phys., 25 (1984), 2404.

[25]

S. Ulam, "A Collection of Mathematical Problems,", Interscience, (1960).

show all references

References:
[1]

C. Beck and Schlögl, "Thermodynamics of Chaotic Systems, an Introduction,", Cambridge University Press, (1993).

[2]

P. Biswas, H. Shimoyama and L. Mead, Lyaponov exponent and natural invariant density determination of chaostic maps: An iterative maximum entropy ansatz,, J. Phys., 43 (2010). doi: 10.1088/1751-8113/43/12/125103.

[3]

C. Bose and R. Murray, The exact rate of approximation in Ulam's method,, Disc. Cont. Dynam. Sys., 7 (2001), 219.

[4]

A. Boyarsky and P. Góra, "Laws of Chaos: Invariant Measures and Dynamical Systems in One Dimension,", Birkhäuser, (1997).

[5]

C. de Boor, "A Practical Guide to Splines,", Revised edition, (2001).

[6]

J. M. Borwein and A. S. Lewis, On the convergence of moment problems,, Trans. Amer. Math. Soc., 325 (1991), 249. doi: 10.1090/S0002-9947-1991-1008695-8.

[7]

J. M. Borwein and A. S. Lewis, Convergence of the best entropy estimates,, SIAM J. Optimi., 1 (1991), 191. doi: 10.1137/0801014.

[8]

J. Ding, A maximum entropy method for solving Frobenius-Perron operator equations,, Appl. Math. Comput., 93 (1998), 155. doi: 10.1016/S0096-3003(97)10061-3.

[9]

J. Ding, Q. Du and T.-Y. Li, High order approximation of the Frobenius-Perron operator,, Appl. Math. Comput., 53 (1993), 151. doi: 10.1016/0096-3003(93)90099-Z.

[10]

J. Ding, C. Jin, N. Rhee and A. Zhou, A maximum entropy method based on piecewise linear functions for the recovery of a stationary density of interval mappings,, J. Stat. Phys., 145 (2011), 1620. doi: 10.1007/s10955-011-0366-9.

[11]

J. Ding and T.-Y. Li, Markov approximations of Frobenius-Perron operator,, Nonlinear Anal. TMA, 17 (1991), 759. doi: 10.1016/0362-546X(91)90211-I.

[12]

J. Ding and T.-Y. Li, Projection solutions of Frobenius-Perron operator equations,, Inter. J. Math. Math. Sci., 16 (1993), 465. doi: 10.1155/S0161171293000584.

[13]

J. Ding and L. Mead, Maximum entropy approximation for Lyaponov exponents of chaotic maps,, J. Math. Phys., 43 (2002), 2518. doi: 10.1063/1.1465100.

[14]

J. Ding and N. Rhee, A modified piecewise linear Markov approximation of Markov operators,, Applied Math. Comput., 174 (2006), 236.

[15]

J. Ding and N. Rhee, A maximum entropy method based on orthogonal polynomials for Frobenius-Perron operators,, Adv. Applied Math. Mech., 3 (2011), 204.

[16]

J. Ding and N. Rhee, Birkhoff's ergodic theorem and the piecewise constant maximum entropy method for Frobenius-Perron operators,, Inter. J. Computer Math., 89 (2012), 1083. doi: 10.1080/00207160.2012.680446.

[17]

J. Ding and N. Rhee, On the norm convergence of a piecewise linear least squares method for Frobenius-Perron operators,, J. Math. Anal. Appl., 386 (2012), 91. doi: 10.1016/j.jmaa.2011.07.053.

[18]

J. Ding and A. Zhou, "Statistical Properties of Deterministic Systems,", Springer, (2009).

[19]

G. Froyland, Ulam's method for random interval maps,, Nonlinearity, 12 (1999), 1029. doi: 10.1088/0951-7715/12/4/318.

[20]

M. Keane, R. Murrary and L.-S. Young, Computing invariant measures for expanding circle maps,, Nonlinearity, 11 (1998), 27. doi: 10.1088/0951-7715/11/1/004.

[21]

A. Lasota and M. Mackey, "Chaos, Fractals, and Noises,", 2nd Edition, (1994).

[22]

T.-Y. Li, Finite approximation for the Frobenius-Perron operator, a solution to Ulam's conjecture,, J. Approx. Theory, 17 (1976), 177. doi: 10.1016/0021-9045(76)90037-X.

[23]

C. Liverani, Rigorous numerical investigation of the statistical properties of piecewise expanding maps. A feasibility study,, Nonlinearity, 14 (2001), 463. doi: 10.1088/0951-7715/14/3/303.

[24]

L. R. Mead and N. Papanicolaou, Maximum entropy in the problem of moments,, J. Math. Phys., 25 (1984), 2404.

[25]

S. Ulam, "A Collection of Mathematical Problems,", Interscience, (1960).

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