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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). Google Scholar

[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. Google Scholar

[3]

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

[4]

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

[5]

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

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[19]

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

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[24]

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

[25]

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

show all references

References:
[1]

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

[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. Google Scholar

[3]

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

[4]

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

[5]

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

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[14]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[18]

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

[19]

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

[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. Google Scholar

[21]

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

[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. Google Scholar

[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. Google Scholar

[24]

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

[25]

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

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