# American Institute of Mathematical Sciences

• Previous Article
Linearized alternating direction method of multipliers with Gaussian back substitution for separable convex programming
• NACO Home
• This Issue
• Next Article
Subspace trust-region algorithm with conic model for unconstrained optimization
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., A 43 (2010), 125103. 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), 219-235.  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, Springer, 2001. Google Scholar [6] J. M. Borwein and A. S. Lewis, On the convergence of moment problems, Trans. Amer. Math. Soc., 325 (1991), 249-271. 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), 191-205. doi: 10.1137/0801014.  Google Scholar [8] J. Ding, A maximum entropy method for solving Frobenius-Perron operator equations, Appl. Math. Comput., 93 (1998), 155-168. 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), 151-171. 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), 1620-1639. 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), 759-772. 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), 465-484. 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), 2518-2522. 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), 236-251.  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), 204-218.  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), 1083-1091. 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), 91-102. 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), 1029-1052. 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), 27-46. doi: 10.1088/0951-7715/11/1/004.  Google Scholar [21] A. Lasota and M. Mackey, "Chaos, Fractals, and Noises," 2nd Edition, Springer-Verlag, New York, 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), 177-186. 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), 463-490. 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), 2404-2417.  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., A 43 (2010), 125103. 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), 219-235.  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, Springer, 2001. Google Scholar [6] J. M. Borwein and A. S. Lewis, On the convergence of moment problems, Trans. Amer. Math. Soc., 325 (1991), 249-271. 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), 191-205. doi: 10.1137/0801014.  Google Scholar [8] J. Ding, A maximum entropy method for solving Frobenius-Perron operator equations, Appl. Math. Comput., 93 (1998), 155-168. 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), 151-171. 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), 1620-1639. 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), 759-772. 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), 465-484. 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), 2518-2522. 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), 236-251.  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), 204-218.  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), 1083-1091. 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), 91-102. 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), 1029-1052. 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), 27-46. doi: 10.1088/0951-7715/11/1/004.  Google Scholar [21] A. Lasota and M. Mackey, "Chaos, Fractals, and Noises," 2nd Edition, Springer-Verlag, New York, 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), 177-186. 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), 463-490. 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), 2404-2417.  Google Scholar [25] S. Ulam, "A Collection of Mathematical Problems," Interscience, 1960. Google Scholar
 [1] Stefan Klus, Péter Koltai, Christof Schütte. On the numerical approximation of the Perron-Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (1) : 51-79. doi: 10.3934/jcd.2016003 [2] Marc Kesseböhmer, Sabrina Kombrink. A complex Ruelle-Perron-Frobenius theorem for infinite Markov shifts with applications to renewal theory. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 335-352. doi: 10.3934/dcdss.2017016 [3] Stefan Klus, Christof Schütte. Towards tensor-based methods for the numerical approximation of the Perron--Frobenius and Koopman operator. Journal of Computational Dynamics, 2016, 3 (2) : 139-161. doi: 10.3934/jcd.2016007 [4] Martin Lustig, Caglar Uyanik. Perron-Frobenius theory and frequency convergence for reducible substitutions. Discrete & Continuous Dynamical Systems, 2017, 37 (1) : 355-385. doi: 10.3934/dcds.2017015 [5] Gary Froyland, Ognjen Stancevic. Escape rates and Perron-Frobenius operators: Open and closed dynamical systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 457-472. doi: 10.3934/dcdsb.2010.14.457 [6] Marianne Akian, Stéphane Gaubert, Antoine Hochart. A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 207-231. doi: 10.3934/dcds.2020009 [7] Julia Piantadosi, Phil Howlett, Jonathan Borwein, John Henstridge. Maximum entropy methods for generating simulated rainfall. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 233-256. doi: 10.3934/naco.2012.2.233 [8] Dušan M. Stipanović, Claire J. Tomlin, George Leitmann. A note on monotone approximations of minimum and maximum functions and multi-objective problems. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 487-493. doi: 10.3934/naco.2011.1.487 [9] James P. Kelly, Kevin McGoff. Entropy conjugacy for Markov multi-maps of the interval. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2071-2094. doi: 10.3934/dcds.2020353 [10] András Bátkai, Istvan Z. Kiss, Eszter Sikolya, Péter L. Simon. Differential equation approximations of stochastic network processes: An operator semigroup approach. Networks & Heterogeneous Media, 2012, 7 (1) : 43-58. doi: 10.3934/nhm.2012.7.43 [11] Alexander Schaub, Olivier Rioul, Jean-Luc Danger, Sylvain Guilley, Joseph Boutros. Challenge codes for physically unclonable functions with Gaussian delays: A maximum entropy problem. Advances in Mathematics of Communications, 2020, 14 (3) : 491-505. doi: 10.3934/amc.2020060 [12] Fritz Colonius. Invariance entropy, quasi-stationary measures and control sets. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2093-2123. doi: 10.3934/dcds.2018086 [13] Frank Neubrander, Koray Özer, Lee Windsperger. On subdiagonal rational Padé approximations and the Brenner-Thomée approximation theorem for operator semigroups. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3565-3579. doi: 10.3934/dcdss.2020238 [14] Yunkyong Hyon, José A. Carrillo, Qiang Du, Chun Liu. A maximum entropy principle based closure method for macro-micro models of polymeric materials. Kinetic & Related Models, 2008, 1 (2) : 171-184. doi: 10.3934/krm.2008.1.171 [15] Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247 [16] Gregorio Díaz, Jesús Ildefonso Díaz. On the free boundary associated with the stationary Monge--Ampère operator on the set of non strictly convex functions. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1447-1468. doi: 10.3934/dcds.2015.35.1447 [17] Scott W. Hansen. Controllability of a basic cochlea model. Evolution Equations & Control Theory, 2016, 5 (4) : 475-487. doi: 10.3934/eect.2016015 [18] Yubo Yuan, Weiguo Fan, Dongmei Pu. Spline function smooth support vector machine for classification. Journal of Industrial & Management Optimization, 2007, 3 (3) : 529-542. doi: 10.3934/jimo.2007.3.529 [19] Amir Averbuch, Pekka Neittaanmäki, Valery Zheludev. Periodic spline-based frames for image restoration. Inverse Problems & Imaging, 2015, 9 (3) : 661-707. doi: 10.3934/ipi.2015.9.661 [20] Claude Carlet, Serge Feukoua. Three basic questions on Boolean functions. Advances in Mathematics of Communications, 2017, 11 (4) : 837-855. doi: 10.3934/amc.2017061

Impact Factor:

## Metrics

• PDF downloads (85)
• HTML views (0)
• Cited by (5)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]