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Volume 3, Issue 2
A Maximum Entropy Method Based on Orthogonal Polynomials for Frobenius-Perron Operators

Jiu Ding & Noah H. Rhee

DOI: 10.4208/aamm.10-m1022

Adv. Appl. Math. Mech., 3 (2011), pp. 204-218.

Published online: 2011-03

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  • Abstract

Let $S$: [0, 1]→[0, 1] be a chaotic map and let $f^∗$ be a stationary density of the Frobenius-Perron operator $P_S$: $L^1$→$L^1$ associated with $S$. We develop a numerical algorithm for approximating $f^∗$, using the maximum entropy approach to an under-determined moment problem and the Chebyshev polynomials for the stability consideration. Numerical experiments show considerable improvements to both the original maximum entropy method and the discrete maximum entropy method. 

  • Keywords

Frobenius-Perron operator, stationary density, maximum entropy, orthogonal polynomials, Chebyshev polynomials.

  • AMS Subject Headings

41A35, 65D07, 65J10

  • Copyright

COPYRIGHT: © Global Science Press

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  • TXT
@Article{AAMM-3-204, author = {Ding , Jiu and Rhee , Noah H.}, title = {A Maximum Entropy Method Based on Orthogonal Polynomials for Frobenius-Perron Operators}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2011}, volume = {3}, number = {2}, pages = {204--218}, abstract = {

Let $S$: [0, 1]→[0, 1] be a chaotic map and let $f^∗$ be a stationary density of the Frobenius-Perron operator $P_S$: $L^1$→$L^1$ associated with $S$. We develop a numerical algorithm for approximating $f^∗$, using the maximum entropy approach to an under-determined moment problem and the Chebyshev polynomials for the stability consideration. Numerical experiments show considerable improvements to both the original maximum entropy method and the discrete maximum entropy method. 

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.10-m1022}, url = {http://global-sci.org/intro/article_detail/aamm/165.html} }
TY - JOUR T1 - A Maximum Entropy Method Based on Orthogonal Polynomials for Frobenius-Perron Operators AU - Ding , Jiu AU - Rhee , Noah H. JO - Advances in Applied Mathematics and Mechanics VL - 2 SP - 204 EP - 218 PY - 2011 DA - 2011/03 SN - 3 DO - http://doi.org/10.4208/aamm.10-m1022 UR - https://global-sci.org/intro/article_detail/aamm/165.html KW - Frobenius-Perron operator, stationary density, maximum entropy, orthogonal polynomials, Chebyshev polynomials. AB -

Let $S$: [0, 1]→[0, 1] be a chaotic map and let $f^∗$ be a stationary density of the Frobenius-Perron operator $P_S$: $L^1$→$L^1$ associated with $S$. We develop a numerical algorithm for approximating $f^∗$, using the maximum entropy approach to an under-determined moment problem and the Chebyshev polynomials for the stability consideration. Numerical experiments show considerable improvements to both the original maximum entropy method and the discrete maximum entropy method. 

Ding , Jiu and Rhee , Noah H.. (2011). A Maximum Entropy Method Based on Orthogonal Polynomials for Frobenius-Perron Operators. Advances in Applied Mathematics and Mechanics. 3 (2). 204-218. doi:10.4208/aamm.10-m1022
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