A Maximum Entropy Method Based on Piecewise Linear Functions for the Recovery of a Stationary Density of Interval Mappings

Authors

Jiu Ding, University of Southern MississippiFollow
Congming Jin, Zheijang Sci-Tech UniversityFollow
Noah H. Rhee, University of Missouri, Kansas CityFollow
Aihui Zhou, Chinese Academy of SciencesFollow

Document Type

Article

Publication Date

12-1-2011

Department

Mathematics

School

Mathematics and Natural Sciences

Abstract

Let S:[0,1]→[0,1] be a nonsingular transformation such that the corresponding Frobenius-Perron operator P _S :L ¹(0,1)→L ¹(0,1) has a stationary density f ^∗. We propose a maximum entropy method based on piecewise linear functions for the numerical recovery of f ^∗. An advantage of this new approximation approach over the maximum entropy method based on polynomial basis functions is that the system of nonlinear equations can be solved efficiently because when we apply Newton’s method, the Jacobian matrices are positive-definite and tri-diagonal. The numerical experiments show that the new maximum entropy method is more accurate than the Markov finite approximation method, which also uses piecewise linear functions, provided that the involved moments are known. This is supported by the convergence rate analysis of the method.

Publication Title

Journal of Statistical Physics

Volume

145

Issue

6

First Page

1620

Last Page

1639

Recommended Citation

Ding, J., Jin, C., Rhee, N. H., Zhou, A. (2011). A Maximum Entropy Method Based on Piecewise Linear Functions for the Recovery of a Stationary Density of Interval Mappings. Journal of Statistical Physics, 145(6), 1620-1639.
Available at: https://aquila.usm.edu/fac_pubs/375