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

摘要

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.