A Maximum Entropy Method Based on Piecewise Linear Functions for the Recovery of a Stationary Density of Interval Mappings
A Maximum Entropy Method Based on Piecewise Linear Functions for the Recovery of a Stationary Density of Interval Mappings
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DOI：
10.1007/s10955-011-0366-9
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发表时间：
2011-09
影响因子：
1.6
通讯作者：
Jiu Ding;Congming Jin;N. Rhee;Aihui Zhou
中科院分区：
文献类型：
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作者：
Jiu Ding;Congming Jin;N. Rhee;Aihui Zhou
设：[0,1]→[0,1]是一个非奇异变换，使得对应的Frobenius-Perron算子ps:L1（0,1）→L1（0,1）具有平稳密度f∗。我们提出了一种基于分段线性函数的最大熵方法来求解off∗的数值恢复。与基于多项式基函数的最大熵方法相比，这种新的近似方法的一个优点是，当我们应用牛顿方法时，雅可比矩阵是正定的三对角线矩阵，因此可以有效地求解非线性方程组。数值实验表明，在已知矩的情况下，最大熵法比同样使用分段线性函数的马尔可夫有限逼近法更精确。该方法的收敛速度分析支持了这一点。
LetS:[0,1]→[0,1] be a nonsingular transformation such that the corresponding Frobenius-Perron operatorPS:L1(0,1)→L1(0,1) has a stationary densityf∗. We propose a maximum entropy method based on piecewise linear functions for the numerical recovery off∗. 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.