计算不变密度的一种二次样条最大熵方法（英文）
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摘要
用二次样条函数来数值逼近对应于非奇异变换的Frobenius-Perron算子的不变密度.所提出的方法消除了使用多项式函数的最大熵方法中出现的坏条件性.只要不变密度有足够的光滑度,由于算法的高阶收敛速率,随着矩量函数个数的增加,数值计算的精度会迅速增加.给出的数值例子验证了算法收敛速度的理论分析.
The numerical recovery of an invariant density of the Frobenius-Perron operator corresponding to a nonsingular transformation is depicted by using quadratic spline functions.We implement a maximum entropy method to approximate the invariant density.The proposed method removes the ill-conditioning in the maximum entropy method,which arises by the use of polynomials.Due to the smoothness of the functions and a good convergence rate,the accuracy in the numerical calculation increases rapidly as the number of moment functions increases.The numerical results from the proposed method are supported by the theoretical analysis.
The numerical recovery of an invariant density of the Frobenius-Perron operator corresponding to a nonsingular transformation is depicted by using quadratic spline functions.We implement a maximum entropy method to approximate the invariant density.The proposed method removes the ill-conditioning in the maximum entropy method,which arises by the use of polynomials.Due to the smoothness of the functions and a good convergence rate,the accuracy in the numerical calculation increases rapidly as the number of moment functions increases.The numerical results from the proposed method are supported by the theoretical analysis.
引文
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[6]LI TY,YORKE JA.Ergodic transformations from an interval into itself[J].Trans Amer Mam Soc,1978,235:177-186.
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[10]FROYLAND G.Finite approximation of Sinai-Bowen-Ruelle measures for Anosov systems in two dimensions[J].Random Comput Dynam,1995,3:251-264.
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[12]DING J.A maximum entropy method for solving Frobenius-Perron operator equations[J].Appl Math Compt,1998,93:155-168.
[13]DING J,RHEE N.A maximum entropy method based on orthogonal polynomials for Frobenius-Perron operators[J].Adv Appl Math Mech,2011,3:204-218.
[14]DING J,JIN C,RHEE N,et al.A maximum entropy method based on piecewise linear functions for the recovery of a stationary density of interval mappings[J].J Stat Phys,2011,145:1620-1639.
[15]DING J,RHEE N.Birkhoff's ergodic theorem and the piecewise maximum entropy method for Frobenius-Perron operators[J].Int J Comput Math,2012,89:1083-1091.
[16]UPADHYAY T,DING J,RHEE N.A piecewise quadratic maximum entropy method for the statistical study of chaos[J].J Math Anal Appl,2015,421:1487-1501.
[17]DING J,LI TY.Markov finite approximation of Frobenius-Perron operator[J].Nonlinear Anal,1991,17:759-772.
[18]UFFINK J.Boltzmann's Work in Statistical Physics[M/OL]//EDWARD N Z.Stanford Encyclopedia of Philosophy.http://ploto.stamford.edu/avchieves/fall2014/entries/statphy-Boltzmann,2014.
[19]GIFFIN A.Maximum Entropy:The Universal Method for Inference[M].New York:PhD Dissertation,University at Albany,State University of New York,2008.
[20]BORWEIN JM,LEWIS AS.Convergence of the best entropy esthnates[J].SIAM J Optim,1991,1:191-205.
[21]SCHUMAKER LL.Spline Functions:Basic Theory[M].New York:John wiley&Sons,1980.
[22]DE BOOR C.A Practical Guide to Splines[M].Revised ed.New York:Springer,2001.
[23]DING J,RHEE N.A unified maximum entropy method via spline functions for Frobenius-Perron operators[J].Numer Algb Control Optim,2013,3:235-245.
[24]BORWEIN JM,LEWIS AS.On the convergence of moment problems[J].Trans Amer Math Soc,1991,325:249-271.
[25]BOYARSKY A,GORA P.Laws of Chaos:Invariant Measures and Dynamical Systems in One Dimension[M].Boston:Birkh(a|¨)user,Mass.,1997.
[2]ASTON PJ,DELLNITZ M.The computation of Lypunov exponents via spatial integration with application to blowdout bifurcation[J].Comput Methods Appl Mech Eng,1999,170:223-237.
[3]CHOE GH.Computational Ergodic Theory,Algorithms and Computation in Mathematics[M].Berlin:Springer-Verlag,2005.
[4]LASOTA A,YORKE JA.On the existence of invariant measures for piecewise monotonic transformation[J].Trans Amer Math Soc,1973,186:481-488.
[5]WONG S.Some metric properties of piecewise monotonic mappings of the unit interval[J].Trans Amer Math Soc,1978,246:493-500.
[6]LI TY,YORKE JA.Ergodic transformations from an interval into itself[J].Trans Amer Mam Soc,1978,235:177-186.
[7]ULAM SM.A Collection of Mathematical Problems[M].New York:Interscience,1960.
[8]LI TY.Finite approximation for the Frobenius-Perron operator:A solution to Ulam's conjecture[J].J Approx Theo,1976,17:177-186.
[9]DING J,ZHOU A.Finite approximations of Frobenius-Perron operators:a solution of Ulam's conjecture to multidimensional transformations[J].Physica D,1996,92:61-68.
[10]FROYLAND G.Finite approximation of Sinai-Bowen-Ruelle measures for Anosov systems in two dimensions[J].Random Comput Dynam,1995,3:251-264.
[11]ASTON PJ,JUNGE O.Computing the invariant measure and the Lyapunov exponent for one-dimensional maps using a measure-preserving polynomial basis[J].Math Comput,2014,83:1869-1902.
[12]DING J.A maximum entropy method for solving Frobenius-Perron operator equations[J].Appl Math Compt,1998,93:155-168.
[13]DING J,RHEE N.A maximum entropy method based on orthogonal polynomials for Frobenius-Perron operators[J].Adv Appl Math Mech,2011,3:204-218.
[14]DING J,JIN C,RHEE N,et al.A maximum entropy method based on piecewise linear functions for the recovery of a stationary density of interval mappings[J].J Stat Phys,2011,145:1620-1639.
[15]DING J,RHEE N.Birkhoff's ergodic theorem and the piecewise maximum entropy method for Frobenius-Perron operators[J].Int J Comput Math,2012,89:1083-1091.
[16]UPADHYAY T,DING J,RHEE N.A piecewise quadratic maximum entropy method for the statistical study of chaos[J].J Math Anal Appl,2015,421:1487-1501.
[17]DING J,LI TY.Markov finite approximation of Frobenius-Perron operator[J].Nonlinear Anal,1991,17:759-772.
[18]UFFINK J.Boltzmann's Work in Statistical Physics[M/OL]//EDWARD N Z.Stanford Encyclopedia of Philosophy.http://ploto.stamford.edu/avchieves/fall2014/entries/statphy-Boltzmann,2014.
[19]GIFFIN A.Maximum Entropy:The Universal Method for Inference[M].New York:PhD Dissertation,University at Albany,State University of New York,2008.
[20]BORWEIN JM,LEWIS AS.Convergence of the best entropy esthnates[J].SIAM J Optim,1991,1:191-205.
[21]SCHUMAKER LL.Spline Functions:Basic Theory[M].New York:John wiley&Sons,1980.
[22]DE BOOR C.A Practical Guide to Splines[M].Revised ed.New York:Springer,2001.
[23]DING J,RHEE N.A unified maximum entropy method via spline functions for Frobenius-Perron operators[J].Numer Algb Control Optim,2013,3:235-245.
[24]BORWEIN JM,LEWIS AS.On the convergence of moment problems[J].Trans Amer Math Soc,1991,325:249-271.
[25]BOYARSKY A,GORA P.Laws of Chaos:Invariant Measures and Dynamical Systems in One Dimension[M].Boston:Birkh(a|¨)user,Mass.,1997.