Papers - PENG Linyu

Relations between symmetries and conservation laws for difference systems

L Peng

Journal of Difference Equations and Applications 20 ( 12 ) 1609 - 1626 2014.12

Research paper (scientific journal), Accepted,  ISSN  10236198

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© 2014, Taylor & Francis. This paper presents a relation between divergence variational symmetries for difference variational problems on lattices and conservation laws for the associated Euler–Lagrange system provided by Noether's theorem. This inspires us to define conservation laws related to symmetries for arbitrary difference equations with or without Lagrangian formulations. These conservation laws are constrained by partial differential equations obtained from the symmetries generators. It is shown that the orders of these partial differential equations have been reduced relative to those used in a general approach. Illustrative examples are presented.

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The α-geometric structures on manifold of positive definite Hermite matrices

XM Duan, HF Sun, LY Peng

Acta Mathematica Sinica, English Series 30 ( 12 ) 2137 - 2145 2014.11

Research paper (scientific journal), Accepted,  ISSN  14398516

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© 2014, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg. Geometric structures of a manifold of positive definite Hermite matrices are considered from the viewpoint of information geometry. A Riemannian metric is defined and dual α-connections are introduced. Then the fact that the manifold is ±1-flat is shown. Moreover, the divergence of two points on the manifold is given through dual potential functions. Furthermore, the optimal approximation of a point onto the submanifold is gotten. Finally, some simulations are given to illustrate our results.

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A Natural Gradient Algorithm for Stochastic Distribution Systems

Z Zhang, H Sun, L Peng, L Jiu

Entropy 16 ( 8 ) 4338 - 4352 2014.08

Research paper (scientific journal), Joint Work, Accepted,  ISSN  1099-4300

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In this paper, we propose a steepest descent algorithm based on the natural gradient to design the controller of an open-loop stochastic distribution control system (SDCS) of multi-input and single output with a stochastic noise. Since the control input vector decides the shape of the output probability density function (PDF), the purpose of the controller design is to select a proper control input vector, so that the output PDF of the SDCS can be as close as possible to the target PDF. In virtue of the statistical characterizations of the SDCS, a new framework based on a statistical manifold is proposed to formulate the control design of the input and output SDCSs. Here, the Kullback-Leibler divergence is presented as a cost function to measure the distance between the output PDF and the target PDF. Therefore, an iterative descent algorithm is provided, and the convergence of the algorithm is discussed, followed by an illustrative example of the effectiveness.

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The arc length variational formula on the exponential manifold

F Zhang, H Sun, L Jiu, L Peng

Mathematica Slovaca 63 ( 5 ) 1101 - 1112 2013.10

Research paper (scientific journal), Joint Work, Accepted,  ISSN  01399918

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In this paper, we mainly consider the first and second arc length variational problems on the exponential statistical manifold, and give the variational formulae. (C) 2013 Mathematical Institute Slovak Academy of Sciences

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Natural gradient algorithm for stochastic distribution systems with output feedback

Z Zhang, H Sun, L Peng

Differential Geometry and Its Applications (ELSEVIER SCIENCE BV)  31 ( 5 ) 682 - 690 2013.10

Research paper (scientific journal), Joint Work, Accepted,  ISSN  09262245

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In this paper, we use natural gradient algorithm to control the shape of the conditional output probability density function for the stochastic distribution systems from the viewpoint of information geometry. The considered system here is of multi-input and single output with an output feedback and a stochastic noise. Based on the assumption that the probability density function of the stochastic noise is known, we obtain the conditional output probability density function whose shape is only determined by the control input vector under the condition that the output feedback is known at any sample time. The set of all the conditional output probability density functions forms a statistical manifold (M), and the control input vector and the output feedback are considered as the coordinate system. The Kullback divergence acts as the distance between the conditional output probability density function and the target probability density function. Thus, an iterative formula for the control input vector is proposed in the sense of information geometry. Meanwhile, we consider the convergence of the presented algorithm. At last, an illustrative example is utilized to demonstrate the effectiveness of the algorithm. (c) 2013 Elsevier B.V. All rights reserved.

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A natural gradient descent algorithm for the solution of discrete algebraic Lyapunov equations based on the geodesic distance

X Duan, H Sun, L Peng, X Zhao

Applied Mathematics and Computation (ELSEVIER SCIENCE INC)  219 ( 19 ) 9899 - 9905 2013.05

Research paper (scientific journal), Joint Work, Accepted,  ISSN  00963003

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A new framework to calculate the numerical solution of the discrete algebraic Lyapunov equation is proposed by using the geometric structures on the Riemannian manifold. Specifically, two algorithms based on the manifold of positive definite symmetric matrices are provided. One is a gradient descent algorithm with an objective function of the classical Euclidean distance. The other is a natural gradient descent algorithm with an objective function of the geodesic distance on the curved Riemannian manifold. Furthermore, these two algorithms are compared with a traditional iteration method. Simulation examples show that the convergence speed of the natural gradient descent algorithm is the fastest one among three algorithms. (C) 2013 Elsevier Inc. All rights reserved.

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Information geometry of power spectral

Syed F.S, Peng L.-Y, Sun H.-F, Duan X.-M

Beijing Ligong Daxue Xuebao/Transaction of Beijing Institute of Technology 33 ( 3 ) 327 - 330 2013.03

Research paper (scientific journal), Joint Work, Accepted,  ISSN  10010645

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The information geometric structure of power spectral is given based on information geometry theory. The set of power spectral is taken as a differential manifold. Moreover, a Riemannian metric is defined and the dual connections are introduced. Further, the geometric metrics of three stochastic process models are founded in this paper.

From differential to difference: The variational bicomplex and invariant Noether's theorems

L Peng

 2013

Doctoral thesis, Accepted

Riemannian means on special Euclidean group and unipotent matrices group

X Duan, H Sun, L Peng

The Scientific World Journal  2013   292787 2013

Research paper (scientific journal), Joint Work, Accepted,  ISSN  1537-744X

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Among the noncompact matrix Lie groups, the special Euclidean group and the unipotent matrix group play important roles in both theoretic and applied studies. The Riemannian means of a finite set of the given points on the two matrix groups are investigated, respectively. Based on the left invariant metric on the matrix Lie groups, the geodesic between any two points is gotten. And the sum of the geodesic distances is taken as the cost function, whose minimizer is the Riemannian mean. Moreover, a Riemannian gradient algorithm for computing the Riemannian mean on the special Euclidean group and an iterative formula for that on the unipotent matrix group are proposed, respectively. Finally, several numerical simulations in the 3-dimensional case are given to illustrate our results.

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Information geometric characterization of the complexity of fractional Brownian motions

L Peng, H Sun, G Xu

Journal of mathematical physics (AMER INST PHYSICS)  53 ( 12 ) 123305 2012.12

Research paper (scientific journal), Joint Work, Accepted,  ISSN  00222488

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The complexity of the fractional Brownian motions is investigated from the viewpoint of information geometry. By introducing a Riemannian metric on the space of their power spectral densities, the geometric structure is achieved. Based on the general construction, for an example, whose power spectral density is obtained by use of the normalized Mexican hat wavelet, we show its information geometric structures, e. g., the dual connections, the curvatures, and the geodesics. Furthermore, the instability of the geodesic spreads on this manifold is analyzed via the behaviors of the length between two neighboring geodesics, the average volume element as well as the divergence (or instability) of the Jacobi vector field. Finally, the Lyapunov exponent is obtained. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4770047]

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Geometry of Hamiltonian Dynamics with Conformal Eisenhart Metric

L Peng, H Sun, X Sun

International Journal of Mathematics and Mathematical Sciences 2011 ( 1 ) 710274 2011.09

Research paper (scientific journal), Joint Work, Accepted,  ISSN  01611712

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We characterize the geometry of the Hamiltonian dynamics with a conformal metric. After investigating the Eisenhart metric, we study the corresponding conformal metric and obtain the geometric structure of the classical Hamiltonian dynamics. Furthermore, the equations for the conformal geodesics, for the Jacobi field along the geodesics, and the equations for a certain flow constrained in a family of conformal equivalent nondegenerate metrics are obtained. At last the conformal curvatures, the geodesic equations, the Jacobi equations, and the equations for the flow of the famous models, an N degrees of freedom linear Hamiltonian system and the Hénon-Heiles model are given, and in a special case, numerical solutions of the conformal geodesics, the generalized momenta, and the Jacobi field along the geodesics of the Hénon-Heiles model are obtained. And the numerical results for the Hénon-Heiles model show us the instability of the associated geodesic spreads. © 2011 Linyu Peng et al.

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The geometric structures and instability of entropic dynamical models

L Peng, H Sun, D Sun, J Yi

Advances in Mathematics (ELSEVIER)  227 ( 1 ) 459 - 471 2011.05

Research paper (scientific journal), Joint Work, Accepted,  ISSN  00018708

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In this paper, we characterize two entropic dynamical (ED) models from the viewpoint of information geometry and give the geometric structures of the associated statistical manifolds of the models. The scalar curvatures and the geodesics are obtained. Also the instability of entropic dynamical models is studied from the behavior of the geodesics lengths, statistical volume elements and Jacobi vector fields. Crown Copyright (C) 2011 Published by Elsevier Inc. All rights reserved.

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Information geometry and its applications

Sun H, Peng L, Zhang Z

Advances in Mathematics (China) 40 ( 3 ) 257 - 269 2011

Research paper (scientific journal), Joint Work, Accepted

3D Object Recognition Based on Linear Lie Algebra Model

FX Li, PD Wu, HF Sun, LY Peng

Journal of Beijing Institute of Technology 18 ( 1 ) 46 - 50 2009.03

Research paper (scientific journal), Joint Work, Accepted,  ISSN  10040579

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A surface model called the fibre bundle model and a 3D object model based on linear Lie algebra model are proposed. Then an algorithm of 3D object recognition using the linear Lie algebra models is presented. It is a convenient recognition method for the objects which are symmetric about some axis. By using the presented algorithm, the representation matrices of the fibre or the base curve from only finite points of the linear Lie algebra model can be obtained. At last some recognition results of practicalities are given. © A paper published in JBIT is protected by copyright.

On Fibre Bundle Surfaces and Their Curvatures

FX Li, LY Peng, HF Sun

Journal of Beijing Institute of Technology 17 ( 4 ) 451 - 454 2008.12

Research paper (scientific journal), Joint Work, Accepted,  ISSN  10040579

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A surface model called the fibre bundle model is proposed. This model represents a surface locally as a direct product of two curves: a base curve and a fibre curve. We introduce the fibre bundle model and then obtain the Gaussian curvatures and the mean curvatures of a certain kind of fibre bundle surface models using 1-parameter groups of a linear Lie algebra as fibres. Some examples are given to verify our results.

The geometric structure of the inverse gamma distribution

T Li, L Peng, H Sun

Contributions to Algebra and Geometry 49 ( 1 ) 217 - 225 2008.04

Research paper (scientific journal), Joint Work, Accepted,  ISSN  01384821

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In the present paper we study the geometric structure of the inverse Gamma manifold from the viewpoint of information geometry and give the Kullback divergence, the J-divergence and the geodesic equations. Also, some applications of the inverse Gamma distribution are provided. © 2008 Heldermann Verlag.

Surface model based on a fibre bundle of Hamiltonian Lie algebra and Gaussian curvature

Li F, Wu P, Peng L, Sun H

Science & Technology Review 26 ( 8 ) 37 - 39 2008

Research paper (scientific journal), Joint Work, Accepted

The geometric structure of the Pareto distribution

L Peng, H Sun, L Jiu

Boletın de la Asociación Matemática Venezolana 14 ( 1-2 ) 5 - 13 2007

Research paper (scientific journal), Joint Work, Accepted