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Papers - PENG Linyu

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Moving frames and Noether’s finite difference conservation laws I

EL Mansfield, A Rojo-Echeburua, PE Hydon, L Peng

Transactions of Mathematics and its Applications 3 ( 1 ) tnz004 2019

Research paper (scientific journal), Joint Work, Accepted, ISSN 2398-4945

We consider the calculation of Euler–Lagrange systems of ordinary difference equations, including the difference Noether’s theorem, in the light of the recently-developed calculus of difference invariants and discrete moving frames. We introduce the difference moving frame, a natural discrete moving frame that is adapted to difference equations by prolongation conditions. For any Lagrangian that is invariant under a Lie group action on the space of dependent variables, we show that the Euler–Lagrange equations can be calculated directly in terms of the invariants of the group action. Furthermore, Noether’s conservation laws can be written in terms of a difference moving frame and the invariants. We show that this form of the laws can significantly ease the problem of solving the Euler–Lagrange equations, and we also show how to use a difference frame to integrate Lie group invariant difference equations. In this Part I, we illustrate the theory by applications to Lagrangians invariant under various solvable Lie groups. The theory is also generalized to deal with variational symmetries that do not leave the Lagrangian invariant. Apart from the study of systems that are inherently discrete, one significant application is to obtain geometric (variational) integrators that have finite difference approximations of the continuous conservation laws embedded a priori. This is achieved by taking an invariant finite difference Lagrangian in which the discrete invariants have the correct continuum limit to their smooth counterparts. We show the calculations for a discretization of the Lagrangian for Euler’s elastica, and compare our discrete solution to that of its smooth continuum limit.

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A Stochastic Analytical Modeling Framework on ISP-P2P Collaborations in Multi domain Environments

Zhang Xu, Wang Ning, Cao Yue, Peng Linyu, Meng Haining

IEEE Systems Journal 12 ( 3 ) 2320 - 2331 2018.09

Research paper (scientific journal), Joint Work, Accepted, ISSN 19328184

Cooperation between peer-to-peer (P2P) overlays and underlying networks has been proposed as an effective approach to improve the efficiency of both the applications and the underlying networks. However, fundamental characteristics with respect to Internet service providers (ISP) business relationships and inter-ISP routing information are not sufficiently investigated in the context of collaborative ISP-P2P paradigms in multidomain environments. In this paper, we focus on such issues and develop an analytical modeling framework for analyzing optimized interdomain peer selection schemes concerning ISP policies, with the main purpose of mitigating cross-ISP traffic and enhancing service quality of end users. In addition, we introduce an advanced hybrid scheme for peer selections based on the proposed analytical theory framework, in accordance with practical network scenarios, wherein cooperative and noncooperative behaviors coexisting. Numerical results show that the proposed scheme incorporating ISP policies is able to achieve desirable network efficiency as well as great service quality for P2P users. Our analytical modeling framework can be used as a guide for analyzing and evaluating future network-aware P2P peer selection paradigms in general multidomain scenarios.

Towards autonomy: Cost-effective scheduling for long-range autonomous valet parking (lavp)

M Khalid, Y Cao, X Zhang, C Han, L Peng, N Aslam, N Ahmad

2018 IEEE Wireless Communications and Networking Conference (WCNC) (IEEE) 1 - 6 2018.06

Research paper (international conference proceedings), Joint Work, Accepted

© 2018 IEEE. Continuous and effective developments in Autonomous Vehicles (AVs) are happening on daily basis. Industries nowadays, are interested in introducing less costly and highly controllable AVs to public. Current so-called AVP solutions are still limited to a very short range (e.g., even only work at the entrance of car parks). This paper proposes a parking scheduling scheme for long-range AVP (LAVP) case, by considering mobility of Autonomous Vehicles (AVs), fuel consumption and journey time. In LAVP, Car Parks (CPs) are used to accommodate increasing numbers of AVs, and placed outside city center, in order to avoid traffic congestions and ensure road safety in public places. Furthermore, with positioning of reference points to guide user-centric long-term driving and drop-off/pick-up passengers, simulation results under the Helsinki city scenario shows the benefits of LAVP. The advantage of LAVP system is also reflected through both analysis and simulation.

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Discrete Crum's theorems and integrable lattice equations

C Zhang, L Peng, D Zhang

arXiv preprint arXiv:1802.10044 2018

Research paper (other academic)

In this paper, we develop discrete versions of Darboux transformations and
Crum's theorems for two second order difference equations. The difference
equations are discretised versions (using Darboux transformations) of the
spectral problems of the KdV quation, and of the modified KdV equation or
sine-Gordon equation. Considering the discrete dynamics created by Darboux
transformations for the difference equations, one obtains the lattice potential
KdV equation, the lattice potential modified KdV equation and the lattice
Schwarzian KdV equation, that are prototypes of integrable lattice equations.
It turns out that, along the discretisation processes using Darboux
transformations, two families of integrable systems (the KdV family, and the
modified KdV or sine-Gordon family), including their continuous, semi-discrete
and lattice versions, are explicitly constructed. As direct applications of the
discrete Crum's theorems, multi-soliton solutions of the lattice equations are
obtained.

Variational Integrators for Interconnected Systems with Holonomic Constraints

PENG Linyu, MOMOSE Hiroki, YOSHIMURA Hiroaki

日本応用数理学会年会講演予稿集(CD-ROM) 2017 273‐274 2017.09

Research paper (other academic), Joint Work, Accepted, ISSN 1345-3378

Applying DTN routing for reservation-driven EV Charging management in smart cities

Y Cao, X Zhang, R Wang, L Peng, N Aslam, X Chen

13th International Wireless Communications and Mobile Computing Conference, IWCMC 2017 1471 - 1476 2017.07

Research paper (international conference proceedings), Accepted

© 2017 IEEE. Charging management for Electric Vehicles (EVs) on-the-move (moving on the road with certain trip destinations) is becoming important, concerning the increasing popularity of EVs in urban city. However, the limited battery volume of EV certainly influences its driver's experience. This is mainly because the EV needed for intermediate charging during trip, may experience a long service waiting time at Charging Station (CS). In this paper, we focus on CS-selection decision making to manage EVs' charging plans, aiming to minimize drivers' trip duration through intermediate charging at CSs. The anticipated EVs' charging reservations including their arrival time and expected charging time at CSs, are brought for charging management, in addition to taking the local status of CSs into account. Compared to applying traditionally applying cellular network communication to report EVs' charging reservations, we alternatively study the feasibility of applying Vehicle-to-Vehicle (V2V) communication with Delay/Disruption Tolerant Networking (DTN) nature, due primarily to its flexibility and cost-efficiency in Vehicular Ad hoc NETworks (VANETs). Evaluation results under the realistic Helsinki city scenario show that applying the V2V for reservation reporting is promisingly cost-efficient in terms of communication overhead for reservation making, while achieving a comparable performance in terms of charging waiting time and total trip duration.

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Symmetries, Conservation Laws, and Noether's Theorem for Differential‐Difference Equations

L Peng

Studies in Applied Mathematics 139 ( 3 ) 457 - 502 2017

非ホロノミック拘束を受ける力学系と離散ラグランジュ・ディラック構造

百瀬宏樹, 彭林玉, 吉村浩明

日本応用数理学会年会講演予稿集(CD-ROM) 2016 2016

Research paper (other academic), Joint Work, Accepted, ISSN 1345-3378

Self-adjointness and conservation laws of difference equations

L Peng

Communications in Nonlinear Science and Numerical Simulation 23 ( 1-3 ) 209 - 219 2015.06

Research paper (scientific journal), Accepted, ISSN 10075704

© 2014 Elsevier B.V. A general theorem on conservation laws for arbitrary difference equations is proved. The theorem is based on an introduction of an adjoint system related with a given difference system, and it does not require the existence of a difference Lagrangian. It is proved that the system, combined by the original system and its adjoint system, is governed by a variational principle, which inherits all symmetries of the original system. Noether's theorem can then be applied. With some special techniques, e.g. self-adjointness properties, this allows us to obtain conservation laws for difference equations, which are not necessary governed by Lagrangian formalisms.

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Entropic dynamical models with unstable Jacobi fields

C Li, L Peng, H Sun

Romanian Journal of Physics 60 ( 9-10 ) 1249 - 1262 2015

Research paper (scientific journal), Accepted, ISSN 1221146X

© 2015, Editura Academiei Romane. All rights reserved. The instability of an entropic dynamical model is considered via Jacobi vector field and the Lyapunov exponent. From the viewpoint of information geometry, geometric structure of the statistical manifold underlying this model is investigated, and we conclude that it is a manifold with constant negative scalar curvature. By use of the Jacobi vector field associated with the geodesics, we study the asymptotic behavior of the geodesic spread on the statistical manifold and reach that it is described by an exponentially divergent Jacobi vector field with respect to time. A positive Lyapunov exponent is also obtained, that explains the local instability of the system as well. Furthermore, submanifolds are studied similarly.

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

© 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

© 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

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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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

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

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

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

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

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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