論文

The difference variational bicomplex, which is the natural setting for systems of difference equations, is constructed and used to examine the geometric and algebraic properties of various systems. Exactness of the bicomplex gives a coordinate-free setting for finite difference variational problems, Euler--Lagrange equations and Noether's theorem. We also examine the connection between the condition for existence of a Hamiltonian and the multisymplecticity of systems of partial difference equations. Furthermore, we define difference multimomentum maps of multisymplectic systems, which yield their conservation laws. To conclude, we demonstrate how multisymplectic integrators can be comprehended even on non-uniform meshes through a generalized difference variational bicomplex.

This paper deals with the problem of detecting maritime targets embedded in nonhomogeneous sea clutter, where limited number of secondary data is available due to the heterogeneity of sea clutter. A class of linear discriminant analysis (LDA)-based matrix information geometry (MIG) detectors is proposed in the supervised scenario. As customary, Hermitian positive-definite (HPD) matrices are used to model the observational sample data, and the clutter covariance matrix of received dataset is estimated as geometric mean of the secondary HPD matrices. Given a set of training HPD matrices with class labels, that are elements of a higher-dimensional HPD matrix manifold, the LDA manifold projection learns a mapping from the higher-dimensional HPD matrix manifold to a lower-dimensional one subject to maximum discrimination. In the current study, the LDA manifold projection, with the cost function maximizing between-class distance while minimizing within-class distance, is formulated as an optimization problem in the Stiefel manifold. Four robust LDA-MIG detectors corresponding to different geometric measures are proposed. Numerical results based on both simulated radar clutter with interferences and real IPIX radar data show the advantage of the proposed LDA-MIG detectors against their counterparts without using LDA as well as the state-of-art maritime target detection methods in nonhomogeneous sea clutter.

A novel family of geometric signal detectors are proposed through medians of the total Bregman divergence (TBD), which are shown advantageous over the conventional methods and their mean counterparts. By interpreting the observation data as Hermitian positive-definite matrices, their mean or median play an essential role in signal detection. As is difficult to be solved analytically, we propose numerical solutions through Riemannian gradient descent algorithms or fixed-point algorithms. Beside detection performance, robustness of a detector to outliers is also of vital importance, which can often be analyzed via the influence functions. Introducing an orthogonal basis for Hermitian matrices, we are able to compute the corresponding influence functions analytically and exactly by solving a linear system, which is transformed from the governing matrix equation. Numerical simulations show that the TBD medians are more robust than their mean counterparts.

In this paper, a stochastic Hamiltonian formulation (SHF) is proposed and applied to dissipative particle dynamics (DPD) simulations. As an extension of Hamiltonian dynamics to stochastic dissipative systems, the SHF provides necessary foundations and great convenience for constructing efficient numerical integrators. As a first attempt, we develop the Störmer–Verlet type of schemes based on the SHF, which are structure-preserving for deterministic Hamiltonian systems without external forces, the dissipative forces in DPD. Long-time behaviour of the schemes is shown numerically by studying the damped Kubo oscillator. In particular, the proposed schemes include the conventional Groot–Warren's modified velocity-Verlet method and a modified version of Gibson–Chen–Chynoweth as special cases. The schemes are applied to DPD simulations and analysed numerically.

Principal component analysis (PCA) is a common used pattern analysis method that maps high-dimensional data into a lower-dimensional space maximizing the data variance, that results in the promotion of separability of data. Inspired by the principle of PCA, a novel type of learning discriminative matrix information geometry (MIG) detectors in the unsupervised scenario are developed, and applied to signal detection in nonhomogeneous environments. Hermitian positive-definite (HPD) matrices can be used to model the sample data, while the clutter covariance matrix is estimated by the geometric mean of a set of secondary HPD matrices. We define a projection that maps the HPD matrices in a high-dimensional manifold to a low-dimensional and more discriminative one to increase the degree of separation of HPD matrices by maximizing the data variance. Learning a mapping can be formulated as a two-step mini-max optimization problem in Riemannian manifolds, which can be solved by the Riemannian gradient descent algorithm. Three discriminative MIG detectors are illustrated with respect to different geometric measures, i.e., the Log-Euclidean metric, the Jensen–Bregman LogDet divergence and the symmetrized Kullback–Leibler divergence. Simulation results show that performance improvements of the novel MIG detectors can be achieved compared with the conventional detectors and their state-of-the-art counterparts within nonhomogeneous environments.

In this paper, we propose a modified formal Lagrangian formulation by introducing dummy dependent variables and prove the existence of such a formulation for any system of differential equations. The corresponding Euler–Lagrange equations, consisting of the original system and its adjoint system about the dummy variables, reduce to the original system via a simple substitution for the dummy variables. The formulation is applied to study conservation laws of differential equations through Noether’s Theorem and in particular, a nontrivial conservation law of the Fornberg–Whitham equation is obtained by using its Lie point symmetries. Finally, a correspondence between conservation laws of the incompressible Euler equations and variational symmetries of the relevant modified formal Lagrangian is shown.

The first part of this paper develops a geometric setting for differential-difference equations that resolves an open question about the extent to which continuous symmetries can depend on discrete independent variables. For general mappings, differentiation and differencing fail to commute. We prove that there is no such failure for structure-preserving mappings, and identify a class of equations that allow greater freedom than is typical. For variational symmetries, the above results lead to a simple proof of the differential-difference version of Noether's Theorem. We state and prove the differential-difference version of Noether's Second Theorem, together with a Noether-type theorem that spans the gap between the analogues of Noether's two theorems. These results are applied to various equations from physics.

In this paper, we study the novel nonlinear wave structures of a (2+1)-dimensional variable-coefficient Korteweg-de Vries (KdV) system by its analytic solutions. Its $N$-soliton solution are obtained via Hirota's bilinear method, and in particular, the hybrid solution of lump, breather and line soliton are derived by the long wave limit method. In addition to soliton solutions, similarity reduction, including similarity solutions (also known as group-invariant solutions) and non-autonomous third-order Painlevé equations, is achieved through symmetry analysis. The analytic results, together with illustrative wave interactions, show interesting physical features, that may shed some light on the study of other variable-coefficient nonlinear systems.

Information divergences are commonly used to measure the dissimilarity of two elements on a statistical manifold. Differentiable manifolds endowed with different divergences may possess different geometric properties, which can result in totally different performances in many practical applications. In this paper, we propose a total Bregman divergence-based matrix information geometry (TBD-MIG) detector and apply it to detect targets emerged into nonhomogeneous clutter. In particular, each sample data is assumed to be modeled as a Hermitian positive-definite (HPD) matrix and the clutter covariance matrix is estimated by the TBD mean of a set of secondary HPD matrices. We then reformulate the problem of signal detection as discriminating two points on the HPD matrix manifold. Three TBD-MIG detectors, referred to as the total square loss, the total log-determinant and the total von Neumann MIG detectors, are proposed, and they can achieve great performances due to their power of discrimination and robustness to interferences. Simulations show the advantage of the proposed TBD-MIG detectors in comparison with the geometric detector using an affine invariant Riemannian metric as well as the adaptive matched filter in nonhomogeneous clutter.

© 1967-2012 IEEE. The proliferation of electric vehicles (EVs) has posed significant challenges to the existing power grid infrastructure. It thus becomes of vital importance to efficiently manage the Electro-Mobility for large demand from EVs. Due to limited cruising range of EVs, vehicles have to make frequent stops for recharging, while long charging period is one major concern under plug-in charging. We herein leverage battery swapping (BS) technology to provide an alternative charging service, which substantially reduces the charging duration (from hours down to minutes). Concerning in practice that various battery is generally not compatible with each other, we thus introduce battery heterogeneity into the swapping service, concerning the case that different types of EVs co-exist. A battery heterogeneity-based swapping service framework is then proposed. Further with reservations for swapping service enabled, the demand load can be anticipated at BS stations as a guidance to alleviate service congestion. Therefore, potential hotspots can be avoided. Results show the performance gains under the proposed scheme by comparing to other benchmarks, in terms of service waiting time, etc. In particular, the diversity of battery stock across the network can be effectively managed.

© 2020, Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature. In this paper, the Riemannian gradient algorithm and the natural gradient algorithm are applied to solve descent direction problems on the manifold of positive definite Hermitian matrices, where the geodesic distance is considered as the objective function. The first proposed problem is the control for positive definite Hermitian matrix systems whose outputs only depend on their inputs. The geodesic distance is adopted as the difference of the output matrix and the target matrix. The controller to adjust the input is obtained such that the output matrix is as close as possible to the target matrix. We show the trajectory of the control input on the manifold using the Riemannian gradient algorithm. The second application is to compute the Karcher mean of a finite set of given Toeplitz positive definite Hermitian matrices, which is defined as the minimizer of the sum of geodesic distances. To obtain more efficient iterative algorithm than traditional ones, a natural gradient algorithm is proposed to compute the Karcher mean. Illustrative simulations are provided to show the computational behavior of the proposed algorithms.

© 2019 IEEE. The introduction of Electric Vehicles (EVs) leads to new concern on the E-Mobility. Making charging reservation, by considering the EV's arrival time and its expected charging time at Charging Stations (CSs) has been studied to predict the dynamic status of CSs. In this paper, we propose a Mobile Edge computer Geared v2x for E-mobility Ecosystem (MEGEE), as a decentralized alternative to the conventional centralized cloud based architecture. MEGEE enables the Vehicular Delay/Disruption Tolerant Networking (VDTN)-driven anycasting for information delivery, and Mobile Edge Computing (MEC) functioned CSs for information mining and aggregation. MEGEE efficiently and timely processes essential charging reservations and charging control information, through the Internet of EVs and MEC servers. Our studies show that MEGEE can achieve the close charging performance as performed by the centralized system, while offers a significant saving in communications cost.

© 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.

© 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.

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. © 2014 by the authors; licensee MDPI, Basel, Switzerland.

In this paper, we mainly consider the first and second arc length variational problems on the exponential statistical manifold, and give the variational formulae. © 2013 Versita Warsaw and Springer-Verlag Wien.

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.

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. © 2013 Xiaomin Duan et al.

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 curvaturesthe 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. © 2012 American Institute of Physics.