Poisson schemes for Hamiltonian systems on Poisson manifolds

In this paper, we construct Poisson difference schemes of any order accuracy based on Padé approximation for linear Hamiltonian systems on Poisson manifolds with constant coefficients. For nonlinear Hamiltonian systems on Poisson manifolds, we point out that symplectic diagonal implicit Runge-Kutta methods are also Poisson schemes. The preservation of distinguished functions and quadratic first integrals of the original Hamiltonian systems of these schemes are also discussed.     

Characterization and Detection of Toric Loops in <Emphasis Type="Italic">n</Emphasis>-Dimensional Discrete Toric Spaces

Since a toric space is not simply connected, it is possible to find in such spaces some loops which are not homotopic to a point: we call them toric loops. Some applications, such as the study of the relationship between the geometrical characteristics of a material and its physical properties, rely on three-dimensional discrete toric spaces and require detecting objects having a toric loop. In this work, we study objects embedded in discrete toric spaces, and propose a new definition of loops and equivalence of loops. Moreover, we introduce a characteristic of loops that we call wrapping vector: relying on this notion, we propose a linear time algorithm which detects whether an object has a toric loop or not.     

Fundamental Constraints on Uncertainty Evolution in Hamiltonian Systems

A realization of Gromov's nonsqueezing theorem and its applications to uncertainty analysis in Hamiltonian systems are studied in this note. Gromov's nonsqueezing theorem describes a fundamental property of symplectic manifolds, however, this theorem is usually started in terms of topology and its physical meaning is vague. In this note we introduce a physical interpretation of the linear symplectic width, which is the lower bound in the nonsqueezing theorem, in terms of the eigenstructure of a positive-definite, symmetric matrix. Since uncertainty is often represented in terms of a positive definite, symmetric matrix in control theory, our study can be applied to uncertainty analysis by applying the nonsqueezing theorem to the uncertainty ellipsoid. We find a fundamental inequality for the evolving uncertainty in a linear dynamical system and provide some numerical examples     

High-order closed Newton-Cotes trigonometrically-fitted formulae for long-time integration of orbital problems

In this paper we investigate the connection between closed Newton-Cotes, trigonometrically-fitted differential methods and symplectic integrators. From the literature we can see that several one step symplectic integrators have been obtained based on symplectic geometry. However, the investigation of multistep symplectic integrators is very poor. The well-known open Newton-Cotes differential methods presented as multilayer symplectic integrators by Zhu et al. [W. Zhu, X. Zhao, Y. Tang, Journal of Chem. Phys. 104 (1996) 2275]. The construction of multistep symplectic integrators based on the open Newton-Cotes integration methods is investigated by Chiou and Wu [J.C. Chiou, S.D. Wu, J. Chem. Phys. 107 (1997) 6894]. In this paper we investigate the closed Newton-Cotes formulae and we write them as symplectic multilayer structures. After this we construct trigonometrically-fitted symplectic methods which are based on the closed Newton-Cotes formulae. We apply the symplectic schemes in order to solve Hamilton's equations of motion which are linear in position and momentum. We observe that the Hamiltonian energy of the system remains almost constant as integration proceeds.     

Initial ideals of unimodular integer programs

This article concerns the initial ideal computation for toric ideals, which arises in the computational algebraic approach for integer programs. It is known that the initial ideals of toric ideals contain many information about original toric ideals that are beneficial for solving the integer programs. Therefore, their analyses may bring further comprehensions for structure of integer programs and can help us to construct a new type of algorithm. There are two known algorithms for initial ideal computation. One is based on the theory of Gröbner bases and the other is proposed by Thomas and Hos¸ten. In this article, we propose another type of algorithm to perform the initial ideal computation for the toric ideals defined by unimodular matrices. The unimodular matrices is one of the important matrix classes, that are often looming as the coefficient matrices of some combinatorial optimization problems with the forms of integer programs. Ours are based on a combinatorial relation between the vector matroids of defining matrices and the minimal generators of initial ideals.     

Symplectic feedback using Hamiltonian Lie algebra and its applications to an inverted pendulum

From the symplectic representation of an autonomous nonlinear dynamical system with holonomic constraints, i.e., those that can be represented through a symplectic form derived from a Hamiltonian, we present a new proof on the realization of the symplectic feedback action, which has several theoretical advantages in demonstrating the uniqueness and existence of this type of solution. Also, we propose a technique based on the interpretation, construction and characterization of the pull-back differential on the symplectic manifold as a member of a one-parameter Lie group. This allows one to synthesize the control law that governs a certain system to achieve a desired behavior; and the method developed from this is applied to a classical system such as the inverted pendulum.     

Learning independent components on the orthogonal group of matrices by retractions

Neural independent component learning algorithms based on optimization on manifolds have attracted interest in the neural network community. In the past years, we have developed learning algorithms specialized for the orthogonal group of matrices as parameters manifold. Here, we sketch a view of these algorithms by the help of ‘retractions’ on manifolds.     

三类保辛摄动及其数值比较

摄动法近似应当保辛．本文指出,有限元位移法自动保辛,有限元混合能表示也保辛．摄动法的刚度阵Taylor级数展开能证明保辛;混合能的Taylor级数展开摄动也证明了保辛．但传递辛矩阵的Taylor级数展开摄动却不能保辛．辛矩阵只能在乘法群下保辛,故传递辛矩阵的保辛摄动必须采用正则变换的乘法．虽然刚度阵加法摄动、混合能矩阵加法摄动与传递辛矩阵正则变换乘法摄动都保辛,但这3种摄动近似并不相同．最后通过数值例题给出了对比．     

大阻尼杆振动的广义多辛算法

本文基于Bridges教授建立的多辛算法理论及其Hamilton变分原理,采用广义多辛算法研究了大阻尼杆的阻尼振动特性.引入正交动量后,首先将描述大阻尼杆振动的控制方程降阶为一阶Hamilton近似对称形式,即广义多辛形式;随后采用中点离散方法构造形式广义多辛形式的中点Box广义多辛离散格式;最后通过计算机模拟研究大阻尼杆振动过程中的耗散效应.研究结果表明,本文构造的广义多辛算法不仅能够保持系统守恒型几何性质,同时能够再现系统的耗散效应.     

Local discontinuous Galerkin methods based on the multisymplectic formulation for two kinds of Hamiltonian PDEs

ABSTRACT

This paper examines the novel local discontinuous Galerkin (LDG) discretization for Hamiltonian PDEs based on its multisymplectic formulation. This new kind of LDG discretizations possess one major advantage over other standard LDG method, which, through specially chosen numerical fluxes, states the preservation of discrete conservation laws (i.e. energy), and also the multisymplectic structure while the symplectic time integration is adopted. Moreover, the corresponding local multisymplectic conservation law holds at the units of elements instead of each node. Taking the nonlinear Schrödinger equation and the KdV equation as the examples, we illustrate the derivations of discrete conservation laws and the corresponding numerical fluxes. Numerical experiments by using the modified LDG method are demonstrated for the sake of validating our theoretical results.     

Priors with Coupled First and Second Order Differences for Manifold-Valued Image Processing

We generalize discrete variational models involving the infimal convolution (IC) of first and second order differences and the total generalized variation (TGV) to manifold-valued images. We propose both extrinsic and intrinsic approaches. The extrinsic models are based on embedding the manifold into an Euclidean space of higher dimension with manifold constraints. An alternating direction methods of multipliers can be employed for finding the minimizers. However, the components within the extrinsic IC or TGV decompositions live in the embedding space which makes their interpretation difficult. Therefore, we investigate two intrinsic approaches: for Lie groups, we employ the group action within the models; for more general manifolds, our IC model is based on recently developed absolute second order differences on manifolds, while our TGV approach uses an approximation of the parallel transport by the pole ladder. For computing the minimizers of the intrinsic models, we apply gradient descent algorithms. Numerical examples demonstrate that our approaches work well for certain manifolds.     

Intrinsic Cramér–Rao bounds for distributed Bayesian estimator

This paper addresses the intrinsic Cramér–Rao bounds (CRBs) for a distributed Bayesian estimator whose states and measurements are on Riemannian manifolds. As Euclidean-based CRBs for recursive Bayesian estimator are no longer applicable to general Riemannian manifolds, the bounds need redesigning to accommodate the non-zero Riemannian curvature. To derive the intrinsic CRBs, we append a coordination step to the recursive Bayesian procedure, where the proposed sequential steps are prediction, measurement and coordination updates. In the coordination step, the estimator minimises the Kullback–Liebler divergence to obtain the consensus of multiple probability density functions (PDFs). Employing the PDFs from those steps together with the affine connection on manifolds the Fisher Information Matrix (FIM) and the curvature terms of the corresponding intrinsic bounds are derived. Subsequently, the design of a distributed estimator for Riemannian information manifold with Gaussian distribution – referred to as distributed Riemannian Kalman filter – is also presented to exemplify the application of the proposed intrinsic bounds. Finally, simulations utilising the designed filter for a distributed quaternionic estimation problem verifies that the covariance matrices of the filter are never below the formulated intrinsic CRBs.     

Multilayer Minimum Projection Method with Singular Point Assignment for Nonsmooth Control Lyapunov Function Design

Control Lyapunov function (CLF) design on a manifold is a difficult problem in control theory. To address this problem, we have proposed the multilayer minimum projection method. The method requires CLFs on different manifolds from the manifold where the control problem is defined. In this paper, we relax the requirement by desingularization of the functions on the manifolds. The paper focuses on the problem of desingularization in the multilayer minimum projection method. We show that the functions on other manifolds need not be CLFs by consideration of desingularization. Moreover, we propose a CLF design method by singular point assignment based on the advantage of desingularization. The method enables us to merge local CLFs into the global CLF. This paper proposes two CLF design methods: desingularization and singular point assignment. A CLF design example is provided for each method; the advantages of the proposed methods are confirmed by those two examples.     

Duffing方程的辛精细积分方法研究

辛精细积分方法汲取了辛几何算法保持动力学系统辛结构的优点和精细积分方法高精度的数值优点,其实现过程中涉及到大量矩阵求逆运算.为减小辛精细积分方法的运算量,本文在辛精细积分算法之前先将非齐次方程近似齐次化,使得矩阵求逆部分不显含时间,降低矩阵求逆计算量,并将这一方法应用于无阻尼Duffing方程的数值分析.通过与经典四阶Runge-Kutta格式及精细积分方法对比,发现辛精细积分方法在数值精度、计算耗时、保持系统能量等方面明显优于Runge-Kutta格式.此外,与精细积分方法相比,辛精细积分方法在保持系统能量方面存在明显优势.     

A fourth order modified trigonometrically fitted symplectic Runge–Kutta–Nyström method

In this work we construct a modified trigonometrically fitted symplectic Runge Kutta Nyström method based on the fourth order five stages method of Calvo and Sanz-Serna (1994). We apply the new method on the numerical integration of the two-dimensional harmonic oscillator, the two-body problem, a perturbed two-body problem and two two-dimensional nonlinear oscillatory Hamiltonian systems.     

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

In medical image analysis and high level computer vision, there is an intensive use of geometric features like orientations, lines, and geometric transformations ranging from simple ones (orientations, lines, rigid body or affine transformations, etc.) to very complex ones like curves, surfaces, or general diffeomorphic transformations. The measurement of such geometric primitives is generally noisy in real applications and we need to use statistics either to reduce the uncertainty (estimation), to compare observations, or to test hypotheses. Unfortunately, even simple geometric primitives often belong to manifolds that are not vector spaces. In previous works [1, 2], we investigated invariance requirements to build some statistical tools on transformation groups and homogeneous manifolds that avoids paradoxes. In this paper, we consider finite dimensional manifolds with a Riemannian metric as the basic structure. Based on this metric, we develop the notions of mean value and covariance matrix of a random element, normal law, Mahalanobis distance and χ² law. We provide a new proof of the characterization of Riemannian centers of mass and an original gradient descent algorithm to efficiently compute them. The notion of Normal law we propose is based on the maximization of the entropy knowing the mean and covariance of the distribution. The resulting family of pdfs spans the whole range from uniform (on compact manifolds) to the point mass distribution. Moreover, we were able to provide tractable approximations (with their limits) for small variances which show that we can effectively implement and work with these definitions.     

Computing with toric varieties

A computer algebra package (written by the second author) is described which deals with both affine and projective toric varieties in any number of dimensions (written in both MAGMA and GAP). Among other things, the package implements the desingularization procedure, constructs some error-correcting codes associated with toric varieties, and computes the Riemann–Roch space of a divisor on a toric variety.     

Global one-dimensionality conjecture within quantum general relativity

A simple quantum gravity model, based on a new conjecture within the canonically quantized 3 + 1 general relativity, is presented. The conjecture states that matter fields are functionals of an embedding volume form only and reduces the quantum geometrodynamics. By dimensional reduction the resulting theory is presented in the form of the Dirac equation, and application of the Fock quantization with the diagonalization procedure yields construction of the appropriate quantum field theory. The 1D wave function is derived, the corresponding 3–dimensional manifolds are discussed as well as physical scales associated with quantum correlations.     

On the structure of generalized toric codes

Toric codes are obtained by evaluating rational functions of a nonsingular toric variety at the algebraic torus. One can extend toric codes to the so-called generalized toric codes. This extension consists of evaluating elements of an arbitrary polynomial algebra at the algebraic torus instead of a linear combination of monomials whose exponents are rational points of a convex polytope. We study their multicyclic and metric structure, and we use them to express their dual and to estimate their minimum distance.