Derivatives with respect to metrics and applications: subgradient marching algorithm

This paper introduces a subgradient descent algorithm to compute a Riemannian metric that minimizes an energy involving geodesic distances. The heart of the method is the Subgradient Marching Algorithm to compute the derivative of the geodesic distance with respect to the metric. The geodesic distance being a concave function of the metric, this algorithm computes an element of the subgradient in O(N ² log(N)) operations on a discrete grid of N points. It performs a front propagation that computes a subgradient of a discrete geodesic distance. We show applications to landscape modeling and to traffic congestion. Both applications require the maximization of geodesic distances under convex constraints, and are solved by subgradient descent computed with our Subgradient Marching. We also show application to the inversion of travel time tomography, where the recovered metric is the local minimum of a non-convex variational problem involving geodesic distances.     

Geodesic Distance in Planar Graphs: An Integrable Approach

We discuss the enumeration of planar graphs using bijections with suitably decorated trees, which allow for keeping track of the geodesic distances between faces of the graph. The corresponding generating functions obey non-linear recursion relations on the geodesic distance. These are solved by use of stationary multi-soliton tau-functions of suitable reductions of the KP hierarchy. We obtain a unified formulation of the (multi-) critical continuum limit describing large graphs with marked points at large geodesic distances, and obtain integrable differential equations for the corresponding scaling functions. This provides a continuum formulation of two-dimensional quantum gravity, in terms of the geodesic distance. 2000 Mathematics Subject Classification: Primary—05C30; Secondary—05A15, 05C05, 05C12, 68R05     

Integration of barotropic vorticity equation over spherical geodesic grid using multilevel adaptive wavelet collocation method

In this paper, we present the multilevel adaptive wavelet collocation method for solving non-divergent barotropic vorticity equation over spherical geodesic grid. This method is based on multi-dimensional second generation wavelet over a spherical geodesic grid. The method is more useful in capturing, identifying, and analyzing local structure [1] than any other traditional methods (i.e. finite difference, spectral method), because those methods are either full or partial miss important phenomena such as trends, breakdown points, discontinuities in higher derivatives of the solution. Wavelet decomposition is used for interpolation and adaptive grid refinement on different levels.     

(h, Φ)-entropy differential metric

Burbea and Rao (1982a, 1982b) gave some general methods for constructing quadratic differential metrics on probability spaces. Using these methods, they obtained the Fisher information metric as a particular case. In this paper we apply the method based on entropy measures to obtain a Riemannian metric based on (h, )-entropy measures (Salicrú et al., 1993). The geodesic distances based on that information metric have been computed for a number of parametric families of distributions. The use of geodesic distances in testing statistical hypotheses is illustrated by an example within the Pareto family. We obtain the asymptotic distribution of the information matrices associated with the metric when the parameter is replaced by its maximum likelihood estimator. The relation between the information matrices and the Cramér-Rao inequality is also obtained.     

A characterization of spaces of constant curvature by minimum covering radius of triangles

In this paper, we prove that if in a Riemannian manifold, the minimum covering radius of a point triple of small diameter depends only on the geodesic distances between the points, then the manifold must be of constant curvature. This implies that if in a complete connected Riemannian manifold, the volume of the intersection of three small geodesic balls of equal radii depends only on the distances between the centers and the radius, then it is one of the simply connected spaces of constant curvature. This generalizes an earlier result of the first author and D. Kunszenti-Kovács (2010).     

A geodesic approach to calculating the information sets in dynamic search problems

A new method for calculating the information sets in dynamic search problems is proposed. Representation of the sets is based on the concept of distance fields and operations on these fields, on calculating the geodesic distances via finding a viscous solution to the Hamilton-Jacobi equation. The method is applied to solving the search problems on surfaces in a 3D space.     

On the volume of the intersection of two geodesic balls

The first author and D. Kunszenti-Kovács (2010) [1] proved that if the volume of the intersection of three geodesic balls of a complete connected Riemannian manifold depends only on the center-center distances and the radii of the balls, then the manifold is one of the simply connected spaces of constant curvature. In this paper, we study the geometrical consequences of the analogous condition for pairs of geodesic balls. We show that in a complete, connected and simply connected Riemannian manifold, the volume of the intersection of two small geodesic balls depends only on the distance between the centers and the radii if and only if the space is harmonic. It is also shown that if in a Riemannian manifold the volume of the intersection of two small geodesic balls of equal radii depends only on the distance between the centers and the common value of the radii, then the space is Einstein, and if we assume in addition that the space is symmetric, then it must be Osserman and hence two-point homogeneous.     

Sufficient conditions for a strong relative minimum in an optimal control problem

In this paper, on the basis of Young's method (Ref. 1), sufficient conditions for a strong relative minimum in an optimal control problem are given. Young's method generalizes geodesic coverings and the simplest Hilbert integral from the standard variational calculus. This paper carries Young's method over to nonparametric problems.     

From 1-homogeneous supremal functionals to difference quotients: relaxation and Γ-convergence

In this paper we consider positively 1-homogeneous supremal functionals of the type . We prove that the relaxation $\bar{F}$ is a difference quotient, that is where is a geodesic distance associated to F. Moreover we prove that the closure of the class of 1-homogeneous supremal functionals with respect to Γ-convergence is given exactly by the class of difference quotients associated to geodesic distances. This class strictly contains supremal functionals, as the class of geodesic distances strictly contains intrinsic distances. Mathematics Subject Classification (2000) 47J20, 58B20, 49J45     

Progressive Tuning of Simple Exponential Smoothing Forecasts

The paper outlines a finite sample version of exponential smoothing, and proposes a formula for estimating the smoothing parameter. The resulting method, which can be implemented on a recursive basis over time, is compared with alternative approaches, such as progressive numerical optimization of the smoothing parameter and adaptive forecasting on both synthetic and real data.     

Discrete Lagrangian algorithm for finding geodesics on triangular meshes

The present paper introduces an approximation method for finding open geodesics on triangular surfaces. The algorithm is specifically designed to be able to solve real world problems where geodesic paths are needed. We use the model of geodesic curvature flow for open curves in the Lagrangian formulation. The model is enriched with a tangential term in order to have a control over the quality of the discretization grid during the computation. The governing equation of the flow is solved by a numerical method based on a semi-implicit time discretization and a finite difference space discretization. The paper presents the numerical scheme and various implementation details as well as numerous experiments to demonstrate the performance of the method and to provide comparison with several other well known methods. We also present a Grasshopper component for Rhinoceros for finding optimal paths on surface meshes that we developed and that includes our algorithm.     

A survey of geodesic paths on 3D surfaces

This survey gives a brief overview of theoretically and practically relevant algorithms to compute geodesic paths and distances on three-dimensional surfaces. The survey focuses on three-dimensional polyhedral surfaces. The goal of this survey is to identify the most relevant open problems, both theoretical and practical.     

Unbalanced optimal transport: Dynamic and Kantorovich formulations

This article presents a new class of distances between arbitrary nonnegative Radon measures inspired by optimal transport. These distances are defined by two equivalent alternative formulations: (i) a dynamic formulation defining the distance as a geodesic distance over the space of measures (ii) a static “Kantorovich” formulation where the distance is the minimum of an optimization problem over pairs of couplings describing the transfer (transport, creation and destruction) of mass between two measures. Both formulations are convex optimization problems, and the ability to switch from one to the other depending on the targeted application is a crucial property of our models. Of particular interest is the Wasserstein–Fisher–Rao metric recently introduced independently by [7], [15]. Defined initially through a dynamic formulation, it belongs to this class of metrics and hence automatically benefits from a static Kantorovich formulation.     

Embeddings of Gromov hyperbolic spaces

It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasi-isometric to a convex subset of hyperbolic space. If one is allowed to rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at most an additive constant.?Another embedding theorem states that any -hyperbolic metric space embeds isometrically into a complete geodesic -hyperbolic space.?The relation of a Gromov hyperbolic space to its boundary is further investigated. One of the applications is a characterization of the hyperbolic plane up to rough quasi-isometries. Submitted: October 1998, Revised version: January 1999.     

A two-level nesting smoothed meshfree method for structural dynamic analysis

This paper presents a novel two-level nesting smoothed meshfree method (NSMM), which significantly improves the computational efficiency of the meshfree Galerkin methods without losing their accuracy, thus facilitates the employment of meshfree methods in applications where background integration cells would be prohibitively expensive. In the NSMM, the system stiffness matrix is calculated using the general smoothing strain technique over the two-level nesting smoothing sub-domains where fewer integration points are used and the costly derivative computation of meshfree shape functions is avoided. The accuracy, efficiency and stability of the present method are assessed by virtue of several numerical examples for problems involving free and forced vibration analysis of the linear elastic continua and dynamic crack response of elastic solid. The results reveal that the NSMM stands out and achieves better performance compared to other existing approaches in the literature.     

A nonmonotone smoothing Newton method for circular cone programming

The circular cone programming (CCP) problem is to minimize or maximize a linear function over the intersection of an affine space with the Cartesian product of circular cones. In this paper, we study nondegeneracy and strict complementarity for the CCP, and present a nonmonotone smoothing Newton method for solving the CCP. We reformulate the CCP as a second-order cone programming (SOCP) problem using the algebraic relation between the circular cone and the second-order cone. Then based on a one parametric class of smoothing functions for the SOCP, a smoothing Newton method is developed for the CCP by adopting a new nonmonotone line search scheme. Without restrictions regarding its starting point, our algorithm solves one linear system of equations approximately and performs one line search at each iteration. Under mild assumptions, our algorithm is shown to possess global and local quadratic convergence properties. Some preliminary numerical results illustrate that our nonmonotone smoothing Newton method is promising for solving the CCP.     

An extended edge-based smoothed discrete shear gap method for free vibration analysis of cracked Reissner–Mindlin plate

An extended edge-based smoothed discrete shear gap method (XES-DSG3) is proposed for free vibration analysis of cracked Reissner–Mindlin plate by implementing the edge-based strain smoothing operation into the discrete shear gap-based extended finiteelement method (XFEM-DSG3). In present method, the strain smoothing operation is implemented into the bending strain gradient matrices, in which the enriched functions are included. Then, the derivatives of element shape functions and derivatives of crack-tip singular enriched functions are not required in the computation. The calculation of element matrices is performed over the smoothing domains which are associated with edges of elements. The transverse shear locking of Reissner–Mindlin plate can be avoided by using the integration of discrete shear gap (DSG) method. Several numerical examples are investigated to illustrate the accuracy of XES-DSG3 for the free vibration analysis of cracked Reissner–Mindlin plate. Moreover, numerical results show that the present method is insensitive to mesh distortion and it is more stable than the pervious XFEM-DSG3.     

Symmetric subvarieties in compactifications and the Radon transform on Riemannian symmetric spaces of noncompact type

We investigate the totally geodesic Radon transform which assigns a function to its integration over totally geodesic symmetric submanifolds in Riemannian symmetric spaces of noncompact type. Our main concern is focused on the injectivity and support theorem. Our approach is based on the projection slice theorem relating the totally geodesic Radon transform and the Fourier transforms on symmetric spaces. Our approach also uses the study of geometry concerned with the totally geodesic symmetric subvarieties in Riemannian symmetric spaces in terms of the cell structure of the Satake compactifications.     

Consequences of contractible geodesics on surfaces

The geodesic flow of any Riemannian metric on a geodesically convex surface of negative Euler characteristic is shown to be semi-equivalent to that of any hyperbolic metric on a homeomorphic surface for which the boundary (if any) is geodesic. This has interesting corollaries. For example, it implies chaotic dynamics for geodesic flows on a torus with a simple contractible closed geodesic, and for geodesic flows on a sphere with three simple closed geodesics bounding disjoint discs.

     

飞越北极

本文对“飞机从北京出发、飞越北极直达底特律的所需时间 ,可比原航线节省多少时间”的问题进行讨论 ,并将航线选择归结为寻求曲面上的最短弧 .应用“曲面上最短弧为测地线”的事实进行了讨论 .模型 (一 )假设地球是球体 ,我们可通过单位向量的点乘与夹角的关系 ,加以解决 ;对于模型 (二 )设地球是旋转椭球体 ,我们利用微分几何学中测地线方程加以解决 ,并且把球面的纬度转化为旋转椭球面纬度 .对于 4组较特殊的点 ,纬度几乎相等或相近 ,或者两者之间的经度差过大时 ,用测地线计算比较困难 ,我们用椭圆弧 (长 )代替测地线长 ,结合数学软件 Mathematica的数值积分功能 ,可求得测地线长