Solving the initial value problem of discrete geodesics

A fast propagation scheme for approximate geodesic paths

Computing centroidal Voronoi tessellations (CVT) has many applications in computer graphics. The existing methods, such as the Lloyd algorithm and the quasi-Newton solver, are efficient and easy to implement; however, they compute only the local optimal solutions due to the highly non-linear nature of the CVT energy. This paper presents a novel method, called manifold differential evolution (MDE), for computing globally optimal geodesic CVT energy on triangle meshes. Formulating the mutation operator using discrete geodesics, MDE naturally extends the powerful differential evolution framework from Euclidean spaces to manifold domains. Under mild assumptions, we show that MDE has a provable probabilistic convergence to the global optimum. Experiments on a wide range of 3D models show that MDE consistently out-performs the existing methods by producing results with lower energy. Thanks to its intrinsic and global nature, MDE is insensitive to initialization and mesh tessellation. Moreover, it is able to handle multiply-connected Voronoi cells, which are challenging to the existing geodesic CVT methods.

The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many approximation methods exploit the restrictive requirement of an explicit time-scale separation parameter. Most of those methods are also not formulated covariantly, i.e. in terms of tensorial constructions. We propose an intrinsically coordinate-free differential geometric approximation criterion approximating normally attracting invariant manifolds (NAIMs). We translate some ideas behind existing approximation approaches, the stretching based diagnostics (SBD) and the flow curvature method (FCM) to tensors of Riemannian geometry, specifically to spacetime curvature in extended phase space. For that purpose we derive from flow-generating smooth vector fields a metric tensor such that the original dynamical system is a geodesic flow on a Riemannian manifold. We apply the resulting method to test models.

The large deformation matching (LDM) framework is a method for registration of images and other data structures, used in computational anatomy. We show how to reformulate the large deformation matching framework for registration in a geometric way. The general framework also allows to generalize the large deformation matching framework to include multiple scales by using the iterated semidirect product of groups. An important ingredient in the LDM framework is the choice of a suitable Riemannian metric on the space of diffeomorphisms. Since the space in question is infinite-dimensional, not every choice of the metric is suitable. In particular the geodesic distance, which is defined as the infimum over the length of all paths connecting two points, may vanish. For the family of Sobolev-type Hs-metrics on the diffeomorphism groups of R and S1 we establish that the geodesic distance vanishes for metrics of order 0 ≤ s ≤ [half]. The geodesic distance also vanishes for the L2-metric on the Virasoro-Bott group, which is a central extension of the diffeomorphism group of the circle. Vanishing of geodesic distance implies that the length-functional, which assigns to each curve in the manifold its length, has no global minima, when restricted to paths with fixed endpoints. We show that for the L2-metric on the diffeomorphism group of R and the Virasoro-Bott group doesn't have any local minima either. The large deformation matching framework is not the only approach to the registration and shape comparison. For curves and surfaces it is possible to define a Riemannian metric directly on the space of curves or surfaces and use geodesics with respect to this metric to measure differences in shape. We use the family of Sobolev-type metrics on surfaces from [7]. We show how to discretize the geodesic equations and solve the boundary value problem via a shooting method on the initial velocity. The discrete equations are implemented via the finite element method.

On a two-dimensional compact Riemannian manifold with boundary, we prove that the first nonzero Steklov eigenvalue is nondecreasing along the unnormalized geodesic curvature flow if the initial metric has positive geodesic curvature and vanishing Gaussian curvature. Using the normalized geodesic curvature flow, we also obtain some estimate for the first nonzero Steklov eigenvalue. On the other hand, we prove that the compact soliton of the geodesic curvature flow must be the trivial one.

What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on the manifold SU(2^n). The geodesic curves on these manifolds have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size. For each Finsler metric we give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call \emph{Pauli geodesics}, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.

An optimization-driven approach for computing geodesic paths on triangle meshes

A fast and precise filament winding path planning method based on discrete non-iterative semi-geodesic algorithm for all applicable mandrel

Given a collection of points in the plane, pick an arbitrary horizontal segment and move it vertically until it hits one of the points (if at all). This form ofsegment-dragging is a common operation in computer graphics and motion-planning, it can also serve as a building block for multidimensional data structures. This note describes a new approach to segment-dragging which yields a simple and efficient solution. The data structure requiresO(n) storage andO(n logn) preprocessing time, and each query can be answered inO(logn) time, wheren is the number of points in the collection. The method is best understood as the end result of a sequence of transformations applied to a simple but inefficient starting solution.

Exact solutions are presented for the load-deflection behavior of a free-free beam on an elastic (Winkler type) foundation, which is subected to a concentrated load at midlength. Two constitutive models are considered; Rigid-perfectly-plastic and elastic-perfectly-plastic (ideal section), and in each case the problem is reduced to the solution of a nonlinear ordinary differential equation with associated initial conditions. The initial value problems are solved numerically using the fourth order Runge-Kutta method. For loads greater than the threshold value at which the three-hinge mechanism forms, both exact solutions produce practically identical results for long beams. This leads to the conclusion that elastic effects are insignificant in a long elastic plastic floating beam, after the three-hinge mechanism is formed. The second iteration solution of an approximate method of analysis for the rigid-plastic beam is also presented. Good agreement is found between the approximate and the exact rigid-plastic results.

In this work, the use of probabilistic evolution theory for the solution of two particle classical mechanics problem is under consideration. Using the separation of the mass center and some algebra, it is possible to reduce the problem to the solution of the initial value problem which has two differential equations and two unknown functions. This is the starting point of this manuscript and the steps up to this point is given in a companion paper. Then, space extension, constancy adding space extension, probabilistic evolution theory with squarification are utilized to form a numerical approximation. These steps and the results are detailed. Therefore, a complete application of probabilistic evolution theory is provided and the results are compared to the results obtained by fourth order Runge-Kutta method.

Dual solutions of a mixed convection flow of nanofluids over a moving vertical plate

The motion of an isolated vortex filament with small curvature is governed by the local induction equation. The present work is an attempt to study such an integrable system based on the Riemannian geometry of the solution manifold, by reformulating the governing equation as the geodesic equation on an infinite dimensional Lie group ${C}^{\ensuremath{\infty}}({S}^{1}, \mathrm{SO}(3))$. The sectional curvatures for the vector fields along the geodesic curves and their perturbations are calculated to show their tendency to be positive for some typical filaments such as a straight vortex, a helical vortex, and a vortex soliton.

Given a zone on the unit sphere S2 with periodic undulating boundaries, we consider the motion of a curve in this zone which is driven by its geodesic curvature. First, we give a necessary and sufficient condition for the existence of periodic rotating waves. Then we study how the average rotating speed of the periodic rotating wave depends on the geometry of the boundaries. We find that when the period of the boundaries tends to 0, the homogenization limit of the rotating speed depends only on the maximum slope of the domain boundaries.

A numerical method for singularly perturbed two-point boundary-value problems for second-order ordinary differential equations subject to Neumann boundary conditions is proposed. In this method, the given interval (the domain of definition of the differential equation) is divided into one ‘outer region’ and two ‘inner regions’. Two initial-value problems associated with the inner region will be derived from the given boundary-value problem. One boundary-value problem derived from the given problem will correspond to the outer region. In each of the two inner regions an initial-value problem is solved by the fourth-order Runge-Kutta method. The boundary-value problem in the outer region is solved by a classical finite difference scheme. A combination of the solutions so obtained yields a numerical solution of the boundary-value problem on the whole interval. The implementation of the method on parallel architectures is discussed. Numerical examples are presented in support of the proposed method.