相关论文: Shooting methods for computing geodesics on the St…

The problem of finding the distance from a given $n \times n$ matrix polynomial of degree $k$ to the set of matrix polynomials having the elementary divisor $(\lambda-\lambda_0)^j, \, j \geqslant r,$ for a fixed scalar $\lambda_0$ and $2…

The geodesic motion in a Lorentzian spacetime can be described by trajectories in a $3-$dimensional Riemannian metric. In this article we present a generalized Jacobi metric obtained from projecting a Lorentzian metric over the directions…

Random orthogonal matrices play an important role in probability and statistics, arising in multivariate analysis, directional statistics, and models of physical systems, among other areas. Calculations involving random orthogonal matrices…

In probabilistic modeling, parameter estimation is commonly formulated as a minimization problem on a parameter manifold. Optimization in such spaces requires geometry-aware methods that respect the underlying information structure. While…

This chapter investigates numerical solution of nonlinear two-point boundary value problems. It establishes a connection between three important, seemingly unrelated, classes of iterative methods, namely: the linearization methods, the…

Consider the geometric inverse problem: There is a set of delta-sources in spacetime that emit waves travelling at unit speed. If we know all the arrival times at the boundary cylinder of the spacetime, can we reconstruct the space, a…

This paper first presents a detailed implementation of Newton's method on the indefinite Stiefel manifold. To this end, an intensive analysis of the second-order geometry of the manifold is performed. Specifically, given the two types of…

The paper is concerned with the properties of the distance function from a closed subset of a Riemannian manifold, with particular attention to the set of singularities.

We develop two numerical methods to solve the differential equations with deviating arguments for the motion of two charges in the action-at-a-distance electrodynamics. Our first method uses St\"urmer's extrapolation formula and assumes…

We consider the proximal gradient method on Riemannian manifolds for functions that are possibly not geodesically convex. Starting from the forward-backward-splitting, we define an intrinsic variant of the proximal gradient method that uses…

We detail an approach to develop Stein's method for bounding integral metrics on probability measures defined on a Riemannian manifold $\mathbf M$. Our approach exploits the relationship between the generator of a diffusion on $\mathbf M$…

Fractional calculus with respect to function $\psi$, also named as $\psi$-fractional calculus, generalizes the Hadamard and the Riemann-Liouville fractional calculi, which causes challenge in numerical treatment. In this paper we study…

Symmetric Positive Definite (SPD) matrices have become popular to encode image information. Accounting for the geometry of the Riemannian manifold of SPD matrices has proven key to the success of many algorithms. However, most existing…

Robotic surgery for minimally invasive surgery can reduce the surgeon's workload by autonomously guiding robotic forceps. Movement of the robot is restricted around a fixed insertion port. The robot often encounters angle limitations during…

Analyzing high-dimensional data with manifold learning algorithms often requires searching for the nearest neighbors of all observations. This presents a computational bottleneck in statistical manifold learning when observations of…

We propose a new Riemannian gradient descent method for computing spherical area-preserving mappings of topological spheres using a Riemannian retraction-based framework with theoretically guaranteed convergence. The objective function is…

Geodesic problems involve computing trajectories between prescribed initial and final states to minimize a user-defined measure of distance, cost, or energy. They arise throughout physics and engineering -- for instance, in determining…

The shooting and finite-difference method are both numeric methods that approximate the solution of a BVP to a given accuracy. In this report both methods were implemented in Matlab and compared to each other on a BVP found in the context…

We develop the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions to Schubert problems on Grassmannians and is based on the geometric Littlewood-Richardson rule. One key ingredient of this…

We analyze a variational time discretization of geodesic calculus on finite- and certain classes of infinite-dimensional Riemannian manifolds. We investigate the fundamental properties of discrete geodesics, the associated discrete…