Approximating geodesics via random points

TL;DR

This paper introduces a probabilistic method for approximating geodesic paths and minimum costs in a domain by constructing random geometric graphs from sampled points, with convergence guarantees as sample size increases.

Contribution

It establishes almost sure convergence of discrete geodesic paths and costs to their continuum counterparts using Gamma convergence, a novel result in this context.

Findings

Discrete geodesics converge to continuum geodesics

Minimum costs approximate the true functional values

Method applies to Finsler and other distance functionals

Abstract

Given a `cost' functional $F$ on paths $\gamma$ in a domain $D\subset\mathbb{R}^d$, in the form $F(\gamma) = \int_0^1 f(\gamma(t),\dot\gamma(t))dt$, it is of interest to approximate its minimum cost and geodesic paths. Let $X_1,\ldots, X_n$ be points drawn independently from $D$ according to a distribution with a density. Form a random geometric graph on the points where $X_i$ and $X_j$ are connected when $0<|X_i - X_j|<\epsilon$, and the length scale $\epsilon=\epsilon_n$ vanishes at a suitable rate. For a general class of functionals $F$, associated to Finsler and other distances on $D$, using a probabilistic form of Gamma convergence, we show that the minimum costs and geodesic paths, with respect to types of approximating discrete `cost' functionals, built from the random geometric graph, converge almost surely in various senses to those corresponding to the continuum cost $F$, as the number of sample points diverges. In particular, the geodesic path convergence shown appears to be among the first results of its kind.