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June 2019 Approximating geodesics via random points
Erik Davis, Sunder Sethuraman
Ann. Appl. Probab. 29(3): 1446-1486 (June 2019). DOI: 10.1214/18-AAP1414

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 $0<|X_{i}-X_{j}|<\varepsilon$, and the length scale $\varepsilon=\varepsilon_{n}$ 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.

Citation

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Erik Davis. Sunder Sethuraman. "Approximating geodesics via random points." Ann. Appl. Probab. 29 (3) 1446 - 1486, June 2019. https://doi.org/10.1214/18-AAP1414

Information

Received: 1 December 2017; Revised: 1 April 2018; Published: June 2019
First available in Project Euclid: 19 February 2019

zbMATH: 07057459
MathSciNet: MR3914549
Digital Object Identifier: 10.1214/18-AAP1414

Subjects:
Primary: 60D05
Secondary: 05C82, 49J45, 49J55, 53C22, 58E10, 62-07

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.29 • No. 3 • June 2019
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